Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.

• Wednesday January 23, 2019 at 14:30, Wachman 617

  Surfaces almost transverse to circular pseudo-Anosov flows

  Michael Landry, Yale University

  Let \(M\) be a closed hyperbolic 3-manifold which fibers over \(S^1\), and let \(F\) be a fibered face of the unit ball of the Thurston norm on \(H^1(M;R)\). By results of Fried, there is a nice flow on \(M\) naturally associated to \(F\). We study surfaces which are almost transverse to \(F\) and give a new characterization of the set of homology directions of \(F\) using Agol’s veering triangulation of an auxiliary cusped 3-manifold.

• Wednesday January 30, 2019 at 14:30, Wachman 617

  Circle packings and Delaunay circle patterns for complex projective structures

  Andrew Yarmola, Princeton University

  Abstract: At the interface of discrete conformal geometry and the study of Riemann surfaces lies the Koebe-Andreev-Thurston theorem. Given a triangulation of a surface \(S\), this theorem produces a unique hyperbolic structure on \(S\) and a geometric circle packing whose dual is the given triangulation. In this talk, we explore an extension of this theorem to the space of complex projective structures - the family of maximal \(CP^1\)-atlases on \(S\) up to Möbius equivalence. Our goal is to understand the space of all circle packings on complex projective structures with a fixed dual triangulation. As it turns out, this space is no longer a unique point and evidence suggests that it is homeomorphic to Teichmüller space via uniformization - a conjecture by Kojima, Mizushima, and Tan. In joint work with Jean-Marc Schlenker, we show that this projection is proper, giving partial support for the conjectured result. Our proof relies on geometric arguments in hyperbolic ends and allows us to work with the more general notion of Delaunay circle patterns, which may be of separate interest. I will give an introductory overview of the definitions and results and demonstrate some software used to motivate the conjecture. If time permits, I will discuss additional ongoing work with Wayne Lam.

• Wednesday February 6, 2019 at 14:30, Wachman 617

  CAT(0) cubical groups with uniform exponential growth

  Thomas Ng, Temple University

  Abstract: A group is said to have uniform exponential growth if the number of elements that can be spelled with words of bounded length is bounded below by a single exponential function over all generating sets. In 1981, Gromov asked whether all groups with exponential growing group in fact have uniform exponential growth. While this was shown not to be the case in general, it has been answered affirmatively for many natural classes of groups such as hyperbolic groups, linear groups, and the mapping class groups of a surface. In 2018, Kar-Sageev show that groups acting properly on 2-dimensional CAT(0) cube complexes by loxodromic isometries either have uniform exponential growth or are virtually abelian by explicitly exhibiting free semigroups whose generators have uniformly bounded word length whenever they exist. These free semigroups witness the uniform exponential growth of the group. I will explain how certain arrangements of hyperplane orbits can be used to build loxodromic isometries generating free semigroups and then describe how to use the convex hull of their axes and the Bowditch boundary to extend Kar and Sageev's result to CAT(0) cube complexes with isolated flats. This is joint work with Radhika Gupta and Kasia Jankiewicz.

• Wednesday February 13, 2019 at 14:30, Wachman 617

  Simplicial complexes, configuration spaces, and ‘chromatic’ invariants

  Andrew Cooper, NC State

  Given a space \(X\), the configuration space \(F(X,n)\) is the space of possible ways to place \(n\) points on \(X\), so that no two occupy the same position. But what if we allow some of the points to coincide?

  The natural way to encode the allowed coincidences is as a simplicial complex \(S\). I will describe how the configuration space \(M(S,X)\) obtained in this way gives rise to polynomial and homological invariants of \(S\), how those invariants are related to the cohomology ring \(H^*(X)\), and what this has to do with the topology of spaces of maps into \(X\).

  I will also mention some potential applications of this structure to problems arising from international relations and economics.

  This is joint work with Vin de Silva, Radmila Sazdanovic, and Robert J Carroll.

• Friday February 15, 2019 at 14:30, Haverford College, room TBA

  TBA

  Oleg Lazarev, Columbia University

  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

  Abstract TBA

• Friday February 15, 2019 at 16:00, Haverford College, room TBA

  TBA

  Francesco Lin, Princeton University PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

  Abstract TBA

• Wednesday February 20, 2019 at 14:30, Wachman 617

  Coherence and lattices 

  Matthew Stover, Temple University

  I will survey (in)coherence of lattices in semisimple Lie groups, with a view toward open problems and connections with the geometry of locally symmetric spaces. Particular focus will be placed on rank one lattices, where I will discuss connections with reflection groups, "algebraic" fibrations of lattices, and analogies with classical low-dimensional topology.

• Wednesday March 13, 2019 at 14:30, Wachman 617

  Free products and random walks in acylindrically hyperbolic groups

  Carolyn Abbott, University of California Berkley Imagine you are standing at the point 0 on a number line, and you take a step forward or a step backwards, each with probability 1/2. If you take a large number of steps, is it likely that you will end up back where you started? What if you are standing at a vertex of an 4-valent tree, and you take a step in each of the 4 possible directions with probability 1/4? This process is special case of what is called a random walk on a space. If the space you choose is the Cayley graph of a group (as these examples are), then a random walk allows you to choose a "random" or "generic" element of the group by taking a large number of steps and considering the label of the vertex where you end up. One can ask what properties a generic element of the group is likely to have: for example, is it likely that the element you land on has infinite order? In this talk, I will focus on the class of the class of so-called acylindrically hyperbolic groups, which contains many interesting groups, such as mapping class groups, outer automorphism groups of free groups, and right-angled Artin and Coxeter groups, among many others. I will discuss the algebraic and geometric properties of subgroups generated by a random element and a fixed subgroup.

• Wednesday March 13, 2019 at 16:00, Wachman 527

  Local to global morse properties, convexity and hierarchically hyperbolic spaces. 

  Davide Spiriano, ETH Zurich

  In a Gromov hyperbolic space, geodesics satisfies the so-called Morse property. This means that if a geodesic and a quasi-geodesic share endpoints, then their Hausdorff distance is uniformly bounded. Remarkably, this is an equivalent characterization of hyperbolic spaces, meaning that all consequences of hyperbolicity can be ascribed to this property. Using this observation to understand hyperbolic-like behaviour in spaces which are not Gromov hyperbolic has been a very successful idea, which led to the definition of important geometric objects such as the Morse boundary and stable subgroups. Another strong consequence of hyperbolicity is the fact that local quasi-geodesics are global quasi-geodesics. This allows detecting global properties on a local scale, which has far-reaching consequences. The goal of this talk is twofold. Firstly, we will prove results that are known for hyperbolic groups in a class of spaces satisfying generalizations of the above properties. Secondly, we show that the set of such spaces is large and contains several examples of interest, i.e. CAT(0) spaces and hierarchically hyperbolic spaces.

• Thursday March 21, 2019 at 14:30, Wachman 617

  Exploring algebraic rigidity in mapping class groups 

  Nicholas Vlamis, CUNY Queen's College

  A classical theorem of Powell (with roots in the work of Mumford and Birman) states that the pure mapping class group of a connected, orientable, finite-type surface of genus at least 3 is perfect, that is, it has trivial abelianization. We will discuss how this fails for infinite-genus surfaces and give a complete characterization of all homomorphisms from pure mapping class groups of infinite-genus surfaces to the integers. This characterization yields a direct connection between algebraic invariants of pure mapping class groups and topological invariants of the underlying surface. This is joint work with Javier Aramayona and Priyam Patel.

• Friday March 22, 2019 at 14:00, Wachman 617

  Commensurability classes of fully augmented pretzel links 

  Christian Millichap, Furman University PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  Abstract: Fully augmented links (FALs) are a large class of links whose complements admit hyperbolic structures that can be explicitly described in terms of combinatorial information coming from their respective link diagrams. In this talk, we will examine an infinite subclass of FALs that are constructed by fully augmenting pretzel links and describe how to build their hyperbolic structures. We will then discuss how we can use the geometries of these link complements to analyze arithmetic properties and commensurability classes of these links. This is joint work with Jeff Meyer (CSSB) and Rollie Trapp (CSSB).

  The morning background talk, at 9:30am, will be an exploration of hyperbolic structures on link complements.

• Friday March 22, 2019 at 15:30, Wachman 617

  Augmentations and immersed Lagrangian fillings 

  Dan Rutherford, Ball State University PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  Abstract: This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let \(\Lambda \subset \mathbb{R}^{3}\) be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, \(\mathcal{A}(\Lambda)\), of \(\Lambda\) is functorial for exact Lagrangian cobordisms in the symplectization of \(\mathbb{R}^3\), i.e. a cobordism \(L \subset \mathit{Symp}(\mathbb{R}^3)\) from \(\Lambda_-\) to \(\Lambda_+\) induces a DG-algebra map, \(f_L:\mathcal{A}(\Lambda_+) \rightarrow \mathcal{A}(\Lambda_-).\) In particular, if \(L\) is an exact Lagrangian filling (\(\Lambda_-= \emptyset\)) the induced map is an augmentation \(\epsilon_L: \mathcal{A}(\Lambda_+) \rightarrow \mathbb{Z}/2.\)

  In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift \(\Sigma\) of \(L\). When \(L\) is an immersed, exact Lagrangian filling a choice of augmentation \(\alpha\) for \(\Sigma\) produces an induced augmentation \(\epsilon_{(L, \alpha)}\) for \(\Lambda_+\). Using the cellular formulation of LCH, we are able to show that any augmentation of \(\Lambda\) may be induced by such a filling.

  In the morning background talk, at 11:00am, I will cover augmentations and immersed Lagrangian fillings.

• Wednesday April 3, 2019 at 14:30, Wachman 617

  The Shape of Phylogenetic Treespace

  Katherine St. John

  City University of New York & American Museum of Natural History

  Trees are a canonical structure for representing evolutionary histories. Many popular criteria used to infer optimal trees are computationally hard, and the number of possible tree shapes grows super-exponentially in the number of taxa. The underlying structure of the spaces of trees yields rich insights that can improve the search for optimal trees, both in accuracy and running time, and the analysis and visualization of results. We review the past work on analyzing and comparing trees by their shape as well as recent work that incorporates trees with weighted branch lengths. This talk will highlight some of the elegant questions that arise from improving search and visualizing the results in this highly structured space. All are welcome.

• Thursday April 4, 2019 at 14:30, Wachman 617

  Gonality and the character variety

  Kate Petersen, Florida State University

• Thursday April 4, 2019 at 16:30, University of Pennsylvania, room DRL 4C8

  Periodic Geodesics and Geodesic Nets on Riemannian Manifolds

  Regina Rotman, University of Toronto and IAS

  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  Abstract: I will talk about periodic geodesics, geodesic loops, and geodesic nets on Riemannian manifolds. More specifically, I will discuss some curvature-free upper bounds for compact manifolds and the existence results for non-compact manifolds. In particular, geodesic nets turn out to be useful for proving results about geodesic loops and periodic geodesics.

• Thursday April 4, 2019 at 17:45, University of Pennsylvania, room DRL 4C8

  Filling metric spaces

  Alex Nabutovsky, University of Toronto and IAS

  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  Abstract: The Uryson \(k\)-width of a metric space \(X\) measures how close \(X\) is to being \(k\)-dimensional. Several years ago Larry Guth proved that if \(M\) is a closed \(n\)-dimensional manifold, and the volume of each ball of radius 1 in \(M\) does not exceed a certain small constant \(e(n)\), then the Uryson \((n-1)\)-width of \(M\) is less than 1. This result is a significant generalization of the famous Gromov inequality relating the volume and the filling radius that plays a central role in systolic geometry.

  Guth asked if a much stronger and more general result holds true: Is there a constant \(e(m)>0\) such that each compact metric space with \(m\)-dimensional Hausdorff content less than \( e(m)\) always has \((m-1)\)-dimensional Uryson width less than 1? Note that here the dimension of the metric space is not assumed to be \(m\), and is allowed to be arbitrary.

  Such a result immediately leads to interesting new inequalities even for closed Riemannian manifolds. In my talk I am are going to discuss a joint project with Yevgeny Liokumovich, Boris Lishak and Regina Rotman towards the positive resolution of Guth's problem.

• Wednesday April 10, 2019 at 14:30, Wachman 617

  Effective Special Covers of Alternating Links

  Edgar Bering, Temple University

  In 1982 Thurston stated the "virtual conjectures" for 3-manifolds: that every hyperbolic 3-manifold has finite covers that are Haken and fibered, with large Betti numbers. These conjectures were resolved by Agol and Wise in 2012, using the machinery of special cube complexes. Even before the work of Agol and Wise, but especially after, mathematicians have been interested in understanding the degree of these covers in terms of a manifold's invariants.

  In joint work, David Futer and I give the first steps of a quantitative answer to this question in the setting of alternating link complements. Given an alternating link with n crossings we construct a special cover of degree less than n!. As a corollary, we bound the degree of a cover with Betti number at least k.

• Wednesday April 17, 2019 at 14:30, Wachman 617

  Spectral Rigidity of q-differential Metrics

  Marissa Loving, University of Illinois at Urbana Champaign

  When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.

• Tuesday April 23, 2019 at 14:30, Wachman 617

  Distance formulae and quasi-cube complexes

  Mark Hagen, University of Bristol

  Abstract: Masur and Minsky's work on the geometry of mapping class groups, combined with more recent results about the geometry of CAT(0) cube complexes, motivated the introduction of the class of hierarchically hyperbolic spaces. A metric space \(X\) is hierarchically hyperbolic if there is a set of (uniformly) Gromov-hyperbolic spaces \(U\), each equipped with a projection from \(X\) to \(U\), satisfying various axioms that amount to saying that the geometry of \(X\) is recoverable, up to quasi-isometry, from this projection data. Working in this context often allows one to promote facts about hyperbolic spaces to conclusions about highly non-hyperbolic spaces: mapping class groups, Teichmuller space, "most" 3-manifold groups, etc. In particular, many CAT(0) cube complexes -- including those associated to right-angled Artin and Coxeter groups -- are hierarchically hyperbolic.

  The relationship between CAT(0) cube complexes and hierarchically hyperbolic spaces is intriguing. Just as, in a hyperbolic space, a collection of n points has quasiconvex hull quasi-isometric to a finite tree (i.e. 1-dimensional CAT(0) cube complex), in a hierarchically hyperbolic space, there is a natural notion of the quasiconvex hull of a set of n points, and it is quasi-isometric to a CAT(0) cube complex, by a result of Behrstock-Hagen-Sisto. The quasi-isometry constants depend on n in general. However, when each hyperbolic space U is quasi-isometric to a tree, it turns out that this dependence disappears. From this one deduces that, if \(X\) is a metric space that is hierarchically hyperbolic with respect to quasi-trees, then \(X\) is quasi-isometric to a CAT(0) cube complex. I will discuss this theorem and some of its group-theoretic consequences. This is joint work with Harry Petyt.

• Wednesday May 1, 2019 at 14:30, Wachman 617

  Exotic real projective Dehn surgery space

  Jeff Danciger, University of Texas at Austin

  We study properly convex real projective structures on closed 3-manifolds. A hyperbolic structure is one special example, and in some cases the hyperbolic structure may be deformed non-trivially as a convex projective structure. However, such deformations seem to be exceedingly rare. By contrast, we show that many closed hyperbolic manifolds admit a second convex projective structure not obtained through deformation. We find these examples through a theory of properly convex projective Dehn filling, generalizing Thurston’s picture of hyperbolic Dehn surgery space. Joint work with Sam Ballas, Gye-Seon Lee, and Ludovic Marquis.

• Monday May 6, 2019 at 14:00, Wachman 617

  Separability properties of finitely generated groups

  Mark Pengitore, The Ohio State University

  This talk will be an introduction to separability of finitely generated groups. The premise is that we can detect membership of interesting subsets of finitely generated groups such as the identity subgroup, finitely generated subgroups, and conjugacy classes via surjective group morphisms to finite groups. This idea can be interpreted in many distinct ways such as lifting of closed loops of manifolds to finite covers, topological properties of a totally disconnected compact topological group, and well approximation of elements in a metric space. One of the many applications of these ideas is a quantitative solution to the word problem, conjugacy problem, and other decision problems. In a more topological direction, another application is the constructing lifts of an immersed submanifold to an embedded submanifold in a finite cover. This talk will be expository and will explain connections between all of above ideas and motivate interest in separability.

• Monday May 6, 2019 at 16:00, Wachman 617

  Lower bounds for separability of nilpotent and solvable groups

  Mark Pengitore, The Ohio State University

  In this talk, we introduce quantitative approaches to the study of separability in nilpotent and solvable groups. In particular, we will describe effective residual finiteness, effective subgroup separability, and effective conjugacy separability and discuss various results for asymptotic lower bounds of these properties for these classes of groups. Moreover, we introduce the algebraic, number theoretic, and geometric methods used in the construction of these lower bounds.

• Wednesday September 4, 2019 at 14:30, Wachman 617

  Finiteness of geodesic submanifolds of hyperbolic manifolds

  Matthew Stover, Temple University Hyperbolic manifolds, n≥3, that are arithmetic were characterized by Borel and Margulis as being infinite index in their commensurator. One can use this to show that an arithmetic hyperbolic n-manifold either contains no totally geodesic hypersurfaces or they are everywhere dense. Reid and McMullen (for n= 3) asked whether having infinitely many totally geodesic hypersurfaces conversely implies arithmeticity. I will discuss work with Bader, Fisher, and Miller that answers this question in the positive.

• Wednesday September 11, 2019 at 14:30, Wachman 617

  Algebraic K-theory and G-manifolds

  Mona Merling, University of Pennsylvania The "stable parametrized h-cobordism theorem" provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a characterization of the stable h-cobordism space of M in terms of Waldhausen's algebraic K-theory of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold.

• Wednesday September 18, 2019 at 14:30,

  Detecting covers from simple closed curves

  Tarik Aougab, Haverford College

  Given two finite degree regular covers (not necessarily of the same degree) Y, Z of a surface S, suppose that for any closed curve gamma on S, gamma lifts to a simple closed curve on Y if and only if it does to Z. We prove that Y and Z must be equivalent covers. The proof uses some Teichmuller theory and the curve complex. This represents joint work with Max Lahn, Marissa Loving, and Sunny Yang Xiao.

• Friday September 20, 2019 at 14:30, Wachman 617

  Maps between braid groups

  Dan Margalit, Georgia Tech

  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn About 100 years ago, Artin showed that any homomorphism from the braid group \(B_n\) to the symmetric group \(S_n\) is either cyclic or conjugate to the standard homomorphism. Much more recently, Castel showed that any endomorphism of \(B_n\) is either cyclic or conjugate to (a transvection of) the identity map. With Lei Chen and Kevin Kordek, we extend Castel's result by showing that any homomorphism from \(B_n\) to \(B_{2n}\) is either cyclic or conjugate to (a transvection of) one of the standard maps.

  In the morning background talk (9:30am in room 527) I will review braid groups, mapping class groups, canonical reduction systems, and totally symmetric sets.

• Friday September 20, 2019 at 16:00, Wachman 617

  Exotic Mazur manifolds and Property R

  Kyle Hayden, Columbia University

  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from the knot Floer homology concordance invariant nu, which we prove is an invariant of a simple 4-manifold associated to a knot, called the knot trace. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct \(S^1 \times S^2\) surgeries, resolving a question from Problem 1.16 in Kirby's list. We also resolve a related question about the knot concordance invariants tau and epsilon. This is joint work with Tom Mark and Lisa Piccirillo.

  In the morning background talk (at 11:00am), we'll review the key constructive ingredients (Dehn surgery, handlebody structures, and cork twists), and provide some extra historical context for the results described above. In the afternoon talk, I'll explain our main results, present examples demonstrating our constructive results, and discuss the key ideas in the proofs. In particular, I'll focus on how the main arguments take tools from smooth 3- and 4-dimensional topology, hyperbolic geometry, and Heegaard Floer homology and play them off one another.

• Wednesday September 25, 2019 at 14:30, Wachman 617

  The outer automorphism group of a free product of finite groups

  Rylee Lyman, Tufts University

  Mapping class groups, \(GL(n,\mathbb{Z})\), and \(Out(F_n)\), the outer automorphism group of a free group are among some of the most well-studied infinite discrete groups. One facet they have in common is that, although finitely presented, they are "big" groups, in the sense that their elements exhibit a rich and wide array of dynamical behavior. The Nielsen–Thurston normal form, Jordan normal form and relative train track representative, respectively, all attempt to expose and present this information in an organized way to aid reasoning about this behavior.

  The group of outer automorphisms of a finite free product of finite groups is closely related to \(Out(F_n)\), but is comparatively understudied. In this talk we will introduce these groups, related geometric structures they act on, and review some of the known results. We would like to argue that these groups are also "big": to this end we have shown how to extend work of Bestvina, Feighn and Handel to construct relative train track representatives for outer automorphisms of free products.

• Wednesday October 2, 2019 at 14:30, Wachman 617

  Polynomial or not? Twisting rabbits and lifting trees

  Justin Lanier, Georgia Tech

  A polynomial can be viewed as a branched cover of the sphere over itself that is compatible with a complex structure. If handed a topological branched cover of the sphere, we can ask whether it can arise from a polynomial, and if so, which one? In 2006, Bartholdi and Nekrashevych used group theoretic methods to explicitly solve this problem in special cases, including Hubbard’s twisted rabbit problem. We introduce a new topological approach that draws from the theory of mapping class groups of surfaces. By iterating a lifting map on a complex of trees, we are able to certify whether or not a given branched cover arises as a polynomial. This is joint work with Jim Belk, Dan Margalit, and Becca Winarski.

