Abstract: Fluid flows in the presence of free surfaces occur in a great many situations in nature; examples include waves on the ocean and the flow of groundwater. In this talk, I will discuss my contributions to the understanding of the systems of nonlinear partial differential equations which model such phenomena. The most important step in these results is making a suitable formulation of the problem. Influenced by the computational work of Hou, Lowengrub, and Shelley, we formulate the problems in natural, geometric variables. I will discuss my proofs (most of which are joint with Nader Masmoudi) of existence of solutions to the initial value problems for vortex sheets and water waves. I will also discuss computational results, including work with Jon Wilkening on the computation of special solutions, especially time- periodic interfacial flows

ABSTRACT: Bochner’s tube theorem states that a holomorphic function of several complex variables defined on an open tube can be extended to the convex hull of the tube. We will describe some recent CR generalizations of this result.

Title/abstract tba

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

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

• Wednesday January 31, 2024 at 14:30, Wachman 617

  Transverse surfaces to pseudo-Anosov flows

  Samuel Taylor (Temple University)
   

  Our goal will be to describe some work-in-progress (joint with Landry and Minsky) that classifies transverse surfaces to pseudo-Anosov flows on 3—manifolds. This is an introductory talk and there’ll be lots of background and examples. 

   

• Wednesday February 21, 2024 at 14:30, Wachman 617

  Quotients of free products

  Thomas Ng, Brandeis University

  Abstract: Quotients of free products are natural combinations of groups that have been exploited to study embedding problems. These groups have seen a resurgence of attention from a more geometric point of view following celebrated work of Haglund--Wise and Agol.  I will discuss a geometric model for studying quotients of free products. We will use this model to adapt ideas from Gromov's density model to this new class of quotients, their actions on CAT(0) cube complexes, and combination theorems for residual finiteness.  Results discussed will be based on ongoing work with Einstein, Krishna MS, Montee, and Steenbock.

• Friday February 23, 2024 at 15:00, Swarthmore College, Science Center room 104

  An eye towards understanding of smooth mapping class groups of 4-manifolds

  Anubhav Mukherjee, Princeton University
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: The groundbreaking research by Freedman, Kreck, Perron, and Quinn provided valuable insights into the topological mapping class group of closed simply connected 4-manifolds. However, the development of gauge theory revealed the exotic nature of the smooth mapping class group of 4-manifolds in general. While gauge theory can at times obstruct smooth isotopy between two diffeomorphisms, it falls short of offering a comprehensive understanding of the existence of diffeomorphisms that are topologically isotopic but not smoothly so. In this talk, I will elucidate some fundamental principles and delve into the origins of such exotic diffeomorphisms. This is my upcoming work joint with Slava Krushkal, Mark Powell, and Terrin Warren.
  
  In the morning background talk (at 10am), I will give an overview of mapping class groups of 4-manifolds.

• Friday February 23, 2024 at 16:30, Swarthmore College Science Center room 104

  Canonical hierarchical decompositions of free-by-cyclic groups

  Jean-Pierre Mutanguha, Princeton University and IAS
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss various dynamical properties of automorphisms that turn out to be group invariants of the corresponding free-by-cyclic groups (e.g. growth type). In particular, certain dynamical hierarchical decompositions of an automorphism determine canonical hierarchical decompositions of its mapping torus. In the intro talk, I will discuss how Bass-Serre theory (actions on simplicial trees) gives us a grip on these groups.
  
  In the morning background talk,  at 11:30am, I will introduce free-by-cyclic groups from a tree's point of view.

• Wednesday February 28, 2024 at 14:30, Wachman 617

  Dilatations of pseudo-Anosov maps and standardly embedded train tracks 

  Chi Cheuk Tsang (UQAM)

  The minimum dilatation problem asks for the minimum value of the dilatation among all pseudo-Anosov maps defined on a fixed surface. This value can be thought of as the smallest amount of mixing one can perform on the surface while still doing something topologically interesting. In this talk, we will present some recent progress on the fully-punctured version of this problem. The strategy for proving these results involves something called standardly embedded train tracks. We will explain what these are and formulate some future directions that may be tackled using this technology. This is joint work with Eriko Hironaka, and with Erwan Lanneau and Livio Liechti.

• Wednesday March 13, 2024 at 14:30, Wachman 617

  Conformal dimension for Bowditch boundaries of Coxeter groups

  Rylee Lyman
  Rutgers University, Newark
  
  Abstract: Coxeter groups, defined by a labeled simplicial graph, are a beautiful family of groups which are in a certain precise sense generated by reflections. With Elizabeth Field, Radhika Gupta and Emily Stark, we study the family of Coxeter groups whose defining graph is complete with all edges labels at least three. We show that they fall into infinitely many quasi-isometry classes. These groups were previously studied by Haulmark, Hruska and Sathaye, who showed that generically they all have visual boundary the Menger curve and posed the question of quasi-isometric classification. Along the way to our proof, we show that these groups have a geometrically finite action on a CAT(-1) space, whose geometry can be studied by reasoning about hyperbolic 3-space. I think this space and the geometric reasoning about it are really