• Wednesday October 16, 2019 at 14:30, Wachman 617

  A study of subgroups of right-angled Coxeter groups via Stallings-like techniques

  Ivan Levcovitz, Technion

  Associated to any simplicial graph K is the right-angled Coxeter group (RACG) whose presentation consists of an order 2 generator for each vertex of K and relations stating that two generators commute if there is an edge between the corresponding vertices of K. RACGs contain a rich class of subgroups including, up to commensurability, hyperbolic 3-manifold groups, surface groups, free groups, Coxeter groups and right-angled Artin groups to name a few. I will describe a procedure which associates a cube complex to a given subgroup of RACG. I will then present some results regarding structural and algorithmic properties of subgroups of RACGs whose proofs follow from this viewpoint. This is joint work with Pallavi Dani.

• Thursday October 24, 2019 at 16:30, DRL 4C8, David Rittenhouse Labs, University of Pennsylvania

  Quantum representations and geometry of mapping class groups 

  Effie Kalfagianni, Michigan State University PATCH Seminar, at UPenn

  Abstract: The generalization of the Jones polynomial for links and 3-manifolds, due to Witten-Reshetiking-Turaev in the late 90’s, led to constructions of Topological Quantum Field Theory in dimensions (2+1). These theories also include representations of surface mapping class groups. The question of how much of the Thurston geometric picture of 3-manifolds is reflected in these theories is open. I will report on recent work in this direction, with emphasis on the corresponding mapping class group representations. The talk is based on joint work with R. Detcherry and G. Belletti, R. Detcherry, T. Yang.

• Thursday October 24, 2019 at 17:45, Room DRL 4C8, David Rittenhouse Labs, UPenn

  Decomposable cobordisms of legendrian knots 

  Roberta Guadagni, University of Pennsylvania

  PATCH Seminar, at UPenn

  Abstract: The standard notion of concordance and cobordism of smooth knots translates into a notion of Lagrangian concordance and cobordism for Legendrian knots. A natural question is then: can we interpret the cobordism relation as a sequence of moves in the front diagrams of the knots? We will look at the elementary handle attachments that yield a "decomposable" cobordism (Ekholm-Honda-Kalman). We will then construct cobordisms and concordances that are not decomposable in the EHK sense (this is a new result) and end with some currently open questions.

• Wednesday October 30, 2019 at 14:30, Wachman 617

  A central limit theorem for random closed geodesics on surfaces

  Samuel Taylor, Temple University

  In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic lengths of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem. I will discuss a proof of this conjecture obtained in joint work with I. Gekhtman and G. Tiozzo, and its generalizations to all negative curved surfaces.

• Wednesday November 6, 2019 at 14:30, Wachman 617

  Automorphisms with exotic growth

  Rémi Coulon Université de Rennes 1

  Let G be a group. Given an (outer) automorphism f of G, one can study its properties by considering the dynamics induced by the action of f on the set of conjugacy classes of G. A classical problem is to understand how the length of a conjugacy class grows under the iterations of f. For many groups (e.g, free groups, free abelian groups, surface groups, etc) one observes a strong dichotomy : the length of any conjugacy class grows either polynomially or at least exponentially. In this talk, we will explain how to build examples of outer automorphisms of finitely generated groups for which this dichotomy fail.

• Thursday November 7, 2019 at 15:30, Wachman 527

  Random Walks and CAT(0) Cube Complexes 

  Talia Fernós, UNC Greensboro

  Let \(G\) be a group acting on a finite dimensional CAT(0) cube complex \(X\). By studying equivariant maps from the Furstenberg-Poisson boundary to the Roller boundary, we deduce a variety of phenomena concerning the push-forward of the random walk from \(G\) to an orbit in \(X\). Under mild and natural assumptions, we deduce positivity of the drift, sublinear tracking, and the central limit theorem. Along the way we prove that regular elements are plentiful and establish a homeomorphism between the boundary of the contact graph of \(X\) with a special subset of the Roller boundary called the regular points. This is joint work with Jean Lécureux and Frédéric Mathéus.

• Thursday November 14, 2019 at 15:30, Wachman 527

  Dynamics on geodesic currents and atoroidal subgroups of Out(F_N)

  Caglar Uyanik Yale University

  Geodesic currents on surfaces are measure theoretic generalizations of closed curves on surfaces and they play an important role in the study of the Teichmuller spaces. I will talk about their analogs in the setting of free groups, and try to illustrate how the dynamics and geometry of the Out(F_N) action reflects on the algebraic structure of Out(F_N).

• Friday November 15, 2019 at 14:30, Bryn Mawr College, Park Science Building

  Stein domains in complex 2-plane with prescribed boundary 

  Bulent Tosun, University of Alabama
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
  
   

  Title: Stein domains in complex 2-plane with prescribed boundary

  Abstract:  A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider existence of Stein structures in “ambient” setting (Stein manifolds/domains as open/compact subsets of a fixed complex manifold, e.g. complex 2-plane). In particular, I would like to discuss the following question that has been circulating among contact and symplectic topologist for some time: "which integral homology spheres embed in complex 2-plane as the boundary of a Stein domain". This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that: No non-trivial Brieskorn homology sphere, with either orientation, embed in complex 2-plane as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.

  In the morning background talk (9:30am), I will talk about embedding 3-manifolds in 4-manifolds.

   

• Friday November 15, 2019 at 16:00, Bryn Mawr College, Park Science Building

  The triangulation complexity of fibred 3-manifolds 

  Jessica Purcell, Monash University

  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

  Abstract: In the 1950s, Moise showed that any 3-manifold decomposes into tetrahedra. But how many tetrahedra? The triangulation complexity of a 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. While knowing this number can be useful for computing and algorithmic topology, it seems to be difficult to determine. In this talk, I will discuss recent work giving upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3-manifold that fibres over the circle. We show that the triangulation complexity of the manifold is equal to the translation length of the monodromy action on the mapping class group of the fibre, up to a bounded factor, where the bound depends only on the genus of the fibre. This is joint work with Marc Lackenby.

  In the morning background talk (11:30am), I'll give a little background on triangulation complexity, and then describe various terms in the abstract, particularly fibred manifolds, and translation length in the mapping class group. I'll discuss related work, and give an outline of why this is the "right" result in the context of various things we know about fibred manifolds and their geometry.

• Wednesday November 20, 2019 at 16:00, Wachman 527

  Sphere Packings and Arithmetic

  Alex Kontorovich Rutgers University, New Brunswick

  We will discuss recent work on "crystallographic" sphere packings (defined in work with Nakamura), and the subclass of "superintegral" such. (A quintessential example is the classical Apollonian Circle Packing.) These exist in finitely many dimensions, and in fact in finitely many commensurability classes in each dimension. This is a consequence of the Arithmeticity Theorem, that such packings come from arithmetic hyperbolic reflection groups.

• Wednesday December 4, 2019 at 14:30, Wachman 617

  Michelle Chu, UIC

   

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.

• Wednesday January 31, 2018 at 14:30, Wachman 617

  Distances between hyperbolic tubes

  David Futer, Temple University

  Abstract: The study of hyperbolic manifolds often begins with the thick-thin decomposition. Given a number \(\epsilon > 0\), we decompose a manifold into the \(\epsilon\)-thin part (points on essential loops of length less than \(\epsilon\)), and the \(\epsilon\)-thick part (everything else). The Margulis lemma says that there is a universal number \(\epsilon_n\), depending only on the dimension, such that the thin part of every hyperbolic \(n\)-manifold has very simple topology.

  In dimension 3, we still do not know the optimal Margulis constant \(\epsilon_3\). Part of the problem is that while the topology is simple, the geometry of \(\epsilon\)-thin tubes can be quite complicated. I will describe some results that control and estimate the geometry, which has applications to narrowing down the value of the Margulis constant. This is joint work with Jessica Purcell and Saul Schleimer.

• Wednesday February 7, 2018 at 14:30, Wachman 617

  On the rank of hyperbolic group extensions

  Sam Taylor, Temple University

  The rank of a group is the minimal cardinality of a generating set. While simple to define, this quantity is notoriously difficult to calculate, even for well behaved groups. In this talk, I’ll introduce some algebraic and geometric properties of hyperbolic group extensions and discuss how their bundle structure can be used to understand rank in this setting.

• Wednesday February 14, 2018 at 14:45, Wachman 617

  Moduli of Curves: from GT theory to Arithmetic Geometry

  
  Benjamin Collas, Bayreuth
  
  
  The goal of Grothendieck-Teichmüller theory is to lead an arithmetic study of the moduli spaces of curves via their geometric fundamental group. Once identified to the profinite orbifold fundamental group, the latter provides a computational framework in terms of braid and mapping class groups.
  While the classical GT theory, as developed by Drinfel'd, Lochak, Nakamura, Schneps et al., essentially deals with the schematic or topological properties of the spaces ``at infinity'', the moduli spaces of curves also admit a stack or orbifold structure that encodes the automorphisms of curves. The goal of this talk is to show how fundamental group theoretic properties of the mapping class groups and Hatcher-Thurston pants decomposition lead to orbifold arithmetic results, then to potential finer GT groups.
  We will present in detail this analytic Teichmüller approach and indicate the essential obstacles encountered, before briefly explaining how they can be circumvent in terms of arithmetic geometry.

• Wednesday February 21, 2018 at 14:30, Wachman 617

  Some hyperbolic actions of subgroups of Aut(F_n) 

  Lee Mosher, Rutgers Newark

  In the course of our theorem on the \(H^2_b\)-alternative for \(Out(F_n)\) — every finitely generated subgroup of \(Out(F_n)\) is either virtually abelian or has second bounded cohomology of uncountable dimension — the case of subgroups of natural embeddings of \(Aut(F_k)\) into \(Out(F_n)\) led us to subgroups of \(Aut(F_k)\) which have interesting new hyperbolic actions arising from “suspension” constructions, generalizing a thread of hyperbolic suspension constructions which goes back to a theorem of W. Thurston. In this talk we will describe these suspension constructions, and we will speculate on what may unify them.

  This is joint work with Michael Handel.

• Friday February 23, 2018 at 14:00, Wachman 617

  Understanding quantum link invariants via surfaces in 3-manifolds

  Christine Lee, University of Texas

  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  Abstract: Quantum link invariants lie at the intersection of hyperbolic geometry, 3-dimensional manifolds, quantum physics, and representation theory, where a central goal is to understand its connection to other invariants of links and 3-manifolds. In this talk, we will introduce the colored Jones polynomial, an important example of quantum link invariants. We will discuss how studying properly embedded surfaces in a 3-manifold provides insight into the topological and geometric content of the polynomial. In particular, we will describe how relating the definition of the polynomial to surfaces in the complement of a link shows that it determines boundary slopes and bounds the hyperbolic volume of many links, and we will explore the implication of this approach on these classical invariants.

  In the background talk (9:30, AM) I'll introduce the colored Jones polynomial and discuss the many conjectures/open problems surrounding the polynomial, to give the research talk more context.

• Friday February 23, 2018 at 15:30, Wachman 617

  Planar open books and singularities

  Olga Plamenevskaya, SUNY Stony Brook
  
  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
   

  Abstract: Due to work of Giroux, contact structures on 3-manifolds can be topologically described by their open books decompositions (which in turn can be encoded via fibered links). A contact structure is called planar if it admits an open book with fibers of genus 0. Symplectic fillings of such contact structures can be understood, by a theorem of Wendl, via Lefschetz fibrations with the same planar fiber. Using this together with topological considerations, we prove a new obstruction to planarity (in terms of intersection form of fillings) and obtain a few corollaries. In particular, we consider contact structures that arise in a canonical way on links of surface singularities, and show that the canonical contact structure on the link is planar only if the singularity is rational. (Joint work with P. Ghiggini and M. Golla.) 

  In the background talk (11:00 AM), I will discuss topological properties of Lefschetz fibrations over a disk, focusing on the case where fiber is a surface of genus 0. The boundary of the 4-manifold given by Lefschetz fibration has an induced open book and a contact structure. This will be the setting for my second talk.
   

   

• Tuesday March 20, 2018 at 16:00, Wachman 617

  Genus bounds in right-angled Artin groups

  Jing Tao, University of Oklahoma

  Abstract: In this talk, I will describe an elementary and topological argument that gives bounds for the stable commutator lengths in right-angled Artin groups.

• Wednesday March 28, 2018 at 14:30, Wachman 617

  The geodesic flow on Infinite type hyperbolic surfaces

  Ara Basmajian, CUNY Graduate Center

  Abstract: In this talk we first describe some of the known results on the geometry and topology of infinite (topological) type surfaces and then we investigate the relationship between Fenchel-Nielsen coordinates and when the geodesic flow on such a surface is ergodic. Ergodicity of the geodesic flow is equivalent to the surface being of so called parabolic type (the surface does not carry a Green's function), and hence this problem is intimately connected to a version of the classical type problem in the study of Riemann surfaces. Specifically, we study so called tight flute surfaces -- (possibly incomplete) hyperbolic surfaces constructed by linearly gluing infinitely many tight pairs of pants along their cuffs -- and the relationship between their type and geometric structure. This is joint work with Hrant Hakobyan and Dragomir Saric.

• Wednesday April 4, 2018 at 14:30, Wachman 617

  Symmetry and self-similarity in Riemannian geometry

  Wouter van Limbeek, University of Michigan
  
  Abstract: In 1893, Hurwitz showed that a Riemann surface of genus \(g \geq 2\) admits at most \(84(g-1)\) automorphisms; equivalently, any 2-dimensional hyperbolic orbifold \(X\) has $\(Area(X)\geq \pi / 42\). In contrast, such a lower bound on volume fails for the n-dimensional torus \(T^n\), which is closely related to the fact that \(T^n\) covers itself nontrivially. Which geometries admit bounds as above? Which manifolds cover themselves? In the last decade, more than 100 years after Hurwitz, powerful tools have been developed from the simultaneous study of symmetries of all covers of a given manifold, tying together Lie groups, their lattices, and their appearances in differential geometry. In this talk I will explain some of these recent ideas and how they lead to progress on the above (and other) questions.

• Wednesday April 11, 2018 at 14:30, Wachman 617

  Sphere packings and arithmetic lattices

  Kei Nakamura, Rutgers University
  
  Abstract: It has been known for sometime that the Apollonian circle packing, as well as certain other infinite circle/sphere packings, are "integral" packings, i.e. they can be realized so that the bends (the reciprocal of radii) of constituent circles/spheres are integers. Most of the known integral packings exhibit a stronger integral property, and we refer to them as "super-integral" packings. Relating them to the theory of arithmetic reflection lattices, we show that super-integral packings exists only in finitely many dimensions, and only in finitely many commensurability classes.

• Wednesday April 18, 2018 at 14:30, Wachman 617

  Surface bundles, monodromy, and arithmetic groups 

  Bena Tshishiku, Harvard University
  
  Abstract: In the 1960s Atiyah and Kodaira constructed surface bundles over surfaces with many interesting properties (e.g. they're holomorphic with closed base and the total space has nonzero signature). Many questions remain about these examples, including a precise description of their monodromy, viewed as a subgroup of the symplectic group. In this talk I will discuss some recent progress toward this question. The main result is that the monodromy is arithmetic (as opposed to being thin). This is ongoing joint work with Nick Salter. 

• Friday April 20, 2018 at 15:00, 119 Dalton Hall, Bryn Mawr College

  Legendrian satellite knots, DGA representations, and the colored HOMFLY-PT polynomial

  Caitlin Leverson, Georgia Tech
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
  
  Abstract: Legendrian knots are topological knots which satisfy extra geometric conditions. Two classes of invariants of Legendrian knots in \(S^3\) are ruling polynomials and representations of the Chekanov-Eliashberg differential graded algebra (DGA). Given a knot \(K\) and a positive permutation braid \(\beta\), we give a precise formula relating a specialization of the ruling polynomial of the satellite \(S(K,\beta)\) with certain counts of representations of the DGA of the original knot \(K\). We also introduce an \(n\)-colored ruling polynomial, defined analogously to the \(n\)-colored HOMFLY-PT polynomial, and show that the 2-graded version of it arises as a specialization of the \(n\)-colored HOMFLY-PT polynomial. This is joint work with Dan Rutherford.
  
  In the morning background talk (at 10:00 AM), I will give an introduction to Legendrian satellite knots, ruling polynomials, and representations of the Chekanov-Eliashberg DGA.

• Friday April 20, 2018 at 16:30, 119 Dalton Hall, Bryn Mawr College

  Discrete conformal geometry of polyhedral surfaces

  Feng Luo, Rutgers University
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
  
  Abstract: We discuss some of the recent work on discrete conformalgeometry of polyhedral surfaces. The relationship among discrete conformal geometry, the work of Thurston and Alexandrov on convex surfaces in hyperbolic 3-space, and the Koebe circle domain conjecture will be addressed. We also show that the discrete uniformization maps converge to the conformal maps. This is joint work with D. Gu, J. Sun, and T. Wu.
  
  In the morning background talk (at 11:30am), I will review geometric notions such as Delaunay triangulations.

• Wednesday April 25, 2018 at 14:30, Wachman 617

  Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

  Matthew Stover, Temple University

  Reid and McMullen both asked whether or not the presence of infinitely many finite-volume totally geodesic surfaces in a hyperbolic 3-manifold implies arithmeticity of its fundamental group. I will explain why large classes of non-arithmetic hyperbolic n-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. These are the first examples of finite-volume n-hyperbolic manifolds, n>2, for which the collection of all finite-volume totally geodesic hypersurfaces is finite but nonempty. In this talk, I will focus mostly on dimension 3, where one can even construct link complements with this property.

• Tuesday May 1, 2018 at 10:00, Wachman Hall 617

  Poincaré Homology Sphere Symposium

  Final presentations from Math 9072 on the Poincaré homology sphere and related topics.

  Presenters: Thomas Ng, Rebekah Palmer, Khánh Le, and Elham Matinpour.

• Wednesday May 2, 2018 at 16:00, Wachman 617

  Coloring curves on surfaces

  Jonah Gaster, McGill University
  Abstract: In the context of proving that the mapping class group has finite asymptotic dimension, Bestvina-Bromberg-Fujiwara exhibited a finite coloring of the curve graph, i.e. a map from the vertices to a finite set so that vertices of distance one have distinct images. In joint work with Josh Greene and Nicholas Vlamis we give more attention to the minimum number of colors needed. We show: The separating curve graph has chromatic number coarsely equal to g log(g), and the subgraph spanned by vertices in a fixed non-zero homology class is uniquely g-1-colorable. Time permitting, we discuss related questions, including an intriguing relationship with the Johnson homomorphism of the Torelli group.

• Wednesday August 29, 2018 at 14:30, Wachman 617

  How quickly do loops grow as you unwrap a space?

  Jeffrey Meyer, Cal State San Bernardino

  The length of the shortest closed loop in a Riemannian manifold is called the systole. There are deep connections between the systole and the volume of a manifold. Recently there has been interest in how the systole grows as one goes up a tower of covers. Interestingly, this growth is deeply related to number theory. In this talk, I will go over some examples, these deep connections, and recent results. I will start by concretely looking at the systole growth up covers of flat tori. I will then discuss the celebrated result of Buser and Sarnak in which they showed that systolic growth is logarithmic in area up congruence covers of arithmetic hyperbolic surfaces. I will conclude by discussing my results from a recent paper with collaborators Sara Lapan and Benjamin Linowitz in which we show that the systolic growth up congruence p-towers is a least logarithmic in volume for all arithmetic simple locally symmetric spaces.

• Wednesday September 5, 2018 at 14:30, Wachman 617

  Cubulating one-relator products with torsion

  Ben Stucky, University of Oklahoma

  In 2013, Joseph Lauer and Daniel Wise showed that a one-relator group whose defining relator has exponent at least 4 admits a proper, cocompact action on a CAT(0) cube complex, thus verifying a powerful non-positive curvature condition for these groups. To do this, they build a system of nicely-behaved codimension-1 subspaces (“walls”) in the universal cover and invoke a construction due to Sageev. I will describe a generalization of this result to one-relator products, namely, that a one-relator product of locally indicable groups whose defining relator has exponent at least 4 admits a geometric action on a CAT(0) cube complex if the factors do. The main tools are geometric small-cancellation results for van Kampen diagrams over these groups, which allow us to argue that walls are plentiful and geometrically well-behaved in the universal cover. Relative hyperbolicity of these one-relator products and relative quasiconvexity of wall stabilizers both play a central role.

• Wednesday September 12, 2018 at 14:30, Wachman 617

  Using 2-torsion to obstruct topological isotopy

  Hannah Schwartz, Bryn Mawr College

  Two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other (this follows from the classical result that every orientation preserving automorphism of S^3 is isotopic to the identity). In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres.

• Thursday September 20, 2018 at 14:30, Wachman 617

  An introduction to veering triangulations

  Saul Schleimer, University of Warwick

  Singular euclidean structures on surfaces are a key tool in the study of the mapping class group, of Teichmüller space, and of kleinian three-manifolds. François Guéritaud, while studying work of Ian Agol, gave a powerful technique for turning a singular euclidean structure (on a surface) into a triangulation (of a three-manifold). We will give an exposition of some of this work from the point of view of Delaunay triangulations for the L^\infty metric. We will review the definitions in a relaxed fashion, discuss the technique, and then present applications to the study of strata in the space of singular euclidean structures. If time permits, we will also discuss the naturally occurring algorithmic questions.

  This is joint work with Mark Bell and Vaibhav Gadre. Some of our results are independently due to Ian Frankel, who has further applications.

• Friday September 28, 2018 at 14:30, Wachman 617

  Characterizing slopes for hyperbolic and torus knots

  Duncan McCoy, University of Texas at Austin

  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

  Abstract: Given a knot \(K\) in \(S^3\), we say that \(p/q\) is a characterizing slope if the oriented homeomorphism type of the \(p/q\)-surgery on \(K\) is sufficient to uniquely determine the knot \(K\). It is known that for a given torus knot all but finitely many non-integer slopes are characterizing and that for hyperbolic knots all but finitely many slopes with \(q>2\) are characterizing. I will discuss the proofs of both results, which have a surprising amount in common.

  In the background talk, (at 9:30am), I will give an overview of Dehn surgery and some basic 3-manifold topology concepts that will appear in the main talk.

• Friday September 28, 2018 at 16:00, Wachman 617

  Mahler measure and the Vol-Det Conjecture

  Ilya Kofman, CUNY

  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

  Abstract: A basic open problem is to understand how the hyperbolic volume of knots and links is related to diagrammatic knot invariants. The Vol-Det Conjecture relates the volume and determinant of alternating links. We prove the Vol-Det Conjecture for infinite families of alternating links using the dimer model, the Mahler measure of 2-variable polynomials, and the hyperbolic geometry of biperiodic alternating links. This is joint work with Abhijit Champanerkar and Matilde Lalin.

  In the background talk (at 11:00 AM), we will review some classical ways to get geometric invariants of alternating links, and then generalize these ideas to study the geometry of biperiodic alternating links.

• Wednesday October 3, 2018 at 14:30, Wachman 617

  Continuous sections of families of complex algebraic varieties

  Nick Salter, Columbia University

  Families of algebraic varieties exhibit a wide range of fascinating topological phenomena. Even families of zero-dimensional varieties (configurations of points on the Riemann sphere) and one-dimensional varieties (Riemann surfaces) have a rich theory closely related to the theory of braid groups and mapping class groups. In this talk, I will survey some recent work aimed at understanding one aspect of the topology of such families: the problem of (non)existence of continuous sections of “universal” families. Informally, these results give answers to the following sorts of questions: is it possible to choose a distinguished point on every Riemann surface of genus g in a continuous way? What if some extra data (e.g. a level structure) is specified? Can one instead specify a collection of n distinct points for some larger n? Or, in a different direction, if one is given a collection of n distinct points on CP^1, is there a rule to continuously assign an additional m distinct points? In this last case there is a remarkable relationship between n and m. For instance, we will see that there is a rule which produces 6 new points given 4 distinct points on CP^1, but there is no rule that produces 5 or 7, and when n is at least 6, m must be divisible by n(n-1)(n-2). These results are joint work with Lei Chen.

• Wednesday October 3, 2018 at 16:00, Wachman 527

  Finiteness of geodesic hypersurfaces in hyperbolic hybrids

  Nick Miller, Indiana University

  Both Reid and McMullen have independently asked whether a non-arithmetic hyperbolic 3-manifold necessarily contains only finitely many immersed geodesic surfaces. In this talk, I will discuss recent results where we show that a large class of non-arithmetic hyperbolic n-manifolds has only finitely many geodesic hypersurfaces, provided n is at least 3. Such manifolds are called hyperbolic hybrids and include the manifolds constructed by Gromov and Piatetski-Shapiro. These constitute the first examples of hyperbolic n-manifolds where the set of geodesic hypersurfaces is known to be finite and non-empty. Time allowing, I will also discuss the extension of these results to higher codimension. This is joint work with David Fisher, Jean-Francois Lafont, and Matthew Stover.

• Wednesday October 10, 2018 at 14:30, Wachman 617

  Profinite Completions and Representations of Groups

  Ryan Spitler, Purdue University

  The profinite completion of a group $\Gamma$, $\widehat{\Gamma}$, encodes all of the information of the finite quotients of $\Gamma$. When $\Gamma$ is the fundamental group of a 3-manifold $M$, many properties of the group $\widehat{\Gamma}$ have been shown to correspond to geometric and topological properties of $M$. Forthcoming work with Bridson, McReynolds, and Reid establishes that there are certain hyperbolic 3-manifolds and orbifolds whose fundamental groups are determined by their profinite completion, that is if $\Delta$ is any finitely generated, residually finite group with $\widehat{\Delta} \cong \widehat{\Gamma}$, then $\Delta \cong \Gamma$. I will discuss this work and especially the role the representation theory of $\Gamma$ can play in approaching such profinite rigidity questions.

• Wednesday October 17, 2018 at 14:30, Wachman 617

  Cannon--Thurston maps in non-positive curvature

  Emily Stark, Technion - Israel Institute of Technology

  Two far-reaching methods for studying the geometry of a finitely generated group with non-positive curvature are (1) to study the structure of the boundaries of the group, and (2) to study the structure of its finitely generated subgroups. Cannon--Thurston maps, named after foundational work of Cannon and Thurston in the setting of fibered hyperbolic 3-manifolds, allow one to combine these approaches. Mitra (Mj) generalized work of Cannon and Thurston to prove the existence of Cannon--Thurston maps for normal hyperbolic subgroups of a hyperbolic group. I will explain why a similar theorem fails for certain CAT(0) groups. I will also explain how we use Cannon--Thurston maps to obtain structure on the boundary of certain hyperbolic groups. This is joint work with Algom-Kfir--Hilion and Beeker--Cordes--Gardham--Gupta.

• Wednesday October 24, 2018 at 14:30, Wachman 617

  Boundaries of CAT(0) IFP Groups

  Kim Ruane, Tufts University

  CAT(0) IFP groups are a special class of relatively hyperbolic groups where the peripheral groups are virtually abelian. This class includes fundamental groups of hyperbolic 3-manifolds with torus cusp and many more. I will discuss recent joint work with C. Hruska where we give a characterization of when the CAT(0) boundary of such a group is locally connected. This is different than the Bowditch (or relative) boundary of the group which is always locally connected in the one-ended case. I will explain the relationship between the two boundaries and give lots of examples.

• Wednesday October 31, 2018 at 14:30, Wachman 617

  The shape of Out(F): quasi-geodesics in Out(F) and their shadows in sub-factors.

  Yulan Qing, University of Toronto

  We study the behaviour of quasi-geodesics in Out(F) equipped with word metric. Given an element 𝜙 of Out(F), there are several natural paths connecting the origin to 𝜙 in Out(F). We show that these paths are, in general, not quasi-geodesics in Out(F). In fact, we clear up the current misunderstanding about distance estimating in Out(F) by showing that there exists points in Out(F) where all quasi-geodesics between them backtracks in all of the current Out(F) complexes.

• Thursday November 8, 2018 at 14:30, Wachman 617

  Minimal surfaces in hyperbolic 3-manifolds

  Baris Coskunuzer, Boston College

  In this talk, we will discuss the existence question for closed embedded minimal surfaces in 3-manifolds. After reviewing the classical results on the subject, we will show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic 3-manifolds. More details can be found in the preprint arXiv:1806.10549

• Wednesday November 14, 2018 at 14:30, Wachman 617

  RAAGs as normal subgroups of mapping class groups

  Johanna Mangahas, University at Buffalo

  Free normal subgroups of mapping class groups abound, by the result of Dahmani, Guirardel, and Osin that the normal closure of a pseudo-Anosov is often free. At the other extreme, a mapping class supported on too small a subsurface has normal closure the entire mapping class group, by Brendle and Margalit. I'll talk about joint work with Matt Clay and Dan Margalit finding both free and non-free right-angled Artin groups as normal subgroups of mapping class groups. More generally, we can express as free products groups with suitable actions on certain quasi-trees, the latter being the projection complexes introduced by Bestvina, Bromberg, and Fujiwara.

• Friday November 16, 2018 at 14:30, Park Science Building 245, Bryn Mawr College

  New invariants of spatial graphs 

  Erica Flapan, Pomona College

  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  Abstract: We introduce invariants of graphs embedded in \(S^3\) which are related to the Wu invariant and the Simon invariant. Then we use our invariants to prove that \(K_7\), all Möbius ladders with an odd number of rungs, and the Heawood graph are intrinsically chiral in \(S^3\). We also use our invariants to obtain lower bounds for the minimal crossing number of particular embeddings of graphs in \(S^3\).

  The morning background talk, at 11:30 am, will cover an introduction to spatial graph theory.

• Friday November 16, 2018 at 16:00, Park Science Building 245, Bryn Mawr College

  Constructive techniques in knot traces 

  Lisa Piccirillo, UT Austin

  PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

  Abstract: Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the Kirby problem list. I will also use knot traces to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

  In the morning background talk, at 9:15am, I will survey these results, together with their connections to a few major problems in 4-manifold topology. In the afternoon research talk, I will give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces and give (the fun constructive parts of) proofs of the main results.

• Wednesday November 28, 2018 at 14:30, Wachman 617

  Grothendieck-Teichmüller shadows and their action on child's drawings

  Vasily Dolgushev, Temple University

  I will introduce a functor from a poset of certain finite index normal subgroups of the braid group on 4 strands to the category finite groupoids. The limit of this functor coincides with the profinite version of the Grothendieck-Teichmueller group introduced by Vladimir Drinfeld in 1990. My interest in this functor is motivated by the famous question posed by Yasutaka Ihara at the ICM of 1990. This talk is based on a joint work with Khanh Le and Aidan Lorenz.

• Thursday December 6, 2018 at 14:30, Wachman 617

  Real hyperbolic hyperplane complements in the complex hyperbolic plane

  
  Barry Minemyer, Bloomsburg University
  Let M be a finite volume 4 dimensional manifold modeled on the complex hyperbolic plane, and let N be a 2 dimensional totally geodesic submanifold of M modeled on the hyperbolic plane. The main result to be discussed is that M – N admits a complete, finite volume metric whose sectional curvature is bounded above by a negative constant. In this talk we will discuss the motivation for this research and the more important aspects involved in the proof of this result: writing the metric in the complex hyperbolic plane in polar coordinates about a copy of the real hyperbolic plane, and computing curvature formulas for the associated warped-product metric.

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.

• Wednesday January 25, 2017 at 14:30, Wachman 617

  Divergence of CAT(0) cube complexes and right-angled Coxeter groups

  Ivan Levcovitz, CUNY Graduate Center

  Abstract: The divergence function of a metric space, a quasi-isometry invariant, roughly measures the rate that pairs of geodesic rays stray apart. We will present new results regarding divergence functions of CAT(0) cube complexes. Right-angled Coxeter groups, in particular, exhibit a rich spectrum of possible divergence functions, and we will give special focus to applications of our results to these groups. Applications to the theory of random right-angled Coxeter groups will also be briefly discussed.

• Wednesday February 1, 2017 at 14:30, Wachman 617

  Universal acylindrical actions

  Carolyn Abbott, University of Wisconsin

  Abstract: Given a finitely generated group, one can look for an acylindrical action on a hyperbolic space in which all elements that are loxodromic for some acylindrical action of the group are loxodromic for this particular action. Such an action is called a universal acylindrical action and, for acylindrically hyperbolic groups, tends to give a lot of information about the group. I will discuss recent results in the search for universal acylindrical actions, describing a class of groups for which it is always possible to construct such an action as well as an example of a group for which no such action exists.

• Thursday February 9, 2017 at 16:30, David Rittenhouse Labs, UPenn

  Stability and vanishing in the unstable cohomology of SL_n(Z)

  Thomas Church, Stanford/IAS

  PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

  Abstract: Borel proved that in low dimensions, the cohomology of a locally symmetric space can be represented not just by harmonic forms but by invariant forms. This implies that the \(k\)-th rational cohomology of \(SL_n(Z)\) is independent of \( n\) in a linear range \(n \geq c k\), and tells us exactly what this "stable cohomology" is. In contrast, very little is known about the unstable cohomology, in higher dimensions outside this range.

  In this talk I will explain a conjecture on a new kind of stability in the unstable cohomology of arithmetic groups like \(SL_n(Z)\). These conjectures deal with the "codimension-k" cohomology near the top dimension (the virtual cohomological dimension), and for \( SL_n(Z)\) they imply the cohomology vanishes there. Although the full conjecture is still open, I will explain how we proved it for codimension-0 and codimension-1. The key ingredient is a version of Poincare duality for these groups based on the algebra of modular symbols, and a new presentation for modular symbols. Joint work with Benson Farb and Andrew Putman.

• Thursday February 9, 2017 at 17:45, David Rittenhouse Labs, UPenn

  Lagrangian tori, mutations and toric degenerations

  Denis Auroux, UC Berkeley/IAS

  PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

  Abstract: A basic open problem in symplectic topology is to classify Lagrangian submanifolds (up to, say, Hamiltonian isotopy) in a given symplectic manifold. In recent years, ideas from mirror symmetry have led to the realization that even the simplest symplectic manifolds (eg. vector spaces or complex projective spaces) contain many more Lagrangian tori than previously thought. We will present some of the recent developments on this problem, and discuss some of the connections (established and conjectural) between Lagrangian tori, cluster mutations, and toric degenerations, that arise out of this story.

• Wednesday February 15, 2017 at 14:30, Wachman 617

  Experimental statistics of veering triangulations

  William Worden, Temple University

  Abstract: Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. We present experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold. We will begin by discussing essential background material, including hyperbolic manifolds and ideal triangulations, and more particularly fibered hyperbolic manifolds and the construction of the veering triangulation.

• Wednesday February 22, 2017 at 14:30, Wachman 617

  Minimal surfaces with bounded index

  Davi Maximo, University of Pennsylvania

  Abstract: In this talk we will give a precise picture on how a sequence of minimal surfaces with bounded index might degenerate on a given closed three-manifold. As an application, we prove several compactness results.

• Wednesday March 1, 2017 at 14:30, Wachman 617

  The dynamics of classifying geometric structures

  William Goldman, University of Maryland

  The general theory of locally homogeneous geometric structures (flat Cartan connections) originated with Ehresmann's 1936 paper ``Sur les espaces localement homogènes''. Their classification leads to interesting dynamical systems.

  For example, classifying Euclidean geometries on the torus leads to the usual action of the SL(2,Z) on the upper half­plane. This action is dynamically trivial, with a quotient space the familiar modular curve. In contrast, the classification of other simple geometries on the torus leads to the standard linear action of SL(2,Z) on R^2, with chaotic dynamics and a pathological quotient space. This talk describes such dynamical systems, where the moduli space is described by the nonlinear symmetries of cubic equations like Markoff’s equation x^2 + y^2 + z^2 = x y z.

  Both trivial and chaotic dynamics arise simultaneously, relating to possibly singular hyperbolic metrics on surfaces of Euler characteristic ­1.

• Friday March 3, 2017 at 14:30, Wachman 617

  Polyhedra inscribed in quadrics and their geometry

  Sara Maloni, University of Virginia

  PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

  In 1832 Steiner asked for a characterization of polyhedra which can be inscribed in quadrics. In 1992 Rivin answered in the case of the sphere, using hyperbolic geometry. In this talk, I will describe the complete answer to Steiner's question, which involves the study of interesting analogues of hyperbolic geometry including anti de Sitter geometry. Time permitting, we will also discuss future directions in the study of convex hyperbolic and anti de Sitter manifolds. This is joint work with J. Danciger and J.-M. Schlenker.

  In the morning talk (at 9:30am), I will recall the idea of a geometric structure and the definitions of hyperbolic and anti de Sitter geometry. I will also explain hyperbolic quasi-Fuchsian manifolds and their AdS analogues.

• Friday March 3, 2017 at 16:00, Wachman 617

  Manipulating singularities of Weinstein skeleta

  Laura Starkston, Stanford University

  PATCH seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

  Weinstein manifolds are an important class of symplectic manifolds with convex ends/boundary. These 2n dimensional manifolds come with a retraction onto a core n-dimensional stratified complex called the skeleton, which generally has singularities. The topology of the skeleton does not generally determine the smooth or symplectic structure of the 2n dimensional Weinstein manifold. However, if the singularities fall into a simple enough class (Nadler’s arboreal singularities), the whole Weinstein manifold can be recovered just from the data of the n-dimensional complex. We discuss work in progress showing that every Weinstein manifold can be homotoped to have a skeleton with only arboreal singularities (focusing in low-dimensions). This has significance for combinatorially computing deep invariants of symplectic manifolds like the Fukaya category.

  In the morning background talk (at 11:00), I will discuss the original example of a symplectic manifold: the cotangent bundle \(T^*M\) of any smooth manifold \(M\).

• Wednesday March 29, 2017 at 14:30, Wachman 617

  Counting loxodromics for hyperbolic actions

  Samuel Taylor, Yale University

  Abstract: Consider a nonelementary action by isometries of a hyperbolic group \(G\) on a hyperbolic metric space \(X\). Besides the action of \(G\) on its Cayley graph, some examples to bear in mind are actions of \(G\) on trees and quasi-trees, actions on nonelementary hyperbolic quotients of \(G\), or examples arising from naturally associated spaces, like subgroups of the mapping class group acting on the curve graph.

  We show that the set of elements of \(G\) which act as loxodromic isometries of \(X\) (i.e those with sink-source dynamics) is generic. That is, for any finite generating set of \(G\), the proportion of \(X\)-loxodromics in the ball of radius n about the identity in \(G\) approaches 1 as n goes to infinity. We also establish several results about the behavior in \(X\) of the images of typical geodesic rays in \(G\); for example, we prove that they make linear progress in \(X\) and converge to the boundary of \(X\). Our techniques make use of the automatic structure of \(G\), Patterson-Sullivan measure, and the ergodic theory of randoms walks for groups acting on hyperbolic spaces. This is joint work with I. Gekhtman and G. Tiozzo.

• Friday March 31, 2017 at 15:00, Haverford College, room KINSC H108

  Lagrangian handlebodies in R^6

  David Treumann, Boston College

  PATCH Seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

  Abstract: I will discuss an approach using microlocal sheaf theory to study Legendrian surfaces in \(S^5\) and their Lagrangian fillings in \(R^6\). This talk is based on joint work with Eric Zaslow and Linhui Shen about open Gromov-Witten invariants in \(R^6\).

  MORNING BACKGROUND TALK: The first talk will explain some basic notions about sheaves, Legendrians, and Lagrangians. The background talk takes place at 9:30 AM in in the Science Library (KINSC H305C),

• Friday March 31, 2017 at 16:15, Haverford College, room KINSC H108

  Simplicial volume of links from link diagrams

  Anastasiia Tsvietkova, Rutgers University Newark

  PATCH Seminar (organized by Bryn Mawr, Haverford, Penn, and Temple)

  Abstract: Hyperbolic volume is a powerful invariant of hyperbolic 3-manifolds. For 3-manifolds that are not hyperbolic, simplicial volume, that is closely related to Gromov norm, can be seen as a generalization of hyperbolic volume. The hyperbolic volume of a link complement in a 3-sphere is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalize this to the simplicial volume of link complements by analyzing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the (simplicial and hyperbolic) volume in terms of twists of various lengths in a link diagram. The bound found an application in the work relating coefficients of the colored Jones polynomial to volume, in the spirit of the Volume Conjecture. This is a joint work with Oliver Dasbach.

  MORNING BACKGROUND TALK: In this background talk, I will discuss incompressible surfaces in 3-manifolds, decomposing 3-manifolds along spheres and tori, and hyperbolic and simplicial volume. The background talk takes place at 11:30 AM in in the Science Library (KINSC H305C),

• Wednesday April 5, 2017 at 14:30, Wachman 617

  Counting and dynamics on the Markoff-Hurwitz variety

  Michael Magee, Yale University

  Abstract: I'll discuss some recent results on the Markoff-Hurwitz equation. I'll give some explanation about the fundamental relationship between this equation and geometry. We recently obtained a true asymptotic formula for the number of integer points of bounded height on the Markoff-Hurwitz variety in at least 4 variables. The previous best result here was by Baragar (1998) that gives a rough polylogarithmic rate of growth with a mysterious exponent of growth that is not in general an integer. As a consequence of our work we obtain an asymptotic formula for the number of one sided simple closed curves of given length on a certain hyperbolic thrice punctured projective plane. This is joint work with Gamburd and Ronan. If time permits I'll also report on recent work on the dynamics of pseudo-Anosov automorphisms of the Markoff surface over finite fields. This is joint work with undergraduate students Cerbu, Gunther and Peilen. I'll also try to point out some interesting open questions.

• Wednesday April 12, 2017 at 14:30, Wachman 617

  Azumaya algebras and hyperbolic knots

  Matthew Stover, Temple

  I will talk about arithmetic geometry of SL(2,C) character varieties of hyperbolic knots. A simple criterion on roots of the Alexander polynomial determines whether or not a natural construction extends to determine a so-called Azumaya algebra on the so-called canonical component of the character variety, and I'll then explain how this forces significant restrictions on arithmetic invariants of Dehn surgeries on the knot. This is joint work with Ted Chinburg and Alan Reid.

• Wednesday April 19, 2017 at 14:30, Wachman 617

  Weitzenböck Formulae and Sectional Curvature

  Renato Bettiol, University of Pennsylvania

  Classical geometric applications of Weitzenböck formulae establish that manifolds with positive Ricci curvature have vanishing first Betti number, while manifolds with negative Ricci curvature have no nontrivial Killing vector fields. In this talk, I will describe a framework to produce more general Weitzenböck formulae due to Hitchin, and derive two geometric applications that regard sectional curvature. The first implies a certain geometric restriction on 4-manifolds with positive sectional curvature and indefinite intersection form; the second provides a characterization of nonnegative sectional curvature in terms of Weitzenböck formulae for symmetric tensors. These methods potentially yield applications to negatively curved manifolds as well. This is joint work with R. Mendes (WWU Münster).

• Wednesday May 3, 2017 at 14:30, Wachman 1036

  Deligne-Mostow lattices and cone metrics on the sphere

  Irene Pasquinelli, Durham

  Finding lattices in \(PU(n,1)\) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.

  One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle.

  In this talk we will see how this construction of fundamental polyhedra can be extended to almost all Deligne-Mostow lattices with 3-fold symmetry. Time permitting, we will see how this can be extended to Deligne-Mostow lattices with 2-fold symmetry (work in progress).

  Please note the change of location this week.

• Wednesday May 3, 2017 at 16:00, Wachman 1036

  The Poisson boundary for WPD actions

  Giulio Tiozzo, University of Toronto

  Abstract: Let \(G\) be a group of isometries of a hyperbolic space \(X\). If \(X\) is not proper (e.g., a locally infinite graph), a weak form of properness is given by the WPD (weak proper discontinuity) condition, as defined by Bestvina-Bromberg-Fujiwara.

  We consider random walks on groups which act weakly properly discontinuously on a hyperbolic space, and prove that the topological (Gromov) boundary is a model for the measure-theoretic (Poisson) boundary.

  This provides as a corollary an identification of the Poisson boundary of \(Out(F_n)\) without using the theory of outer space. Joint work with J. Maher.

• Wednesday September 6, 2017 at 14:45, Wachman 617

  A McCarthy-type theorem for linearly growing outer automorphisms of F_n

  Edgar Bering, Temple University

  Abstract: In his proof of the Tits alternative for the mapping class group of a surface, McCarthy also proved that given any two mapping classes \(\sigma\) and \(\tau\), there exists an integer \(N\) such that the group generated by \(\sigma^N, \tau^N\) is either free of rank two or abelian. An analogous statement for two-generator subgroups of a linear group is false, due to the presence of the Heisenberg group. In the setting of \(Out(F_n)\), whether or not such a statement is true remains open, though there are many partial results. In this talk I will give an overview of the problem in the context of the analogy among the three families of groups, survey previous work, and give some of the ideas in my proof of a McCarthy-type theorem for linearly growing outer automorphisms.

• Wednesday September 13, 2017 at 14:45, Wachman 617

  Geometrically finite amalgamations of hyperbolic 3-manifold groups are not LERF

  Hongbin Sun, Rutgers University

  Abstract: We will show that, for any two finite volume hyperbolic 3-manifolds, the amalgamations of their fundamental groups along nontrivial geometrically finite subgroups are always not LERF. A consequence of this result is: all arithmetic hyperbolic manifolds with dimension at least 4, with possible exceptions in 7-dimensional manifolds defined by the octonion, their fundamental groups are not LERF.

• Wednesday September 27, 2017 at 14:45, Wachman 617

  Free products and diffeomorphisms of compact manifolds

  Thomas Koberda, University of Virginia

  Abstract: It is a well-known fact that if \(G\) and \(H\) are groups of homeomorphisms of the interval or of the circle, then the free product \(G*H\) is also a group of homeomorphisms of the interval or of the circle, respectively. I will discuss higher regularity of group actions, showing that if \(G\) and \(H\) are groups of \(C^{\infty}\) diffeomorphisms of the interval or of the circle, then \(G*H\) may fail to act by even \(C^2\) diffeomorphisms on any compact one-manifold. As a corollary, we can classify the right-angled Artin groups which admit faithful \(C^2\) actions on the circle, and recover a joint result with H. Baik and S. Kim. This is joint work with S. Kim.

• Friday October 6, 2017 at 15:00, Wachman 617

  Introducing symplectic billiards

  Sergei Tabachnikov, Penn State University

  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

  Abstract: I shall introduce a simple dynamical system called symplectic billiards. As opposed to the usual (Birkhoff) billiards, where length is the generating function, for symplectic billiards the symplectic area is the generating function. I shall explore basic properties and exhibit several similarities, but also differences, of symplectic billiards to Birkhoff billiards. Symplectic billiards can be defined not only in the plane, but also in linear symplectic spaces. In this multi-dimensional setting, I shall discuss the existence of periodic trajectories and describe the integrable dynamics of symplectic billiards in ellipsoids.

  In the morning background talk (at 10am), I shall survey the conventional and outer billiards to provide context for my afternoon talk.

• Friday October 6, 2017 at 16:30, Wachman 617

  Symplectic and exotic 4-manifolds via positive factorizations

  Inanc Baykur, UMass

  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

  Abstract: We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils.

  In the morning talk (11:30am), I will provide background on these topics.

• Wednesday October 11, 2017 at 14:45, Wachman 617

  Deviation inequalities, Martin boundary, and equidistribution for random walks on relatively hyperbolic groups

  Ilya Gekhtman, Yale University

  Abstract: Consider a random walk $\mu$ on a finitely generated group \(G\). The associated Green's metric is defined as minus log of the probability that a random trajectory starting at the first point ever reaches the second. The horofunction boundary of the Green metric is called the Martin boundary of \((G, \mu)\). Identifying the Martin boundary with some geometric boundary of \(G\) is a difficult question with many dynamical applications. We show that the Martin boundary of a relatively hyperbolic group admits an equivariant surjection to the Bowditch boundary, with the preimage of conical points being a singleton. When the relatively hyperbolic group acts properly and cocompactly on a CAT(0) space, we show the Martin boundary coincides with the boundary of the CAT(0) space. The key technical result is that a random path between two points in a relatively hyperbolic group (e.g. a nonuniform lattice in hyperbolic space) has a uniformly high probability of passing any point on a word metric geodesic between them that is not inside a long subsegment close to a translate of a parabolic subgroup.

  We derive some dynamical consequences:

  -For a geometrically finite action with parabolics on a Gromov hyperbolic space the Patterson-Sullivan and harmonic measure are singular.

  -For a geometrically finite action on a negatively curved manifold, the axes of loxodromic elements defined by random walk trajectories equidistribute with respect to a flow invariant measure on the unit tangent bundle (which when there are parabolics is singular to the measure of maximal entropy).

• Wednesday October 25, 2017 at 14:45, Wachman 617

  Price inequality and Betti numbers of manifolds without conjugate points

  Luca Di Cerbo, Stony Brook

  In this talk, I will present a Price type inequality for harmonic forms on manifolds without conjugate points and negative Ricci curvature. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case one can prove a strengthened result. Equipped with these Price type inequalities, I then study the asymptotic behavior of Betti numbers along infinite towers of regular coverings. If time permits, I will discuss the case of hyperbolic manifolds in some detail. This is joint work with M. Stern.

• Wednesday November 1, 2017 at 14:45, Wachman 617

  Recent results about Kauffman bracket skein algebras

  Helen Wong, Institute for Advanced Study

  Abstract: The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial invariant of knots and links in space, and more precisely by Witten's topological quantum field theory interpretation of the Jones invariant. But the skein algebra is also closely related to the \( SL_2 \mathbb C\) -character variety of the surface. We'll describe two seemingly different methods for constructing finite-dimensional representations of the skein algebra --- one uses combinatorial skein theory whereas the other comes from the quantum Teichmuller space. Very recently, Frohman, Le and Kania-Bartoszynska show that for generic representations, the two methods yield exactly the same representations. We'll discuss implications of this result and some of the many questions that remain.

• Wednesday November 8, 2017 at 14:45, Wachman 617

  The geometry of outer automorphism groups of universal right-angled Coxeter groups

  Charles Cunningham, Haverford College

  Abstract: Abstract: We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups. McCullough-Miller space is a polyhedral complex which is virtually a geometric model for the outer automorphism group of a universal right-angled Coxeter group, \(Out(W_n)\). As it is currently an open question as to whether or not \(Out(W_n)\) is CAT(0) or not, it would be helpful to know whether McCullough-Miller space can always be equipped with an \(Out(W_n)\)-equivariant CAT(0) metric. We show that the answer is in the negative. This is particularly interesting as there are very few non-trivial examples of proving that a space of independent interest is not CAT(0).

• Wednesday November 15, 2017 at 14:45, Wachman 617

  Lower bounds on cubical dimension of C(6) groups

  Kasia Jankiewicz, McGill University

  Abstract: I will discuss a construction which for each n gives an example of a finitely presented C(6) small cancellation group that does not act properly on any n-dimensional CAT(0) cube complex.

• Friday November 17, 2017 at 15:00, Sharpless 113, Haverford College

  Exact Lagrangian cobordisms the Augmentation category

  Yu Pan, MIT

  PATCH seminar (joint with Bryn Mawr, Haverford, and Penn). There will be a background talk and an afternoon research talk.

  Abstract (background): I will give a brief introduction of the Legendrian contact homology, which is an invariant of Legendrian knots \(\Lambda\) defined in the spirit of Symplectic Field Theory. With the similar idea applied to a 2-copy of a Lagrangian filling of \(\Lambda), the wrapped Floer homology gives an isomorphism between the linearized contact homology of \(\Lambda\) and the singular homology of the Lagrangian filling. At the end, I would like to mention an on-going project with Dan Rutherford about the wrapped Floer theory for immersed exact Lagrangian fillings.

  Abstract (research): To a Legendrian knot, one can associate an \(A_{\infty}\) category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

• Friday November 17, 2017 at 16:30, Room Sharpless 113, Haverford College

  Duality and semiduality in cohomology of arithmetic groups

  Daniel Studenmund, Notre Dame University

  PATCH seminar (joint with Bryn Mawr, Haverford, and Penn). There will be a morning background talk and an afternoon research talk.

  Abstract (background): An arithmetic group acts naturally on a product of symmetric spaces and Euclidean buildings. We will discuss the examples of SL(2, Z) acting on the hyperbolic plane, SL(2, Z[sqrt(2)]) acting on a product of hyperbolic planes, and SL(2, F_p[t]) acting on a tree.

  Abstract (research talk): A duality group has a pairing exhibiting isomorphisms between its homology and cohomology groups, analogous to Poincare duality for manifolds. Arithmetic groups over number fields form a large class of examples of duality groups, by work of Borel and Serre. Many naturally occurring groups fail to be duality groups, but are morally very close. In this talk we make this precise with the notion of a semiduality group, and sketch a proof that certain arithmetic groups in positive characteristic are semiduality groups, building on the result of Borel--Serre. This talk covers work joint with Kevin Wortman.

• Wednesday December 6, 2017 at 14:45, Wachman 617

  TBA

  Ben Bakker, UGA

• Wednesday December 13, 2017 at 14:45, Wachman 617

  Tessellations from long geodesics on surfaces

  Jenya Sapir, Binghamton University

  Abstract: I will talk about a recent result of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of this tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.

• Wednesday January 20, 2016 at 14:30, Wachman 617

  Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds

  Dave Futer, Temple University

  I will discuss a proof that a cusped hyperbolic 3-manifold M contains an abundant collection of immersed, quasifuchsian surfaces. These surfaces are abundant in the sense that their boundaries separate any pair of points on the sphere at infinity. As a corollary, we recover Wise's theorem that the fundamental group of M is cubulated. This is joint work with Daryl Cooper.

• Wednesday January 27, 2016 at 14:30, Wachman 617

  Topological constructions of manifolds with geometric structures

  Matthew Stover, Temple University

  Classical uniformization implies that the existence of a complete hyperbolic metric on a Riemann surface depends only on its topological type. In dimension 3, Thurston's geometrization program also gives a necessary and sufficient topological condition. I will discuss topological methods for proving existence of a metric of constant holomorphic sectional curvature -1 on the complement of curves in a smooth complex projective surface. I will mainly focus on an interesting example due to Hirzebruch, and hopefully turn to some applications of these topological constructions, e.g., to questions about betti number growth. This is mostly joint with Luca Di Cerbo.

• Wednesday February 3, 2016 at 14:30, Wachman 617

  Spacious knots

  Richard Kent, University of Wisconsin

  Brock and Dunfield showed that there are integral homology spheres whose thick parts are very thick and take up most of the volume. Precisely, they show that, given \(R\) big and \(r\) small, there is an integral homology 3-sphere whose \(R\)-thick part has volume \((1-r) vol(M)\). Purcell and I find knots in the 3-sphere with this property, answering a question of Brock and Dunfield.

• Wednesday February 10, 2016 at 14:30, Wachman 617

  The topology of local commensurability graphs

  Khalid Bou-Rabee, City College of New York

  The p-local commensurability graph (p-local graph) of a group has vertices consisting of all finite-index subgroups, where an edge is drawn between two subgroups if their commensurability index is a power of p. Sitting at the interface between intersection graphs, containment graphs, and commensurability, these p-local graphs give insights to Lubotzky-Segal's subgroup growth functions. In this talk, we connect topological properties of p-local graphs to nilpotence, solvability, and largeness (containing a free subgroup of finite index) of the target group. This talk covers joint work with Daniel Studenmund and Chen Shi.

• Wednesday February 24, 2016 at 14:30, Wachman 617

  Compactifying spaces of Riemannian manifolds, with applications

  Ian Biringer, Boston College

  We will describe how to compactify sets of Riemannian manifolds with constrained geometry (e.g. locally symmetric spaces), where the added limit points are transverse measures on some universal foliated space. As an application, we study the ratio of the \(k\)-th Betti number of a manifold to its volume, and give a strong convergence result for higher rank locally symmetric spaces.

• Friday February 26, 2016 at 15:00, Wachman 617

  Knot contact homology and string topology

  Lenny Ng, Duke University [PATCH seminar, joint with Bryn Mawr, Haverford, and Penn]

  Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe a way to use this approach, combined with the modern theory of Legendrian contact homology (which I'll also introduce), to construct a rather powerful invariant of knots called "knot contact homology".

   

• Friday February 26, 2016 at 16:30, Wachman 617

  Alternating links and definite surfaces

  Josh Greene, Boston College [PATCH seminar, joint with Bryn Mawr, Haverford, and Penn]

  I will describe a characterization of alternating links in terms intrinsic to the link exterior and use it to derive some properties of these links, including algorithmic detection and new proofs of some of Tait's conjectures.

• Wednesday March 9, 2016 at 14:30, Wachman 617

  \(k\)-geodesics and lifting curves simply

  Tarik Aougab, Brown University

  Let \(\gamma\) be a closed curve on a surface \(S\) with negative Euler characteristic, and suppose gamma has at most \(k\) self-intersections. We construct a hyperbolic metric with respect to which \(\gamma\) has length (on the order of) \(\sqrt{k}\), and whose injectivity radius is bounded below by \(1/\sqrt{k}\); these results are optimal. As an application, we give sharp upper bounds on the minimum degree of a cover for which gamma lifts to a simple closed curve. This is joint work with Jonah Gaster, Priyam Patel, and Jenya Sapir.

• Friday March 18, 2016 at 13:30, Wachman 617

  Counting curves on hyperbolic surfaces

  -Note different day and time-

  Viveka Erlandsson, Aalto University

  In this talk I will discuss the growth of the number of closed geodesic of bounded length, and the length grows. More precisely, let \(c\) be a closed curve on a hyperbolic surface \(S=S(g,n)\) and let \(N_c(L)\) denote the number of curves in the mapping class orbit of \(c\) with length bounded by \(L\). Mirzakhani showed that when \(c\) is simple, this number is asymptotic to \(L^{6g-6+2n}\). Here we consider the case when \(c\) is an arbitrary closed curve, i.e. not necessarily simple. This is joint work with Juan Souto.

• Friday March 25, 2016 at 14:00, PATCH seminar, at Bryn Mawr, Park Science Building room 328

  Boundary rigidity

  Genevieve Walsh, Tufts University

• Friday March 25, 2016 at 16:00, PATCH seminar, at Bryn Mawr, Park Science Building room 328

  Embeddings of contact manifolds

  John Etnyre, Georgia Tech

• Wednesday April 6, 2016 at 14:30, Wachman 617

  Veering Dehn surgery

  Saul Schleimer, University of Warwick

  It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced ``ideal triangulations'' for studying manifolds with torus boundary; Lackenby introduced ``taut ideal triangulations'' for studying the Thurston norm ball; Agol introduced ``veering triangulations'' for studying punctured surface bundles over the circle. Veering triangulations are very rigid; one current conjecture is that any fixed three-manifold admits only finitely many veering triangulations.

   

  After giving an overview of these ideas, we will introduce ``veering Dehn surgery''. We use this to give the first infinite families of veering triangulations with various interesting properties. This is joint work with Henry Segerman.

• Thursday April 14, 2016 at 16:30, PATCH seminar, at Penn, DRL room 4C8

  Controlling Ray Bundles with Reflectors

  Andrew Hicks, Drexel University

• Thursday April 14, 2016 at 17:30, PATCH seminar, at Penn, DRL room 4C8

  Loop products, index growth, and closed geodesics

  Nancy Hingston, The College of New Jersey

• Wednesday April 20, 2016 at 14:30, Wachman 617

  Unsmoothable group actions on one-manifolds

  Thomas Koberda, University of Virginia

  I will discuss virtual mapping class group actions on compact one-manifolds. The main result will be that there exists no faithful \(C^2\) action of a finite index subgroup of the mapping class group on the circle, which generalizes results of Farb-Franks and establishes a higher rank phenomenon for the mapping class group, mirroring a result of Ghys and Burger-Monod.

• Wednesday April 27, 2016 at 14:30, Wachman 617

  The simple loop conjecture for 3-manifolds modeled on Sol

  Drew Zemke, Cornell University

  The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We will give a brief history of this problem and outline a solution when the target 3-manifold admits a geometric structure modeled on Sol.

• Thursday August 18, 2016 at 14:30, Wachman 617

  Rips complex for relatively hyperbolic groups

  Piotr Przytycki, McGill University

  We will describe a Rips complex, a thickening of the Cayley graph of a relatively hyperbolic group G, with a graph-theoretic property called dismantlability. This guarantees fixed-point properties and implies that the Rips complex is a classifying space for G (with respect to appropriate family). This is joint work with Eduardo Martinez-Pedroza.

• Wednesday September 14, 2016 at 14:30, Wachman 617

  On Thurston's Euler class one conjecture 

  Mehdi Yazdi, Princeton University

  Abstract: In 1976, Thurston proved that taut foliations on closed hyperbolic 3–manifolds have Euler class of norm at most one, and conjectured that, conversely, any Euler class with norm equal to one is Euler class of a taut foliation. I construct counterexamples to this conjecture and suggest an alternative conjecture.

• Friday September 23, 2016 at 14:30, Bryn Mawr College, Park Science Building 328

  Legendrian graph surfaces (PATCH) 

  Roger Casals, MIT

  In this talk we discuss Legendrian surfaces in the standard contact 5-sphere. The goal is to present ideas relating cubic planar graphs and Legendrian surfaces, elaborating on earlier work of E. Zaslow and D. Treumann. In particular, we will talk about Legendrian singularities, count trees and introduce a combinatorial invariant in graph theory. This is work in progress with E. Murphy.

  There will also be a background talk at 9:30 AM.

• Friday September 23, 2016 at 16:00, Bryn Mawr College, Park Science Building 328

  Packings of hyperbolic surfaces (PATCH) 

  Jason DeBlois, University of Pittsburgh

  In the background talk (11:00 AM), I'll introduce packing problems in general and some famous packing problems in particular. I'll discuss the related meshing problem, some of its standard solutions the Delaunay and Voronoi triangulations, and some of their advantages and shortcomings.

  In the research talk (4:00 PM), I'll specialize to the problem of packing disks on complete hyperbolic surfaces of finite area. I'll exhibit the best density bounds that I know, and I'll show that they are sharp in some cases and not sharp in others.

• Wednesday October 5, 2016 at 14:30, Wachman 527

  Surface group actions on products of trees

  Matthew Stover, Temple University

  Let G be the fundamental group of a closed Riemann surface of genus g > 1. Does G admit a properly discontinuous action on a (finite) product of (finite-valence) trees? This remains open. I will discuss a number of results, joint with David Fisher, Michael Larsen, and Ralf Spatzier, related to this question.

• Wednesday October 19, 2016 at 14:30, Wachman 617

  Finding geodesics in the curve graph 

  Mark Bell, University of Illinois

  The curve graph associated to a surface records the pairs of essential closed curves that are disjoint. The graph is connected but, unfortunately, locally infinite. Thus standard pathfinding algorithms struggle to compute paths through this graph. We will discuss some of the techniques of Leasure, Shackleton, Watanabe and Webb for overcoming this local infiniteness, enabling geodesics to be constructed.

  We will finish with a new refinement that allows such geodesics to be found in polynomial time (in terms of their length). An important corollary of which, is a new (polynomial-time) algorithm to determine the Nielsen--Thurston type of a mapping class via its action on the curve graph. This is joint work with Richard Webb.

• Friday October 21, 2016 at 14:00, Wachman 617

  Computer driven questions, pre-theorems and theorems in geometry (PATCH)

  Moira Chas, Stony Brook University

  In the background talk (9:30-10:30am), I will introduce several numbers can be associated to free homotopy class \(X\) of closed curves on a surface \(S\), with boundary and negative Euler characteristic. Among these are:

  - the self-intersection number of \(X\) (this is the smallest number of times a representative of the X crosses itself),

  - the word length of \(X\) (given a minimal set of generators of the fundamental group, this is the smallest number of generators in a word representing the deformation or conjugacy class) and

  - the length of the geodesic corresponding to \(X\) (given a hyperbolic metric on \(S\) with geodesic boundary)

  - the number of free homotopy classes of a given word length the mapping class group orbit of \(X\).

  The interrelations of these numbers exhibit many patterns when explicitly determined or approximated by running a variety of algorithms in a computer.

  In the research talk (2:00-3:00pm), we will discuss how these computations lead to counterexamples to existing conjectures and to the discovery of new patterns . Some of these new patterns, so intricate and unlikely that they are certainly true (even if not proven yet), are "pre-theorems". Many of these pre-theorems later became theorems. An example of such a theorem states that the distribution of the self-intersection of free homotopy classes of closed curves on a surface, appropriately normalized, sampling among given word length, approaches a Gaussian when the word length goes to infinity. An example of a counterexample (no pun untended!) is that there exists pairs of length equivalent free homotopy classes of curves on a surface S that have different self-intersection number. (Two free homotopy classes \(X\) and \(Y\) are length equivalent if for every hyperbolic metric on \(S\), \(\ell(X)=\ell(Y)\)).

• Friday October 21, 2016 at 15:30, Wachman Hall 617

  Cellular Sheaves in Applications (PATCH)

  Robert Ghrist, University of Pennsylvania

  Background talk (11am-12pm): Homological Inference

  In this background talk, we'll recall what makes homological methods work so well for problems of inference (in Science as well as in Mathematics): the fundamentals of functoriality, exactness, and naturality, are the engines of inference. We'll show what basic commutative diagrams can do by demonstrating a new proof of the classic Hex Theorem from game theory using only exactness and diagram chasing.

  Research talk (3:30-4:30pm): Cellular Sheaves in Applications

  In this talk, I'll argue that the recent advances in applied algebraic topology (persistent homology especially) point to cellular co/sheaves as good structures for modelling data tethered to spaces; and co/homology as an especially useful compression of such data. I'll survey a few simple applications, then dig into one less-simple application from game theory.

• Thursday October 27, 2016 at 11:30, Wachman 527

  Arithmetic progressions in the primitive length spectrum 

  Nick Miller, Purdue University

• Wednesday November 2, 2016 at 14:30, Wachman 617

  Surface group actions on products of trees II

  Matthew Stover, Temple University

  This is part II, where I will talk about character varieties in characteristic p.

• Wednesday November 9, 2016 at 14:30, Wachman 617

  Essential surfaces from intersections in the character variety

  Michelle Chu, University of Texas

  I will describe the SL2(C) character variety for a family of hyperbolic two-bridge knots. These character varieties have multiple components which intersect at points corresponding to non-integral irreducible representations. As such, these points carry lots of interesting topological information. In particular, they are associated to splittings along Seifert surfaces.

• Wednesday November 30, 2016 at 14:30, Wachman 617

  Effective Conjugacy Separability of Lattices in Nilpotent Lie groups

  Mark Pengitore, Purdue University

  In this talk, we give polynomial upper and lower bounds for conjugacy separability of cocompact lattices in nilpotent Lie groups.

• Wednesday December 14, 2016 at 14:30, Waschman 617

  Non-arithmetic lattices

  Martin Deraux, Université Grenoble Alpes

  I will present joint work with Parker and Paupert, that allowed us to exhibit new commensurability classes of non-arithmetic lattices in the isometry group of the complex hyperbolic plane. If time permits, I will also explain close ties between our work and the theory of discrete reflection groups acting on other 2-dimensional complex space forms.

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:45 PM in Room 617 on the sixth floor of Wachman Hall.

• Tuesday January 20, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Ivan Izmestiev, FU Berlin, Variational properties of the discrete Hilbert-Einstein functional

   

  The discrete Hilbert-Einstein functional (also known as Regge action) for a 3-manifold glued from euclidean simplices is the sum of edge lengths multiplied with angular defects at the edges. There is an analog for hyperbolic cone-manifolds; a discrete total mean curvature term appears if the manifold has a non-empty boundary. Variational properties of this functional are similar to those of its smooth counterpart. In particular, critical points correspond to vanishing angular defects, i.e. to metrics of constant curvature. We give a survey on isometric embeddings and rigidity results that can be obtained by studying the second derivative of the discrete Hilbert-Einstein and speak about possible future developments.

   

   

• Tuesday February 3, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Jason Behrstock, CUNY, Curve complexes for cube complexes

   

  For \(CAT(0)\) cubical groups we develop analogues of tools which have played a key role in the study of the mapping class group, namely, the theory of curve complexes and subsurface projections. We will describe these parallel structures and also some new results that can be proven as a result of this new approach. This is joint work with Mark Hagen and Alessandro Sisto.

   

   

• Tuesday February 10, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

  Robert Young, Courant Institute, Filling multiples of embedded curves

   

  Filling a curve with an oriented surface can sometimes be "cheaper by the dozen". For example, L. C. Young constructed a smooth curve drawn on a projective plane in \(\mathbb{R}^n\) which is only about 1.5 times as hard to fill twice as it is to fill once and asked whether this ratio can be bounded below. We will use methods from geometric measure theory to answer this question and pose some open questions about systolic inequalities for surfaces embedded in \(\mathbb{R}^n\).

   

• Tuesday February 10, 2015 at 17:30, Wachman 617

  GeoTop Seminar

  PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

  Elisenda Grigsby, Boston College, (Sutured) Khovanov homology and representation theory

   

  Khovanov homology associates to a link \(L\) in the three-sphere a bigraded vector space arising as the homology groups of an abstract chain complex defined combinatorially from a link diagram. It detects the unknot (Kronheimer-Mrowka) and gives a sharp lower bound (Rasmussen, using a deformation of E.S. Lee) on the 4-ball genus of torus knots.

   

  When \(L\) is realized as the closure of a braid (or more generally, of a "balanced" tangle), one can use a variant of Khovanov's construction due to Asaeda-Przytycki-Sikora and L. Roberts to define its sutured Khovanov homology, an invariant of the tangle closure in the solid torus. Sutured Khovanov homology distinguishes braids from other tangles (joint with Ni) and detects the trivial braid conjugacy class (joint with Baldwin).

   

  In this talk, I will describe some of the representation theory of the sutured Khovanov homology of a tangle closure. It (perhaps unsurprisingly) carries an action of the Lie algebra \(sl(2)\). More surprisingly, this action extends to the action of a slightly larger Lie superalgebra whose structure hints at a unification with the Lee deformation. This is joint work with Tony Licata and Stephan Wehrli.

   

   

• Tuesday February 24, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Genevieve Walsh, Tufts University, Boundaries of Kleinian groups

   

  A hyperbolic group is endowed with a topological space, its boundary, which is well-defined up to homeomorphism. We will discuss hyperbolic groups that have boundaries homeomorphic to the boundaries of different types of Kleinian groups. In particular, we will discuss the boundaries of a type of group which is built up from surface groups, graph-Kleinian groups. This is joint, preliminary work with Peter Haissinsky and Luisa Paoluzzi.

   

   

• Tuesday March 10, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Joseph Maher, CUNY, Random walks on weakly hyperbolic groups

   

  Let \(G\) be a group acting by isometries on a Gromov hyperbolic space, which need not be proper. If \(G\) contains two hyperbolic elements with disjoint fixed points, then we show that a random walk on \(G\) converges to the boundary almost surely. This gives a unified approach to convergence for the mapping class groups of surfaces, \(Out(F_n)\) and acylindrical groups. This is joint work with Giulio Tiozzo.

   

   

• Tuesday March 17, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Brian Rushton, Temple University, Detecting large-scale invariants of infinite groups

   

  Finitely presented groups can be studied geometrically by means of the Cayley graph. The geometry of the Cayley graph has a direct influence on the algebraic properties of the group; for instance, the growth rate of the graph determines if the group is nilpotent. However, it can be difficult to determine the geometric properties of the group. We show how subdivision rules and cube complexes can be used to calculate geometric invariants of infinite groups.

   

   

• Thursday March 19, 2015 at 16:30, PATCH seminar, at Penn, DRL room 4C8

  GeoTop Seminar

  Natasa Sesum, Rutgers University, Ancient solutions in geometric flows

   

  I will discuss ancient solutions in the context of the mean curvature flow, the Ricci flow and the Yamabe flow. I will discuss the classification result in the Ricci flow, construction result of infinitely many ancient solutions in the Yamabe flow. In the last part of the talk I will mention the most recent result about the unique asymptotics of non-collapsed ancient solutions to the mean curvature flow which is a joint work with Daskalopoulos and Angenent.

   

• Thursday March 19, 2015 at 17:30, PATCH seminar, at Penn, DRL room 4C8

  GeoTop Seminar

  Joel Fish, Institute for Advanced Study, Symplectic topology, Hamiltonian flows, and invariant subsets: not just going in going in circles anymore

   

  I will discuss some current joint work with Helmut Hofer in which we make use of symplectic topology and pseudoholmorphic curves to study properties of Hamiltonian flows on compact regular hypersurfaces of symplectic manifolds. In particular, I will show how pseudoholomorphic curve techniques can be used to prove that every non-empty, compact, regular energy surface in \(R^4\) has a trajectory which is not dense in the energy level.

   

• Tuesday March 31, 2015 at 16:30, Wachman 617

  GeoTop Seminar

  -Note different time-

  Christian Millichap, Temple University, Mutations and geometric invariants of hyperbolic 3-manifolds (thesis defense)

   

  In this talk, we will examine how a topological cut and paste operation known as mutation can be used to create geometrically similar hyperbolic manifolds: they are non-isometric yet they have a number of geometric invariants in common. Ruberman has shown that this mutation process preserves the volume of a hyperbolic 3-manifold. Building off of his work, we show that mutations also preserve sufficiently short geodesic lengths. As a result, we are able to construct large classes of hyperbolic knot complements that have the same volume, the same shortest geodesic lengths, but are pairwise incommensurable, i.e., do not share a common finite sheeted cover.

   

   

• Tuesday April 7, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Patricia Cahn, University of Pennsylvania, Knots transverse to a vector field

   

  We study knots transverse to a fixed vector field \(V\) on a 3-manifold \(M\) up to the corresponding isotopy relation. We show this classification is particularly simple when \(V\) is the co-orienting vector field of a tight contact structure, or when \(M\) is irreducible and atoroidal. We also apply our results to study loose Legendrian knots in overtwisted contact manifolds, and generalize results of Dymara and Ding-Geiges. This work is joint with Vladimir Chernov.

   

   

• Friday April 17, 2015 at 15:00, PATCH seminar, at Haverford College, room KINSC H108

  GeoTop Seminar

  Helen Wong, Carleton College, Representations of the Kauffman skein algebra

   

  The Kauffman skein algebra of a surface was originally defined to be a straightforward generalization of the Kauffman bracket polynomial for knots. Only later was it realized as a quantization of the \(PSL(2,\mathbb{C})\) character variety of the surface. The Kauffman skein algebra thus emerged as an important connector between quantum topology and hyperbolic geometry. In this talk, we'll describe how to construct representations of the Kauffman skein algebra and how to construct invariants to help tell them apart. This is joint work with F. Bonahon.

   

• Friday April 17, 2015 at 16:30, PATCH seminar, at Haverford College, room KINSC H108

  GeoTop Seminar

  Ailsa Keating, Columbia University, Lagrangian tori in four-dimensional Milnor fibres

   

  The Milnor fibre of any isolated hypersurface singularity contains exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. I will explain how to construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. Time allowing, I will use these to give examples of fibres whose Fukaya categories are not generated by vanishing cycles, and explain applications to mirror symmetry for those fibres.

   

• Tuesday April 28, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Giulio Tiozzo, Yale University, Random walks and random group extensions

   

  Let us consider a group \(G\) of isometries of a \(\delta\)-hyperbolic metric space \(X\), which is not necessarily proper (e.g. it could be a locally infinite graph). We can define a random walk by picking random products of elements of \(G\), and projecting this sample path to \(X\).

   

  We show that such a random walk converges almost surely to the Gromov boundary of \(X\), and with positive speed.

   

  As an application, we prove that a random k-generated subgroup of the mapping class group is convex cocompact, and a similar statement holds for \(Out(F_n)\).

   

  This is joint work, partially with J. Maher and partially with S. Taylor.

   

   

• Thursday April 30, 2015 at 14:00, Wachman 105D

  GeoTop Seminar

  -Note different day, time, and place-

  Priyam Patel, Purdue University, Separability Properties of Right-Angled Artin Groups.

   

  Right-Angled Artin groups (RAAGs) and their separability properties played an important role in the recent resolutions of some outstanding conjectures in low-dimensional topology and geometry. We begin this talk by defining two separability properties of RAAGs, residual finiteness and subgroup separability, and provide a topological reformulation of each. We then discuss joint work with K. Bou-Rabee and M.F. Hagen regarding quantifications of these properties for RAAGs and the implications of our results for the class of virtually special groups.

   

   

• Tuesday September 1, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Ser-Wei Fu, Temple University, The earthquake deformation of hyperbolic structures

   

  Earthquakes are deformations of a hyperbolic surface introduced by Thurston as generalized Dehn twists. I will describe the earthquake flow on moduli space and discuss some dynamical properties. In particular, there is a cusp excursion result for the once-punctured torus that can be obtained by methods in the study of logarithm laws.

   

   

• Thursday September 10, 2015 at 16:30, PATCH seminar, at Penn, DRL room 4C8

  GeoTop Seminar

  Renato Bettiol, University of Pennsylvania, On the Singular Yamabe Problem on Spheres

   

  The solution to the Yamabe problem of finding a constant scalar curvature metric in a prescribed conformal class on a closed manifold was a major achievement in Geometric Analysis. Among several interesting generalizations to open manifolds, great attention has been devoted to the so-called "singular Yamabe problem". Given a closed Riemannian manifold \(M\) and a submanifold \(S\), this problem consists of finding a complete metric on the complement of \(S\) in \(M\) that has constant scalar curvature and is conformal to the original metric. In other words, these are solutions to the Yamabe problem on \(M\) that blow up along \(S\). A particularly interesting case is the one in which \(M\) is a round sphere and \(S\) is a great circle. In this talk, I will describe how bifurcation techniques and spectral theory of hyperbolic surfaces can be used to prove the existence of uncountably many nontrivial solutions to this problem. This is based on joint work with B. Santoro and P. Piccione.

   

• Thursday September 10, 2015 at 17:30, PATCH seminar, at Penn, DRL room 4C8

  GeoTop Seminar

  Saul Schleimer, University of Warwick, Recognizing three-manifolds

   

  Through the eyes of a topologist, manifolds have no local properties: every point has a small neighborhood that looks like Euclidean space. Accordingly, as initiated by Poincaré, the classification of manifolds is one of the central problems in topology. The ``homeomorphism problem'' is somewhat easier: given a pair of manifolds, we are asked to decide if they are homeomorphic.

   

  These problems are solved for zero-, one-, and two-manifolds. Even better, the solutions are ``effective'': there are complete topological invariants that we can compute in polynomial time. On the other hand, in dimensions four and higher the homeomorphism problem is logically undecidable.

   

  This leaves the provocative third dimension. Work of Haken, Rubenstein, Casson, Manning, Perelman, and others shows that theseproblems are decidable. Sometimes we can do better: for example, if one of the manifolds is the three-sphere then I showed that the homeomorphism problem lies in the complexity class NP. In joint work with Marc Lackenby, we show that recognizing spherical space forms also lies in NP. If time permits, we'll discuss the standing of the other seven Thurston geometries.

   

• Tuesday September 15, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Will Worden, Temple University, Hidden symmetries and commensurability of 2-bridge link complements

   

  The canonical triangulations and symmetry groups of 2-bridge link complements are well understood and relatively easy to describe. We exploit this fact to show that non-arithmetic 2-bridge link complements have no hidden symmetries (i.e., symmetries of a finite cover that do not descend to symmetries of the link complement itself), and are pairwise incommensurable. Much of the talk will focus on understanding 2-bridge links, the canonical triangulations of their complements, and their symmetry groups. From there we will give a sketch of the proof that hidden symmetries do not exist, and touch on the question of pairwise incommensurability.

   

  This is joint work with Christian Millichap.

   

   

• Tuesday September 22, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Amos Nevo, Technion, Actions of arithmetic groups and effective Diophantine approximation

   

  We will describe some recent new developments in Diophantine approximation on algebraic varieties, focusing on some familiar natural examples. The approach we describe utilizes harmonic analysis and ergodic theory on semisimple Lie groups, and provides the best possible solution to many Diophantine approximation problems which were not accessible by previous techniques.

   

  Based on joint work with Alex Gorodnik and on joint work with Anish Ghosh and Alex Gorodnik.

   

   

• Tuesday September 29, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Anastasiia Tsvietkova, UC Davis, The number of surfaces of fixed genus in an alternating link complement

   

  Let \(L\) be a prime alternating link with \(n\) crossings. We show that for each fixed \(g\), the number of genus \(g\) incompressible surfaces in the complement of \(L\) is bounded by a polynomial in \(n\). Previous bounds were exponential in \(n\). This is joint work with Joel Hass and Abigail Thompson.

   

   

• Friday October 23, 2015 at 14:30, Wachman 617

  GeoTop Seminar

  PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

  Effie Kalfagianni, Michigan State University, Non-orientable knot genus and the Jones polynomial.

   

  The non-orientable genus (a.k.a crosscap number) of a knot is the smallest genus over all non-orientable surfaces spanned by the knot. In this talk, I'll describe joint work with Christine Lee, in which we obtain two-sided linear bound of the crosscap number of alternating link in terms of the Jones link polynomial. The bounds are often exact and they allow us to compute the crosscap numbers of infinite families of alternating knots as well as the crosscap number of 283 knots with up to twelve crossings that were previously unknown. Time permitting, we will also discuss generalizations to families of non-alternating links.

   

  The proofs of the results use techniques from angled polyhedral decomposition of 3-manifolds, normal surface theory, and the geometry of augmented links. The background talk, by Jessica Purcell, will explain some of these tools and techniques.

   

• Friday October 23, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

  Tye Lidman, Institute for Advanced Study, Floer homology and symplectic four-manifolds.

   

  Floer homology is a powerful technique in many areas of geometric topology, such as symplectic geometry and three-manifold topology. In the background talk, I will discuss the formal structure of this invariant, as well as its relationships with other objects in low-dimensional topology, including symplectic four-manifolds.

   

   

  Symplectic manifolds are pervasive objects in geometric topology which often give rise to the construction of exotic smooth four-manifolds. We give some new constraints on the topology of symplectic four-manifolds using invariants from Heegaard Floer homology. This is joint work with Jen Hom.

   

• Friday October 30, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  -Note different day-

  Balazs Strenner, Institute of Advanced Study, Construction of pseudo-Anosov maps and a conjecture of Penner

   

  There are many constructions of pseudo-Anosov elements of mapping class groups of surfaces. Some of them are known to generate all pseudo-Anosov mapping classes, others are known not to. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. In this talk I prove that the conjecture is false for most surfaces. (This is joint work with Hyunshik Shin.)

   

   

• Tuesday November 3, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Anja Bankovic, Boston College, Marked length spectral rigidity for flat metrics

   

  In this talk we will introduce the set of non-positively curved Euclidean cone metrics on closed surfaces and explore the lengths of curves in those metrics. We will introduce the techniques we used to show that two such metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity and give the idea of the proof. This is joint work with Chris Leininger.

   

   

• Monday November 9, 2015 at 11:00, Wachman 1036

  GeoTop Seminar

  -Note location, day, and time-

  Daryl Cooper, UC Santa Barbara, Finite-volume hyperbolic 3-manifolds contain immersed quasi-Fuchsian surfaces

   

  will discuss a proof that a complete, non-compact hyperbolic 3- manifold M with finite volume contains an immersed, closed, quasi-Fuchsian surface. Joint with Mark Baker.

   

• Tuesday November 17, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  Jonah Gaster, Boston College, Lifting curves simply

   

  It is a corollary of a celebrated theorem of Scott that every closed curve on a hyperbolic surface \(X\) has a simple lift in a finite cover. In order to discuss a quantitative version of this statement, let the `degree' of a curve be the minimal degree of such a cover. We show: If \(X\) has no punctures, then the maximum degree among curves of length at most \(L\) is coarsely equal to (with constants depending only on the topology of \(X\) the quotient of \(L\) by the length of a systole of \(X\). Time permitting, we will discuss related questions, partial answers, and work in

   

   

   

• Friday November 20, 2015 at 15:00, PATCH seminar, at Bryn Mawr, Park Science Building room 328

  GeoTop Seminar

  Abigail Thompson, UC Davis and IAS, Surgery on fibered knots.

   

  It is a classical result that any closed orientable 3-manifold can be obtained by an operation called surgery on a link in the 3-sphere. The link may have many components. This leads to a natural question: Which 3-manifolds can be obtained by surgery on a knot (i.e. on a 1-component link)? And on which knots? For example, Gordon and Luecke showed that non-trivial surgery on a non-trivial K can't yield the 3-sphere back again.

   

  Which knots have surgeries yielding a lens space? A conjecture of Gordon is that only certain knots, called Berge knots, have such a surgery. The pool of potential counter-examples to this conjecture is slowly diminishing. I'll describe some of what is known so far, and show that some fibered knots can't have lens space surgeries. This is work in progress.

   

   

• Friday November 20, 2015 at 16:30, PATCH seminar, at Bryn Mawr, Park Science Building room 328

  GeoTop Seminar

  Charles Livingston, Indiana University, Heegaard Floer knot homology and its applications.

   

  In 2001, Peter Ozsvath and Zoltan Szabo developed Heegaard Floer theory. Using HF theory, one can associate to each knot K in \(R^3\) a chain complex, CFK(K). From a 3-dimensional perspective, CFK(K) determines the genus of a knot and whether or not it is fibered; from a 4-dimensional perspective, it offers strong constraints on the surfaces the knot can bound in upper 4-space. Applications include new results concerning the classification of complex algebraic curves.

   

  As an algebraic object, CFK(K) has multiple structures: it is a chain complex, it is graded and bifiltered, and it is a module over a polynomial ring. I will begin this talk with a simple example that clarifies the details of these structures. I will then illustrate how, from CFK(K), one can extract a variety of knot invariants. Finally, I will describe families of knots for which the computation of CFK(K) follows from a simple algorithm.

   

   

• Tuesday December 8, 2015 at 16:00, Wachman 617

  GeoTop Seminar

  -Note different day and time-

  Andrew Yarmola, Boston College, Basmajian's identity in higher Teichmuller-Thurston theory

   

  We demonstrate an extension of Basmajian's identity to Hitchin representations of compact bordered surfaces. For 3-Hitchin representations, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that the limit set of an incompressible subsurface of a closed surface has measure zero in the Lebesgue measure on the Frenet curve associated to an n-Hitchin representation. This generalizes a classical result in hyperbolic geometry. Finally, we recall the Labourie-McShane extension of the McShane-Mirzakhani identity to Hitchin representations and note a close connection to Basmajian's identity in both the hyperbolic and the Hitchin setting. This is joint work with Nicholas G. Vlamis.

   

   

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:45 PM in Room 617 on the sixth floor of Wachman Hall.

• Tuesday January 28, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Timothy Susse, CUNY, SCL in Torus Knot Complements

   

  Given a group \(G\) and an element \(g\) of its commutator subgroup, its stable commutator length is the growth rate of the smallest number of commutators whose product is \(g^n\). This quantity is closely related to the topology of surfaces with boundary mapping to a topological space with fundamental group \(G\).

   

   

  When \(G\) is the fundamental group of a torus knot complement, or more generally an amalgamated free product of free abelian groups, we will construct a finite sided polyhedron which parameterizes surfaces with a specified boundary. We will then show that scl is rational in these groups, giving a topological solution to a conjecture of Calegari in this special case.

   

   

• Tuesday February 4, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Chris Hays, University of Pennsylvania, Constructing symplectic 4-manifolds

   

  Symplectic 4-manifolds play an important role in the theory of smooth 4- manifolds for two reasons. First, they typically have a non-trivial Seiberg-Witten invariant. Second, there are methods that allow one to create new symplectic 4-manifolds from known ones. These properties allow one to construct infinitely many smooth 4-manifolds with the same underlying homeomorphism type.

   

  In the talk, I will outline a new program for creating symplectic 4-manifolds. This method relies on creating both interesting concave and convex fillings of contact 3-manifolds, and attaching these fillings together. I will discuss manifolds that can be created in this manner, and the ease with which one can determine that these symplectic manifolds are 'non-standard'.

   

   

• Tuesday February 11, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Tarik Aougab, Yale University, Minimally intersecting filling pairs

   

  Let \(S_{g}\) denote the closed orientable surface of genus \(g\). We show the existence of exponentially many mapping class group orbits of pairs of simple closed curves on \(S_{g}\) which fill the surface, and intersect minimally amongst all filling pairs. We will demonstrate the main idea of the construction, and we'll discuss applications to the complex of curves. This is joint work with S. Huang (applications are joint with S. Taylor and R. Webb).

   

   

• Tuesday February 18, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Tudor Dimofte, Institute for Advanced Study, A Spectral Perspective on Neumann-Zagier

   

  Thurston's gluing equations for ideal hyperbolic triangulations have certain symplectic properties, initially discovered by Neumann and Zagier, that underlie the formulation of many classical and quantum 3-manifold invariants. It has long been suspected that these symplectic properties have an intrinsic topological interpretation. I will explain one such interpretation, which trivializes the symplectic properties, based on branched covers of 3-manifolds and their boundaries. (Joint work with R. van der Veen.)

   

   

• Tuesday February 18, 2014 at 16:30, Wachman 617

  GeoTop Seminar

  -Note different time-

  Martin Bridgeman, Boston College, The Pressure metric for convex Anosov representations

   

  Using thermodynamic formalism, we introduce a notion of intersection for convex Anosov representations. We also produce an Out-invariant Riemannian metric on the smooth points of the deformation space of convex, irreducible representations of a word hyperbolic group \(G\) into \(SL(m, R)\) whose Zariski closure contains a generic element. In particular, we produce a mapping class group invariant Riemannian metric on Hitchin components which restricts to the Weil–Petersson metric on the Fuchsian locus. This is joint work with R. Canary, F. Labourie and A. Sambarino.

   

   

• Tuesday February 25, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Dave Futer, Temple University, Volumes of hyperbolic 3-manifolds I

   

  This will be an introductory talk about estimating the volume of hyperbolic 3-manifolds. By the Mostow rigidity theorem (which I will explain), a 3-dimensional manifold admits at most one complete hyperbolic metric. Hence, the volume of this metric is an important topological invariant.

   

   

  After sketching the background, I will describe a program for obtaining explicit estimates on the volume of a hyperbolic 3-manifold directly from combinatorial data. To date, this program works for the broad class of 3-manifolds that fiber over the circle. All new results mentioned in these talks are joint work with J. Purcell and S. Schleimer.

   

• Tuesday March 4, 2014 at 16:00, Wachman 617

  GeoTop Seminar

  PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

  Kathryn Mann, University of Chicago, Surface groups, representation spaces, and rigidity

   

  Let \(S_g\) denote the closed, genus g surface. In this talk, we'll discuss the space of all circle bundles over \(S_g\), namely \(Hom(\pi_1(S_g), Homeo^+(S^1))\). The Milnor-Wood inequality gives a lower bound on the number of components of this space (\(4g-3\)), but until very recently it was not known whether this bound was sharp. In fact, we still don't know whether the space has infinitely many components!

   

  I'll report on recent work and new tools to understand \(Hom(\pi_1(S_g), Homeo^+(S^1))\). In particular, I use dynamical methods to give a new lower bound on the number of its components, and show that certain geometric representations are surprisingly rigid.

   

• Tuesday March 4, 2014 at 17:30, Wachman 617

  GeoTop Seminar

  PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

  Adam Levine, Princeton University, Non-orientable surfaces in homology cobordisms

   

  We study the minimal genus problem for embeddings of closed, non-orientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the Atiyah-Singer index theorem. For instance, we show that if a non-orientable surface embeds essentially in the product of a lens space with an interval, its genus and normal Euler number are the same as those of a stabilization of a non-orientable surface embedded in the lens space itself. This is joint work with Danny Ruberman and Saso Strle.

   

• Tuesday March 25, 2014 at 11:00, Wachman 447

  GeoTop Seminar

  Special Geometry Day:

  Anton Lukyanenko, University of Illinois, Uniformly quasi-regular mappings on sub-Riemannian manifolds

   

  A quasi-regular (QR) mapping between metric manifolds is a branched cover with bounded dilatation, e.g. \(f(z)=z^2\). The notion generalizes both covering and quasi-conformal mappings and is well-studied for Riemannian manifolds. In a joint work with K. Fassler and K. Peltonen, we define QR mappings of sub-Riemannian manifolds and show that:

  1. Every lens space admits a uniformly QR (UQR) self-mapping.
  2. Every UQR mapping leaves invariant a measurable conformal structure.

  The first result uses an explicit "conformal trap" construction, while the second builds on similar results by Sullivan-Tukia and a connection to higher-rank symmetric spaces.

   

   

• Tuesday March 25, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Special Geometry Day:

  Sara Maloni, Brown University, Combinatorial methods on actions on character varieties

   

  In this talk we consider the \(SL(2,C)\)-character variety \(X = Hom(\pi_1(S), SL(2,C) ) // SL(2,C)\) of the four-holed sphere \(S\), and the natural action of the mapping class group \(MCG(S)\) on it. In particular, we describe a domain of discontinuity for the action of \(MCG(S)\) on the relative character varieties \(X_{(a,b,c,d)}\), which is the set of representations for which the traces of the boundary curves are fixed. Time permitting, in the case of real characters, we show that this domain of discontinuity may be non-empty on the components where the relative Euler class is non-maximal.

   

• Tuesday March 25, 2014 at 17:00, Wachman 617

  GeoTop Seminar

  Special Geometry Day:

  François Guéritaud, Université Lille 1, Spacetimes of constant curvature

   

  I will survey recent results (joint with J. Danciger and F. Kassel) on 3-dimensional complete spacetimes of constant curvature \(K\), also known as quotients of \(PSL(2,R)\) (\(K<0\)) or of its Lie algebra (\(K=0\)). I will emphasize the transition phenomena as \(K\) goes to 0 and, time permitting, discuss the so-called Crooked Plane Conjecture of Charette, Drumm and Goldman.

   

• Thursday March 27, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  -Note different day-

  Christian Millichap, Temple University, Geometric invariants of highly twisted hyperbolic pretzel knots

   

  Given a hyperbolic knot \(K\), the corresponding knot complement \(M\) has a number of interesting geometric invariants. Here, we shall consider the systole of \(M\), which is the shortest closed geodesic in \(M\) and the volume of \(M\). It is natural to ask how bad are these invariants at distinguishing hyperbolic \(3\)-manifolds and how do these invariants interact with one another. In this talk, we shall construct large families of hyperbolic pretzel knot complements with the same volume and the same systole. This construction will rely on mutating pretzel knots along four-punctured spheres, and then showing that such mutations often preserve the volume and the systole of a hyperbolic knot.

   

• Tuesday April 1, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Ara Basmajian, CUNY, Involution generating sets for isometries of hyperbolic n-space

   

  The focus of this talk will be on the word length of the orientation preserving isometries of hyperbolic \(n\)-space (the Mobius group), denoted \(G\), with respect to various generating sets of involutions. If the generating set consists of the conjugacy class of a single orientation preserving \(k\)-involution, we show that the word length of \(G\) is comparable to \(n\). Here a \(k\)-involution is an involution with a fixed point set of codimension \(k\). We also discuss the percentage of involution conjugacy classes for which \(G\) has length two as the dimension \(n\) gets large. Most of this is joint work with Karan Puri.

   

• Tuesday April 8, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Andrew Zimmer, University of Michigan, Rigidity of complex convex divisible sets

   

  An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic dividing group. In this talk I will discuss a notion of convexity in complex projective space and show that every divisible complex convex set with $C^1$ boundary is projectively equivalent to the unit ball. The proof uses tools from dynamics, geometric group theory, and algebraic groups.

   

• Tuesday April 15, 2014 at 17:00, Wachman 617

  GeoTop Seminar

  -Note different time-

  Feng Luo, Rutgers University, Choi's theorem on triangulated 3-manifolds and consequences

   

  In her 2000 Ph.D thesis, Y. Choi proved a very nice theorem concerning Thurston's gluing equations on triangulated 3-manifolds. In this talk, we will give a new simple proof of it and discuss some consequences of Choi's theorem.

   

• Tuesday April 22, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Mark Hagen, University of Michigan, Cubulating hyperbolic free-by-cyclic groups

   

  Let \(G\) be a word-hyperbolic free-by-\(\mathbb{Z}\) group. Then \(G\) acts freely and cocompactly on a CAT(0) cube complex. I'll explain some of the consequences of this fact (notably, \(\mathbb{Z}\)-linearity) and discuss the main ingredients of the proof. This talk is on joint work with Dani Wise.

   

• Tuesday May 6, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Ruth Charney, Brandeis University, Hyperbolic-like geodesics

   

  In spaces of non-positive curvature (CAT(0) spaces), some geodesics act like hyperbolic geodesics and others do not. In joint work with H. Sultan, we use hyperbolic-like geodesics to define a new boundary for a CAT(0) space. In this talk I will give various equivalent characterizations of "hyperbolic-like" geodesics and show how these can be used to understand explicit examples. In addition, I will discuss some recent work of M. Cordes generalizing some of these ideas to geodesic metric spaces with no curvature conditions. (This talk will expand on some ideas from the colloquium talk, but it will be self-contained.)

   

• Tuesday May 20, 2014 at 14:00, Wachman 617

  GeoTop Seminar

  Ben McReynolds, Purdue University, Homology of infinite volume manifolds

   

  I will discuss a homological vanishing result for certain classes of real rank one, locally symmetric, infinite volume manifolds that models well-known homological vanishing results for closed manifolds. The talk will largely focus on the mechanism for the vanishing results which is blends analytic, dynamical, and geometric ideas. This is joint work with Chris Connell and Benson Farb.

   

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday January 30, 2024 at 15:30, Wachman 617

  Log-concavity in 1-d Coulomb gas ensembles 

  Mokshay Madiman, University of Delaware
  
  The ordered elements in several one-dimensional Coulomb gas ensembles arising in probability and mathematical physics are shown to have log-concave distributions. Examples include the beta ensembles with convex potentials (in the continuous setting) and the orthogonal polynomial ensembles (in the discrete setting). In particular, we prove the log-concavity of the Tracy-Widom β distributions, Airy distribution, and Airy-2 process. Log-concavity of last passage times in percolation is proven using their connection to Meixner ensembles. We then obtain the log-concavity of top rows of Young diagrams under Poissonized Plancherel measure, which is the Poissonized version of a conjecture of Chen. This is ongoing joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.

• Tuesday February 6, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Asymptotic topological statistics of Gaussian random zero sets

  Zhengjiang Lin, Courant Institute, NYU
   
  We will briefly discuss some asymptotic topological statistics of Gaussian random zero sets, which include a random distribution on knots as a special case. We will also discuss some results on zero sets of random Laplacian eigenfunctions, which are related to Courant’s nodal domain theorem and Milnor-Thom’s theorem on Betti numbers of real algebraic varieties.

• Tuesday February 13, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

  The spherical mixed p-spin glass at zero temperature

  Yuxin Zhou, University of Chicago
  
  In this talk, I will discuss the spherical mixed p-spin glass model at zero temperature. I will present some recent results that classify the possible structure of the functional ordered parameter. For spherical p+s spin glasses, we classify all possibilities for the Parisi measure as a function of the model. Moreover, for the spherical spin models with n components, the Parisi measure at zero temperature is at most n-RSB or n-FRSB. Some of these results are jointly with Antonio Auffinger.

• Tuesday February 20, 2024 at 15:30, Wachman 617

  A matrix model for conditioned Stochastic Airy

  Brian Rider, Temple University
  
  There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure. What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level.  I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).

• Tuesday February 27, 2024 at 15:30, Wachman 617

  Periodic orbits of stochastic Hamiltonian ODEs

  Fraydoun Rezakhanlou, University of California, Berkeley
  
  According to Conley-Zehnder's theorem, any periodic Hamiltonian ODE in $\mathbb{R}^{2n}$ has at least $2n+1$ geometrically distinct periodic orbits. For a stochastically stationary Hamiltonian ODE, the set of periodic orbits yields a translation invariant random process. In this talk, I will discuss an ergodic theorem for the density of periodic orbits, and formulate some open questions which are the stochastic variants of Conley-Zehnder's theorem.

• Tuesday March 12, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Weak universality in random walks in random environments

  Sayan Das, University of Chicago
  
  We consider one-dimensional simple random walks whose all one-step transition probabilities are iid [0,1]-valued mean 1/2 random variables. In this talk, we will explain how under a certain moderate deviation scaling the quenched density of the walk converges weakly to the Stochastic Heat Equation with multiplicative noise. Our result captures universality in the sense that it holds for all non-trivial laws for random environments. Time permitting, we will discuss briefly how our proof techniques depart from the existing techniques in the literature. Based on a joint work with Hindy Drillick and Shalin Parekh.

• Tuesday March 26, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Contact process on large networks

  Oanh Nguyen, Brown University
   
  The contact process serves as a model for the spread of epidemics on networks, with three popular variations: the Susceptible-Infected-Recovered-Susceptible (SIRS), SIR, and SIS. Our focus lies in understanding the temporal evolution of these processes, especially regarding survival time and its associated phase transitions. I will provide a brief overview of related literature, recent progress, and open problems.

• Tuesday April 2, 2024 at 15:30, Wachman 617

  A generalization of hierarchical exchangeability on trees to Directed Acyclic Graphs

  Paul Jung, Fordham University
  
  We discuss a class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs (DAG). More specifically, such a random array is indexed by N^|V| for some DAG, G = (V,E), and its exchangeability structure is governed by the edge set E. We prove a representation theorem which generalizes the Aldous-Hoover and Austin-Panchenko representation theorems.

• Tuesday April 9, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Scaling limits in dimers and tableaux

  Zhongyang Li, University of Connecticut
  
  We investigate limit shapes and height fluctuations in statistical mechanical models, such as dimers and lecture hall tableaux, through the asymptotics of symmetric polynomials. Confirming a conjecture by Corteel, Keating, and Nicoletti, we show that the rescaled height functions' slopes in the scaling limit of lecture hall tableaux adhere to a complex Burgers equation.

• Tuesday April 16, 2024 at 15:30, Wachman 617

  The shape of the front of multidimensional branching Brownian motion

  Yujin Kim, Courant Institute, NYU
  
  The extremal process of branching Brownian motion (BBM) —i.e., the collection of particles furthest from the origin— has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point— the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.

• Tuesday April 23, 2024 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Programmable matter and emergent computation

  Dana Randall, Georgia Tech
  
  Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior.  We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.

• Tuesday April 30, 2024 at 15:30, Wachman 617

  Homogenization of nonconvex Hamilton-Jacobi equations in stationary ergodic media

  Atilla Yilmaz, Temple University
  
  I will start with a self-contained introduction to the homogenization of inviscid (first-order) and viscous (second-order) Hamilton-Jacobi (HJ) equations in stationary ergodic media in any dimension. After a brief account of the now-classical works that are concerned with periodic media or convex Hamiltonians, I will return to the general setting and outline the results obtained in the last decade that: (i) established homogenization for inviscid HJ equations in one dimension; and (ii) provided counterexamples to homogenization in the inviscid and viscous cases in dimensions two and higher. Finally, I will present my recent joint work with E. Kosygina in which we prove homogenization for viscous HJ equations in one dimension, and also describe how the solution of this problem qualitatively differs from that of its inviscid counterpart.

• Tuesday September 3, 2024 at 15:30, Wachman 617

  Vector-valued concentration on the symmetric group

  Mira Gordin, Princeton University
   
  Concentration inequalities for real-valued functions are well understood in many settings and are classical probabilistic tools in theory and applications -- however, much less is known about concentration phenomena for vector-valued functions. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group. Furthermore, we discuss the implications of this result regarding the distortion of embeddings of the symmetric group into Banach spaces, a question which is of interest in metric geometry and algorithmic applications. We build on prior work of Ivanisvili, van Handel, and Volberg (2020) who proved a vector-valued inequality on the discrete hypercube, resolving a conjecture of Enflo in the metric theory of Banach spaces. This talk is based on joint work with Ramon van Handel.

• Tuesday September 10, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

  Functional limit theorems for local functionals of dynamic point processes

  Efe Onaran, UPenn
   
  I will present functional limit theorems for local, additive, interaction functions of temporally evolving point processes. The dynamics are those of a spatial Poisson process on the flat torus with points subject to a birth-death mechanism, and which move according to Brownian motion while alive. The results reveal the existence of a phase diagram describing at least three distinct structures for the limiting processes, depending on the extent of the local interactions and the speed of the Brownian movements. The proofs, which identify three different limits, rely heavily on Malliavin-Stein bounds on a representation of the dynamic point process via a distributionally equivalent marked point process. Based on a joint work with Omer Bobrowski and Robert J. Adler.

• Tuesday September 17, 2024 at 15:30, Wachman 617

  An involution framework for Metropolis-Hastings algorithms on general state spaces

  Cecilia Mondaini, Drexel University
   
  Metropolis-Hastings algorithms are a common type of Markov Chain Monte Carlo method for sampling from a desired probability distribution. In this talk, I will present a general framework for such algorithms which is based on a fundamental involution structure on a general state space, and encompasses several popular algorithms as special cases, both in the finite- and infinite-dimensional settings. In particular, these include random walk, preconditioned Crank-Nicolson (pCN), schemes based on a suitable Langevin dynamics such as the Metropolis Adjusted Langevin algorithm (MALA), and also ones based on Hamiltonian dynamics including several variants of the Hamiltonian Monte Carlo (HMC) algorithm. In fact, with a slight generalization of our first framework, we are also able to cover algorithms that generate multiple proposals at each iteration. These have the potential of providing efficient sampling schemes through the use of modern parallel computing resources. Here we derive several generalizations of the aforementioned algorithms following as special cases of this multiproposal framework. To illustrate the effectiveness of these sampling procedures, we present applications in the context of some Bayesian inverse problems in fluid dynamics. This is based on joint works with N. Glatt-Holtz (Indiana University), A. Holbrook (UCLA), and J. Krometis (Virginia Tech).

• Tuesday September 24, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

  Coefficientwise Hankel-total positivity in enumerative combinatorics 

  Alan Sokal, University College London
   
  The abstract (which involves quite a bit of TeX) is available here.

• Tuesday October 8, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

  KPZ equation from driven lattice gases

  Kevin Yang, Harvard University 
  
  We will discuss a family of exclusion processes in one spatial dimension, where the random walk particles feel a drift whose speed depends on the local particle configuration. We show that their height fluctuations have a large-N limit given by the KPZ equation with an additional linear transport term. To our knowledge, it is the first KPZ result for a class of particle systems where the invariant measures are not explicit or known. This result also extends a prior series of works on deriving KPZ from stochastic Hamilton-Jacobi equations of Hairer, Quastel, Shen, Xu, and others.

• Tuesday October 22, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

  Reconstruction on hypertrees

  Yuzhou Gu, NYU
  
  We develop new methods for analyzing information propagation in branching processes. Our approach is applied to the broadcasting on hypertrees (BOHT) problem, where we obtain the exact reconstruction threshold for a wide range of parameters. As a consequence, we establish the weak recovery threshold for the hypergraph stochastic block model and the condensation threshold for the random NAE-SAT problem in the corresponding parameter regimes, resolving conjectures made by physicists. Our method introduces a rigorous version of population dynamics and improves robust reconstruction analysis. The core of our analysis relies on information-theoretic methods for channel comparison.

• Tuesday October 29, 2024 at 15:30, Wachman 617

  The diffusion limit of the Aldous chain on the space of continuum trees

  Douglas Rizzolo, University of Delaware
  
  Tree-valued dynamics arise in applications in many areas such as computer science, machine learning, and phylogenetics, often in the context of Markov chain Monte Carlo inference. The immense size of phylogenetic trees has motivated a growing literature on asymptotic properties of such Markov chains and their scaling limits and continuum analogs.  In the 1990's David Aldous conjectured the existence of a scaling limit for a tree-valued Markov chain that can be thought of as the simple random walk on binary trees.  Despite significant interest, constructing the limiting process using traditional methods such as generators or martingale problems is challenging and, indeed, remains open.  In this talk we will discuss the recent resolution of Aldous's conjecture using a novel pathwise construction.  Along the way, we will discuss how some important intermediate processes we construct are related to integrable probability, specifically to up-down chains on branching graphs.

• Tuesday November 12, 2024 at 15:30, Penn (David Rittenhouse Lab 4C6)

  Limits of large contingency tables

  Sumit Mukherjee, Columbia University
  
  We explore the limiting structure of a large contingency table (in cut metric), when the row and column marginals converge in a suitable sense. Our results go beyond the classical contingency table framework, and allows for general matrices with real entries chosen from an arbitrary base measure.

• Tuesday November 19, 2024 at 15:30, Wachman 617

  The dipole phase transition in the 2D Coulomb gas

  Jeanne Boursier, Columbia University
   
  The 2D two-component Coulomb gas is expected to exhibit an infinite sequence of phase transitions driven by the divergence of 2k-poles, accumulating at the Berezinskii-Kosterlitz-Thouless temperature. I will discuss joint work with Sylvia Serfaty in which we provide a rigorous proof of this "dipole transition." Our proof is based on the analysis of dipole pairs and large deviations techniques.

• Tuesday December 3, 2024 at 15:30, Wachman 617

  The No-U-Turn sampler: reversibility and mixing time

  Nawaf Bou-Rabee, Rutgers University
  
  The No-U-Turn sampler (NUTS) is arguably one of the most important Markov chain Monte Carlo methods out there, but its recursive architecture has long made it challenging to understand.  This talk will provide a clearer picture of how NUTS operates and why it performs well in high-dimensional problems.  Specifically, I will present a concise proof of NUTS’ reversibility by interpreting it as an auxiliary variable method, in the sense of Section 2.1 of arxiv.org/abs/2404.15253.  This novel auxiliary variable approach offers significant flexibility for constructing transition kernels that are reversible with respect to a given target distribution, and includes as special cases Metropolis-Hastings, slice samplers, proximal samplers, and existing locally adaptive HMC methods.  Next, I will present the first mixing time guarantee for NUTS, based on couplings and concentration of measure, which aligns well with empirical observations. Specifically, the mixing time of NUTS, when initialized in the concentration region of the canonical Gaussian measure, scales as d^{1/4}, up to log factors, where d is the dimension (see arxiv.org/abs/2410.06978). This scaling is expected to be sharp (see arxiv.org/abs/1001.4460). A key insight in our analysis is that concentration of measure leads to uniformity in NUTS’ locally adapted transitions.  We formalize this uniformity by using an extension of a recent coupling framework (see arxiv.org/abs/2308.04634) to a broader class of accept/reject chains. NUTS is then interpreted as one such chain, with the accept chain showing more uniform behavior.  This is joint work with Bob Carpenter (Flatiron), Milo Marsden (Stanford), and Stefan Oberdörster (Bonn).

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday January 24, 2023 at 15:30, Temple (Wachman Hall 617)

  Rare transitions in noisy heteroclinic networks

  Yuri Bakhtin, Courant Institute
  
  We study white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude, and we describe the underlying mechanism. A metastability picture emerges, with a hierarchy of time scales and clusters of accessibility, similar to the classical Freidlin-Wentzell picture but with shorter transition times. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus. Joint work with Hong-Bin Chen and Zsolt Pajor-Gyulai.

• Tuesday January 31, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

  Infinite cycles in the interchange process in five dimensions

  Dor Elboim, Princeton University 
   
  In the interchange process on a graph $G=(V, E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $\pi _\beta: V\to V$ is formed for any time $\beta >0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. 
   
  We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $\beta$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions. 
   
  In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.
   
  This is a joint work with Allan Sly.

• Tuesday February 7, 2023 at 15:30, Temple (Wachman Hall 617)

  Mean-field games: asymptotics and refined convergence results

  Kavita Ramanan, Brown University 
   
  A mean-field game is a game with a continuum of players,  describing the limit as n tends to infinity of Nash equilibria of certain n-player games, in which agents interact symmetrically through the empirical measure of their state processes. We first study the asymptotic behavior of Nash equilibria in static games with a large number of agents. In particular, we establish law of large number limits and large deviation principles for the set of Nash equilibria and discuss applications to congestion games and the price of anarchy. Then we discuss stochastic differential games, which are often understood via the so-called "master equation", which is an infinite-dimensional PDE for the value function. We will show how analysis of sufficiently smooth solutions to the master equation play a role in analyzing large deviation principles for mean-field games. This is based on joint works with Francois Delarue and Daniel Lacker.

• Tuesday February 14, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

  KPZ on a large torus

  Yu Gu, University of Maryland
   
  I will present the recent work with Tomasz Komorowski and Alex Dunlap in which we derived optimal variance bounds on the solution to the KPZ equation on a large torus, in certain regimes where the size of the torus increases with time. We only use stochastic calculus and I will try to give a heuristic explanation of the 2/3 and 1/3 exponents in the 1+1 KPZ universality class.

• Tuesday February 21, 2023 at 15:30, Temple (Wachman Hall 617)

  The nonlinear stochastic heat equation in the critical dimension

  Alex Dunlap, Courant Institute
  
  I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and earlier joint work with Yu Gu.

• Tuesday February 28, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

  Functional inequalities on the space of d-regular directed graphs, with applications to mixing

  Konstantin Tikhomirov, Carnegie Mellon University
  
  We consider the space of d-regular directed simple graphs, where two graphs are connected whenever there is a simple switching operation transforming one graph to the other. For constant d, we prove optimal bounds on the modified Log-Sobolev constant of the associated Markov chain on the space of graphs. This implies that the total variation mixing time of the chain is of order n log(n), which settles an old open problem. Based on joint work with Pierre Youssef.

• Tuesday March 14, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

  Liouville conformal field theory and the quantum zipper

  Morris Ang, Columbia University
   
  Sheffield showed that conformally welding a \gamma-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter \kappa = \gamma^2 as the interface, and Duplantier-Miller-Sheffield proved similar stories for \kappa = 16/\gamma^2 for \gamma-LQG surfaces with boundaries decorated by looptrees of disks or by continuum random trees. We study these dynamics for LQG surfaces coming from Liouville conformal field theory (LCFT). At stopping times depending only on the curve, we give an explicit description of the surface and curve in terms of LCFT and SLE. This has applications to both LCFT and SLE. We prove the boundary BPZ equation for LCFT, which is crucial to solving boundary LCFT. With Yu we prove the reversibility of whole-plane SLE for \kappa ≥ 8 via a novel radial mating-of-trees.

• Tuesday March 21, 2023 at 15:30, Temple (Wachman Hall 617)

  Algorithmic barriers from intricate geometry in random computational problems

  Eren C. Kızıldağ, Columbia University
   
  Many computational problems involving randomness exhibit a statistical-to-computational gap (SCG): the best known polynomial-time algorithm performs strictly worse than the existential guarantee. In this talk, we focus on the SCG of the symmetric binary perceptron (SBP), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. We establish that the solution space of the SBP exhibits intricate geometrical features, known as the multi Overlap Gap Property (m-OGP). By leveraging the m-OGP, we obtain nearly sharp hardness guarantees against the class of stable and online algorithms, which capture the best known algorithms for the SBP. Our results mark the first instance of intricate geometry yielding tight algorithmic hardness against classes beyond stable algorithms.
  
  Time permitting, I will discuss how the same program extends also to other models, including (a) discrepancy minimization, and (b) random number partitioning problem. 
  
  Based on joint works with David Gamarnik, Will Perkins, and Changji Xu.

• Tuesday March 28, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

  Phase transition in mean-field models

  Wenpin Tang, Columbia University
   
  In this talk, I will discuss two mean-field models in which a certain phase transition occurs. I first describe McKean-Vlasov equations involving hitting times which arise as the mean-field limit of particle systems with annihilation. One such example is the super-cool Stefan problem. It is well known that such a system may have blow-ups. We provide some sufficient conditions on the model data to assure either blow-ups or no blow-ups. In the second part, I will discuss the convergence rate of second-order mean-field games to first-order ones, motivated from numerical challenges in first-order mean-field PDEs and the weak noise theory in KPZ universality. When the Hamiltonian and the coupling function have a certain growth, the rate is independent of the dimension; on the other hand, the rate decays in dimension (curse of dimensionality) when the Hamiltonian and the coupling function have small growth. These are based on joint work with Yuming Paul Zhang.

• Tuesday April 4, 2023 at 15:30, Temple (Wachman Hall 617)

  A Berry-Esseen theorem and Edgeworth expansions for inhomogeneous elliptic Markov chains

  Yeor Hafouta, University of Maryland
   
  We obtain optimal rates in the central limit theorem (CLT) for additive functionals of uniformly elliptic inhomogeneous Markov chains without any assumptions on the growth rates of the variance of the underlying partial sums. (The CLT itself is due to Dobrushin (1956) and it holds in greater generality.)
  
  We will also discuss Edgeworth expansions (i.e., the correction terms in the CLT) of order one for general classes of functionals, which provide a structural characterization of having better than optimal CLT rates.
  
  Finally, for several classes of additive functionals (e.g., Holder continuous), we will provide optimal conditions for Edgeworth expansions of an arbitrary order.
  
  The talk is based on a joint work with Dmitry Dolgopyat.

• Tuesday April 11, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

  Ising model on locally tree-like graphs: uniqueness of solutions to cavity equations

  Qian Yu, Princeton University
  
  In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there is at most one non-trivial fixed point for Ising models with zero or certain random external fields. Previously this was only known for sufficiently ``low-temperature'' models. 
  
  Our result simultaneously closes the following 6 conjectures in the literature: 1) independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-Polyanskiy'21); 6) characterization of entropy (and mutual information) of community labels given the graph in 2-SBM (ibid). 
  
  This is a joint work with Yury Polyanskiy.

• Tuesday April 11, 2023 at 16:30, Penn (David Rittenhouse Lab 4C4)

  Cutoff profile of the colored ASEP: GOE Tracy-Widom

  Lingfu Zhang, UC Berkeley
  
  In this talk, I will discuss the colored Asymmetric Simple Exclusion Process (ASEP) in a finite interval. This Markov chain is also known as the biased card shuffling or random Metropolis scan, and its study dates back to Diaconis-Ram (2000). A total-variation cutoff was proved for this chain a few years ago using hydrodynamic techniques (Labbé-Lacoin, 2016). In this talk, I will explain how to obtain more precise information on its cutoff, specifically to establish the conjectured GOE Tracy-Widom cutoff profile. The proof relies on coupling arguments, as well as symmetries obtained from the Hecke algebra. I will also discuss some related open problems.

• Tuesday April 18, 2023 at 15:30, Temple (Wachman Hall 617)

  Minimal surfaces in random environment

  Ron Peled, Tel Aviv University, IAS and Princeton University
  
  A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. Specializing to a model that we term harmonic MSRE, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations.
  
  Joint work with Barbara Dembin, Dor Elboim and Daniel Hadas.

• Tuesday April 25, 2023 at 15:30, Penn (David Rittenhouse Lab 4C4)

  Stochastic waves on metric graphs and their genealogies

  Louis Fan, Indiana University
   
  Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. These equations describe a quantity (density/concentration of an entity) that evolves over space and time, taking into account random fluctuations. However, for many reaction terms and noises, the solution notion of these equations is still missing in dimension two or above, hindering the study of the spatial effect on stochastic dynamics through these equations.
  
  In this talk, I will discuss a new approach, namely, to study these equations on general metric graphs that flexibly parametrize the underlying space. This enables us to not only bypass the ill-posedness issue of these equations in higher dimensions, but also assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. We will focus on the computation of the probability of extinction, the quasi-stationary distribution, the asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type.

• Tuesday September 5, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Stochastic quantization of Yang-Mills in 2D and 3D

  Hao Shen, University of Wisconsin-Madison
   
  Quantum Yang-Mills model is a type of quantum field theory with gauge symmetry. The rigorous construction of quantum Yang-Mills is a central problem in mathematical physics. Stochastic quantization formulates the problem as stochastic dynamics, which can be studied using tools from analysis, PDE and stochastic PDE. We will discuss stochastic quantization of Yang-Mills on the 2 and 3 dimensional tori. To this end we need to address a number of questions, such as the construction of a singular orbit space, together with class gauge invariant observables (singular holonomies or Wilson loops), solving a stochastic PDE using regularity structures, and projecting the solution to the orbit space. Mostly based on joint work with Chandra, Chevyrev and Hairer.

• Tuesday September 12, 2023 at 15:30, Temple (Wachman Hall 617)

  Statistical mechanics of Log and Riesz interactions

  Luke Peilen, Temple University
  
  We study the statistical mechanics of the log gas, an interacting particle system with applications in random matrix theory and statistical physics, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on a novel next order energy quantity that are valid down to microscopic length scales. Simultaneously, we exhibit a control on fluctuations of linear statistics that is also valid down to microscopic scales. Using these local laws, we exhibit for the first time a CLT at arbitrary mesoscales, improving upon previous results of Bekerman and Lodhia.
   
  The methods we use are suitable for generalization to higher dimensional Riesz interactions; we will discuss some generalizations of the above approach and partial results for the Riesz gas in higher dimensions.

• Tuesday September 19, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Random walks in (Dirichlet) random environments with jumps on $\mathbb{Z}$

  Daniel Slonim, University of Virginia
   
  We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.

• Tuesday September 26, 2023 at 15:30, Temple (Wachman Hall 617)

  Large deviations of the KPZ equation and most probable shapes

  Yier Lin, University of Chicago
  
  The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments.
  
  This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.

• Tuesday October 3, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Geometry of the doubly-periodic Aztec dimer model

  Tomas Berggren, MIT
   
  Random dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly-periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit, we see three types of macroscopic regions — known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.

• Tuesday October 10, 2023 at 15:30, Temple (Wachman Hall 617)

  A rigorous approximation of a certain random Fermi-Pasta-Ulam-Tsingou (FPUT) lattice by the Korteweg-De Vries (KdV) equation

  Joshua McGinnis, UPenn
  
  We review recent results regarding the rigorous approximation of 1D and 2D disordered (random, independent masses and/or springs) harmonic lattices by effective wave equations in the long wave limit. In this linear setting, we show the homogenization argument and highlight the tools used from probability theory to control the stochastic error terms such as the Law of the Iterated Logarithm and Hoeffding’s inequality. With our discussion of the linear problem serving as a springboard, we then present a new result regarding the approximation of an FPUT lattice with random masses by a KdV equation. Specifically, we are able to bound the approximation error in terms of the small parameter from the long wave scaling in an almost sure sense. In our theorem, we require a technical condition on the random masses, which we call transparency. Our proof relies on the incorporation of an auto-regressive process into an approximating ansatz, which itself is approximated by solutions to the KdV equation. We discuss the role of the auto-regressive process as well as the condition of transparency in the proof and give numerical evidence supporting the result. We conclude by discussing open questions such as the apparent lack of KdV dynamics in an FPUT lattice with independent, random masses.

• Tuesday October 17, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Sample duality

  Adrián González-Casanova, UC Berkeley
  
  Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x". Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique is useful, including an application to the Simple Exclusion Process with reservoirs. The last part of the lecture is based in recent joint work with Simone Floriani.

• Tuesday October 24, 2023 at 15:30, Temple (Wachman Hall 617)

  An iterative approach to estimating integrated volatility

  Cooper Boniece, Drexel University
  
  The quadratic variation of a semimartingale plays an important role in a variety of applications, particularly so in financial econometrics, where it is closely linked to volatility.  It contains information pertaining to both continuous and discontinuous path behavior of the underlying process, and separating its continuous and discontinuous parts based on high-frequency observations is a problem that has been tackled through a variety of approaches to-date.
   
  However, despite the favorable asymptotic statistical properties of many of these approaches, their use in practice requires heuristic selection of tuning parameters that can greatly impact their estimation performance.
    
  In this talk, I will discuss some recent work concerning an iterative approach that circumvents the "tuning problem."
  
  This is based on joint work with J. E. Figueroa-López and Y. Han.

• Tuesday October 31, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Fundamental limits of low-rank matrix estimation: information-theoretic and computational perspectives

  Yuchen Wu, Penn
   
  Many statistical estimation problems can be reduced to the reconstruction of a low-rank n×d matrix when observed through a noisy channel. While tremendous positive results have been established, relatively few works focus on understanding the fundamental limitations of the proposed models and algorithms. Understanding such limitations not only provides practitioners with guidance on algorithm selection, but also spurs the development of cutting-edge methodologies. In this talk, I will present some recent progress in this direction from two perspectives in the context of low-rank matrix estimation. From an information-theoretic perspective, I will give an exact characterization of the limiting minimum estimation error. Our results apply to the high-dimensional regime n,d→∞ and d/n→∞ (or d/n→0) and generalize earlier works that focus on the proportional asymptotics n,d→∞, d/n→δ∈(0,∞). From an algorithmic perspective, large-dimensional matrices are often processed by iterative algorithms like power iteration and gradient descent, thus encouraging the pursuit of understanding the fundamental limits of these approaches. We introduce a class of general first order methods (GFOM), which is broad enough to include the aforementioned algorithms and many others. I will describe the asymptotic behavior of any GFOM, and provide a sharp characterization of the optimal error achieved by the GFOM class.

• Tuesday November 7, 2023 at 15:30, Temple (Wachman Hall 617)

  Relative instability and concentration of equilibria in non-gradient dynamics

  Pax Kivimae, Courant Institute, NYU
  
  A classical picture in the theory of complex high-dimensional random functions is that an exponentially large number of critical points causes the gradient dynamics of the function to become slow and "glassy", becoming trapped in local minima. In non-gradient dynamics however, another case is possible. Here, one may have an exponentially large number of equilibria, but have none that are stable, leading to an endless cycle of wandering around saddles. This is believed to occur when the strength of the non-gradient terms is brought past a certain point, a phenomenon coined by Ben Arous, Fyodorov, and Khoruzhenko as the relative-absolute instability transition, and since predicted to occur in a variety of models.
  
  We confirm such a transition occurs in the case of the asymmetric p-spin model, the first such rigorous confirmation of the existence of this transition in any model. To do so, we demonstrate concentration of the quenched complexity of stable and general equilibria around their annealed values. Our methods rely on generalizing the recent framework of Ben Arous, Bourgade, and McKenna on the Kac-Rice formula to the non-relaxational case, as well as a computation of moments of the characteristic polynomial of the elliptic ensemble.

• Tuesday November 14, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Spectral gap estimates for mixed $p$-spin models at high temperature

  Arka Adhikari, Stanford University
  
  We consider general mixed $p$-spin mean-field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the $N$-spin system to that of suitably conditioned subsystems.
  
  Based on joint work w/ C. Brennecke, C. Xu, and H.-T. Yau.

• Tuesday November 28, 2023 at 15:30, Temple (Wachman Hall 617)

  KPZ fluctuations in the planar stochastic heat equation

  Alejandro Ramírez, NYU Shanghai
   
  We consider Wick-ordered solutions to the planar stochastic heat equation, corresponding to a Skorokhod interpretation in the Duhamel integral representation of the equation. We prove that the fluctuations far from the center are given by the stochastic heat equation. This talk is based on a joint work with Jeremy Quastel and Balint Virag.

• Tuesday December 5, 2023 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Stationary measures for integrable polymers on a strip

  Zongrui Yang, Columbia University 
  
  We prove that the stationary measures for the geometric last passage percolation and log-gamma polymer models on a diagonal strip are given by the marginals of objects we call two-layer Gibbs measures. Taking an intermediate disorder limit of the log-gamma polymer stationary measure, we recover the conjectural description of the open KPZ equation stationary measure for all choices of boundary parameters. This is a joint work with Guillaume Barraquand and Ivan Corwin.

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday January 25, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Strong Quantum Unique Ergodicity and its Gaussian fluctuations for Wigner matrices

  Giorgio Cipolloni, Princeton University
   

  We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis, which is a strong form of Quantum Unique Ergodicity (QUE) with optimal speed of convergence. Then, using this a priori bound as an input, we analyze the Stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in the QUE.

  The main methods behind the above results are:

  (i) multi-resolvent local laws established via a novel bootstrap scheme;

  (ii) energy estimates for SEE.

   

• Tuesday February 1, 2022 at 15:30, Temple (Wachman Hall 617)

  On convergence of the cavity and Bolthausen’s TAP iterations to the local magnetization

  Si Tang, Lehigh University
  
  The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work, Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing algorithm, a generalization of Bolthausen’s iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen’s scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated. This is a joint work with Wei-Kuo Chen (University of Minnesota).

• Tuesday February 8, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Lozenge tilings and the Gaussian free field on a cylinder

  Marianna Russkikh, MIT
  
  We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes. 

• Tuesday February 15, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Scaling limits of the Laguerre unitary ensemble

  Xuan Wu, University of Chicago
  
   

  In this talk, we will discuss the LUE, focusing on the scaling limits. On the hard-edge side, we construct the $\alpha$-Bessel line ensemble for all $\alpha \in \mathbb{N}_0$. This novel Gibbsian line ensemble enjoys the $\alpha$-squared Bessel Gibbs property. Moreover, all $\alpha$-Bessel line ensembles can be naturally coupled together in a Bessel field, which enjoys rich integrable structures. We will also talk about work in progress on the soft-edge side, where we expect to have the Airy field as the scaling limit. This talk is based on joint works with Lucas Benigni, Pei-Ken Hung, and Greg Lawler.

   

• Tuesday February 22, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  The local environment of a geodesic in Last-Passage Percolation

  Lingfu Zhang, Princeton University
   

  In exponential Last-Passage Percolation, each vertex in the 2D lattice is assigned an i.i.d. exponential weight, and the geodesic between a pair of vertices refers to the up-right path connecting them, with the maximum total weight along the path. This model was first introduced to model fluid flow through a random medium. It is also a central model in the KPZ universality class and related to various natural processes.

  A classical question asks what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present new results on the convergence of the ‘environment’ as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure. In addition, we obtain an explicit description of the limiting ‘environment’. This in principle enables one to compute all the local statistics of the geodesic, and I will talk about some surprising and interesting examples.

  This is based on joint work with James Martin and Allan Sly.

   

• Tuesday March 8, 2022 at 15:30, Temple (Wachman Hall 617)

  Large deviation estimates for Selberg’s central limit theorem and applications

  Emma Bailey, The Graduate Center, CUNY
  
  Selberg’s celebrated central limit theorem shows that the logarithm of the zeta function at a typical point on the critical line behaves like a complex, centered Gaussian random variable with variance $\log\log T$. This talk will present recent results showing that the Gaussian decay persists in the large deviation regime, at a level on the order of the variance, improving on the best known bounds in that range.  We also present various applications, including on the maximum of the zeta function in short intervals. Whilst the results are number theoretic, the tools used are predominantly probabilistic in nature.  This work is joint with Louis-Pierre Arguin. 

• Tuesday March 15, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Metric growth dynamics in Liouville quantum gravity

  Hugo Falconet, Courant Institute, NYU

  Liouville quantum gravity (LQG) is a canonical model of random geometry. Associated with the planar Gaussian free field, this geometry with conformal symmetries was introduced in the physics literature by Polyakov in the 80’s and is conjectured to describe the scaling limit of random planar maps. In this talk, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The primary focus will be on the dynamics of the trace of the free field on the boundary of growing LQG balls.

  Based on a joint work with Julien Dubédat.

• Tuesday March 22, 2022 at 15:30, Temple (Wachman Hall 617)

  From generalized Ray-Knight theorems to functional limit theorems for some models of self-interacting random walks on integers

  Elena Kosygina, Baruch College & The Graduate Center, CUNY
  
  For several models of self-interacting random walks (SIRWs), generalized Ray-Knight theorems for edge local times are a very useful tool for studying the limiting distributions of these walks. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss recent results (joint work with Thomas Mountford, EPFL, and Jon Peterson, Purdue University) which resolve an open question posed in Toth’s paper. We show that, in the asymptotically free case, the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth), while in the polynomially self-repelling case, the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of perturbed Brownian motions. This negative result was somewhat unexpected. Conjectures on whether there is a suitable limiting process in this case and, if yes, what it might be are welcome.

• Tuesday March 29, 2022 at 15:30, Temple (Wachman Hall 617)

  Localization and delocalization in Erdős–Rényi graphs

  Johannes Alt, Courant Institute, NYU
  
  We consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of the eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized and delocalized phases and our recent progress in establishing it rigorously.
  
  This is based on joint works with Raphael Ducatez and Antti Knowles.

• Tuesday April 5, 2022 at 15:30, Temple (Wachman Hall 617)

  Uniqueness in Cauchy problems for diffusive real-valued strict local martingales

  Kasper Larsen, Rutgers University
  
  For a real-valued one-dimensional diffusive strict local martingale, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution. We exemplify our results using quadratic normal volatility models and the two-dimensional Bessel process. Joint work with Umut Cetin (LSE). 

• Tuesday April 12, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Six-vertex model and the KPZ universality class

  Amol Aggarwal, Columbia University
  
  In this talk, we explain recent results relating the six-vertex model and the Kardar-Parisi-Zhang (KPZ) universality class. In particular, we describe how the six-vertex model can be used to analyze stochastic interacting particle systems, such as asymmetric exclusion processes, and how infinite-volume pure states of the ferroelectric six-vertex model exhibit fluctuations of order $N^{1/3}$, a characteristic feature of systems in the KPZ universality class. 
   

• Tuesday April 19, 2022 at 15:30, Temple (Wachman Hall 617)

  Understanding the upper tail behaviour of the KPZ equation via the tangent method

  Milind Hegde, Columbia University
  
  The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.

• Tuesday April 26, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Multiplicative chaos of the Brownian loop soup

  Antoine Jego, MSRI
  
  On the one hand, the 2D Gaussian free field (GFF) is a log-correlated Gaussian field whose exponential defines a random measure: the multiplicative chaos associated to the GFF, often called Liouville measure. On the other hand, the Brownian loop soup is an infinite collection of loops distributed according to a Poisson point process of intensity $\theta$ times a loop measure. At criticality ($\theta = 1/2$), its occupation field is distributed like half of the GFF squared (Le Jan's isomorphism). The purpose of this talk is to understand the infinitesimal contribution of one loop to Liouville measure in the above coupling. This work is not restricted to the critical intensity and provides the natural notion of multiplicative chaos associated to the Brownian loop soup when $\theta$ is not equal to $1/2$. 
   

• Tuesday September 6, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Fractal Geometry of the KPZ equation

  Promit Ghosal, MIT
    
  The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to the KPZ universality class. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention a work in progress with Jaeyun Yi for the (2+1)-d case. 
   

• Tuesday September 13, 2022 at 15:30, Temple (Wachman Hall 617)

  Ballistic annihilation

  Matthew Junge, Baruch College
   
  In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

• Tuesday September 20, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  A central limit theorem for square ice

  Wei Wu, NYU Shanghai
  
  In the area of statistical mechanics, an important open question is to show that the height function associated with the square ice model (i.e., planar six vertex model with uniform weights), or equivalently with uniform graph homeomorphisms, converges to a continuum Gaussian free field in the scaling limit. I will review some recent results about this model, including that the single point height function, upon renormalization, converges to a Gaussian random variable.

• Tuesday September 27, 2022 at 15:30, Temple (Wachman Hall 617)

  On roots of random trigonometric polynomials and related models

  Hoi Nguyen, Ohio State University
   
  In this talk, we will discuss various basic statistics of the number of real roots of random trigonometric polynomials, as well as the minimum modulus and the nearest roots statistics to the unit circle of Kac polynomials. We will emphasize the universality aspects of all these problems.
   
  Based on joint works with Cook, Do, O. Nguyen, Yakir and Zeitouni.

• Tuesday October 4, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Yaglom-type limits for branching Brownian motion with absorption in the slightly subcritical regime

  Jiaqi Liu, Penn
  
  Branching Brownian motion is a random particle system which incorporates both the tree-like structure and the diffusion process. In this talk, we consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motion with drift, undergo dyadic fission at a constant rate, and are killed upon hitting the origin. We are interested in the asymptotic behaviors of the process conditioned on survival up to a large time t as the process approaches criticality. Results like this are called Yaglom-type results. Specifically, we will talk about the construction of the Yaglom limit law, Yaglom-type limits for the number of particles and the maximal displacement. Based on joint work with Julien Berestycki, Bastien Mallein and Jason Schweinsberg. 
   

• Tuesday October 11, 2022 at 15:30, Temple (Wachman Hall 617)

  Ergodicity and synchronization of the KPZ equation

  Chris Janjigian, Purdue University
   
  The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to have stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some results-in-progress which show that these distributions are totally ergodic and present some progress toward the conjecture that these are the only ergodic stationary distributions of the KPZ equation. The talk will discuss our coupling of Hopf-Cole solutions, which enables us to study the KPZ equation started from any measurable function valued initial condition. Through this coupling, we give a sharp characterization of when such solutions explode, show that all non-explosive functions become instantaneously continuous, and then study the problem of ergodicity on a natural topology on the space of non-explosive continuous functions (mod constants) in which the equation defines a Feller process. We show that any ergodic stationary distribution on this space is either a Brownian motion with drift or a process of a very peculiar form which will be described in the talk. 
  
  Based on joint works with Tom Alberts, Firas Rassoul-Agha, and Timo Seppäläinen.

• Tuesday October 18, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Graph limits and graph homomorphism density inequalities

  Fan Wei, Princeton University
  
  Graph limits is a recently developed powerful theory in studying large (weighted) graphs from a continuous and analytical perspective. It is particularly useful when studying subgraph homomorphism density, which is closely related to graph property testing, graph parameter estimation, and many central questions in extremal combinatorics. In this talk, we will show how the perspective of graph limits helps with graph homomorphism inequalities and how to make advances in a common theme in extremal combinatorics: when is the random construction close to optimal? We will also show some hardness results for proving general theorems in graph homomorphism density inequalities. 
   

• Tuesday October 25, 2022 at 15:30, Temple (Wachman Hall 617)

  Optimal delocalization for generalized Wigner matrices

  Lucas Benigni, Université de Montréal
  
  We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

• Tuesday November 1, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  Sandpiles

  Ahmed Bou-Rabee, Cornell University
  
  I will introduce the Abelian sandpile model and discuss its large-scale behavior in random environments and on different lattices. There are many open questions. 
   

• Tuesday November 8, 2022 at 15:30, Temple (Wachman Hall 617)

  Convergence of densities of spatial averages of the stochastic heat equation

  Şefika Kuzgun, University of Rochester
   
  Let $u$ be the solution to the one-dimensional stochastic heat equation driven by a space-time white noise with constant initial condition. The purpose of this talk is to present a recent result on the uniform convergence of the density of the normalized spatial averages of the solution $u$ on an interval $[-R,R]$, as $R$ tends to infinity, to the density of the standard normal distribution, assuming some non-degeneracy and regularity conditions on the diffusion coefficient. These results are based on the combination of Stein's method for normal approximations and Malliavin calculus techniques. This is a joint work with David Nualart.

• Tuesday November 29, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)

  (at UPenn) Non-backtracking spectra of random hypergraphs and community detection

  Yizhe Zhu, UC Irvine
   
  The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs has become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al (2015). We characterize the spectrum of the non-backtracking operator for sparse random hypergraphs and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix of the hypergraph. Based on joint work with Ludovic Stephan (EPFL). 
   

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday September 7, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

  Delocalization and quantum diffusion of random band matrices in high dimensions

  Fan Yang, UPenn
  
  We consider a Hermitian random band matrix $H$ on the $d$-dimensional lattice of linear size $L$. Its entries are independent centered complex Gaussian random variables with variances $s_{xy}$, that are negligible if $|x-y|$ exceeds the band width $W$. In dimensions eight or higher, we prove that, as long as $W > L^\epsilon$ for a small constant $\epsilon>0$, with high probability, most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Moreover, we also prove a quantum diffusion result of this model in terms of the Green's function of $H$. Joint work with Horng-Tzer Yau and Jun Yin.

• Tuesday September 14, 2021 at 15:30, Temple (Wachman Hall 617)

  Spanning clusters and subcritical connectivity in high-dimensional percolation

  Jack Hanson, City College & The Graduate Center, CUNY
  
  In their study of percolation, physicists have proposed "scaling hypotheses" relating the behavior of the model in the critical ($p = p_c$) and subcritical ($p < p_c$) regimes. We show a version of such a scaling hypothesis for the one-arm probability $\pi(n;p)$ — the probability that the open cluster of the origin has Euclidean diameter at least $n$.
  
  As a consequence of our analysis, we obtain the correct scaling of the lower tail of cluster volumes and the chemical (intrinsic) distances within clusters. We also show that the number of spanning clusters of a side-length $n$ box is tight on scale $n^{d-6}$. A new tool of our analysis is a sharp asymptotic for connectivity probabilities when paths are restricted to lie in half-spaces.

• Tuesday September 21, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

  Hamilton-Jacobi equations for statistical inference problems

  Jiaming Xia, UPenn
  
  In this talk, I will first present the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. We compare the limit with the solution to a certain Hamilton-Jacobi equation, following the recent approach by Jean-Christophe Mourrat. The motivation comes from the averaged free energy solving an approximate Hamilton-Jacobi equation. We consider two notions of solutions which are weak solutions and viscosity solutions. The two types of solutions require different treatments and each has its own advantages. At the end of this part, I will show an example of application of our results to a model with i.i.d. entries and symmetric interactions. If time permits, I will talk about the same problem but with a different model, namely, the multi-layer generalized linear model. I will mainly explain the iteration method as an important tool used in our proof. This is based on joint works with Hong-Bin Chen and J.-C. Mourrat, NYU.

• Tuesday September 28, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

  Wilson loop expectations as sums over surfaces in 2D

  Minjae Park, MIT
  
  Although lattice Yang-Mills theory on $\mathbb{Z}^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb{R}^d$ is a major open problem when $d\ge 3$. Such a theory should assign a Wilson loop expectation to each suitable collection $\mathcal{L}$ of loops in $\mathbb{R}^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal{L}$. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.

  In this talk, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is already understood. We also obtain an alternative proof of the Makeenko-Migdal equation and a version of the Gross-Taylor expansion. Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu. 
   

   

• Tuesday October 5, 2021 at 15:30, Temple (Wachman Hall 617)

  Singularities in the spectrum of random block matrices

  David Renfrew, SUNY Binghamton
  
  We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.

• Tuesday October 12, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

  The local limit theorem on nilpotent groups

  Robert Hough, Stony Brook University
  
  Alexopoulos proved local limit theorems for measures with a density and lattice measures in the general setting of groups of moderate growth. On the Heisenberg group, Breuillard's thesis obtained a local limit theorem for general measures subject to a condition on the characteristic function, and asked if this condition can be removed. I will discuss two new local limit theorems, one joint with Diaconis, that treat local limit theorems on nilpotent Lie groups driven by general measures. We prove Breuillard's conjecture and also solve a problem of Diaconis and Saloff-Coste on the mixing of the central coordinate in unipotent matrix walks modulo $p$. 

• Tuesday October 19, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

  Integrability of boundary Liouville CFT

  Guillaume Remy, Columbia University
  
  Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. Our latest result is derived using conformal welding of random surfaces, in relation with the Schramm-Loewner evolutions. We will also discuss the connection with the conformal blocks of CFT which are fundamental functions determined by conformal invariance that underlie the exact solvability of CFT. Based on joint works with Morris Ang, Promit Ghosal, Xin Sun, Yi Sun and Tunan Zhu.

• Tuesday October 26, 2021 at 15:30, Temple (Wachman Hall 617)

  Gaussian, stable, and tempered stable limiting distributions for random walks in cooling random environments

  Jonathon Peterson, Purdue University
  
  Random walks in cooling random environments are a model of random walks in dynamic random environments where the random environment is re-sampled at a fixed sequence of times (called the cooling sequence) and the environment remains constant between these re-sampling times. We study the limiting distributions of the walk in the case when distribution on environments is such that a walk in a fixed environment has an $s$-stable limiting distribution for some $s \in (1,2)$. It was previously conjectured that for cooling maps whose gaps between re-sampling times grow polynomially that the model should exhibit a phase transition from Gaussian limits to $s$-stable depending on the exponent of the polynomial growth of the re-sampling gaps. We confirm this conjecture, identifying the precise exponent at which the phase transition occurs and proving that at the critical exponent the limiting distribution is a generalized tempered $s$-stable distribution. The proofs require us to prove some previously unknown facts about one-dimensional random walks in random environments which are of independent interest. 

• Tuesday November 2, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

  Loewner chains driven by complex Brownian motion

  Joshua Pfeffer, Columbia University
  
  In my talk I will discuss Loewner chains whose driving functions are complex Brownian motions with general covariance matrices.  This extends the notion of Schramm-Loewner evolution (SLE) by allowing the driving function to be complex-valued and not just real-valued.  We show that these Loewner chains exhibit the same phases (simple, swallowing, and space-filling) as SLE, and we explicitly characterize the values of the covariance matrix corresponding to each phase.  In contrast to SLE, we show that the evolving left hulls are a.s. not generated by curves, and that they a.s. disconnect each fixed point in the plane from infinity before absorbing the point.
  
  This talk is based on a joint work with Ewain Gwynne. 

• Tuesday November 9, 2021 at 15:30, Temple (Wachman Hall 617)

  On the limiting shape of Young diagram associated with Markov random words

  Christian Houdré, Georgia Tech
  
  Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space a totally ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated RSK Young diagrams as a multidimensional Brownian functional. Since the length of the top row of the Young diagrams is also the length of the longest weakly increasing subsequences of $(X_k)_{1\le k \le n}$, the corresponding limiting law follows. We relate our results to a conjecture of Kuperberg by providing, under a cyclic condition, a spectral characterization of the Markov transition matrix precisely characterizing when the limiting shape is the spectrum of the $m \times m$ traceless GUE. For each $m \ge 4$, this characterization identifies a proper, non-trivial class of cyclic transition matrices producing such a limiting shape. However, for $m=3$, all cyclic Markov chains have such a limiting shape, a fact previously only known for $m=2$. For $m$ arbitrary, we also study reversible Markov chains and obtain a characterization of symmetric Markov chains for which the limiting shape is the spectrum of the traceless GUE. To finish, we explore, in this general setting, connections between various limiting laws and spectra of Gaussian random matrices, focusing in particular on the relationship between the terminal points of the Brownian motions, the diagonal terms of the random matrix, and the scaling of its off-diagonal terms, a scaling we conjecture to be a function of the spectrum of the covariance matrix governing the Brownian motion.
  
  Joint work with Trevis Litherland.

• Tuesday November 16, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)

  The skew Brownian permuton

  Jacopo Borga, Stanford University
  
  Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permutons. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations. The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permutons. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of skew Brownian permutons. The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. If time permits, we will present some conjectures on how it should be possible to construct these new limiting permutons directly from the Liouville quantum gravity decorated with two SLE curves. 

• Tuesday November 30, 2021 at 15:30, Temple (Wachman Hall 617)

  Central limit theorem for the characteristic polynomial of general beta-ensembles

  Krishnan Mody, Courant Institute, NYU
  
  I will discuss recent work with P. Bourgade and M. Pain in which we show that the log-characteristic polynomial for general beta ensembles converges to a log-correlated field in the large-dimension limit. The proof of this result relies on a so-called optimal local law, which I will explain and prove in the Gaussian case. I will explain how the local law is useful, and give an outline of the proof of the log-correlated field.

• Tuesday December 7, 2021 at 15:30, Temple (Wachman Hall 617)

  Large values of the Riemann zeta function in short intervals

  Louis-Pierre Arguin, Baruch College & The Graduate Center, CUNY
  
  I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function in small intervals of the critical line. This problem has interesting connections with the extreme value statistics of IID and log-correlated random variables, as well as random matrix theory.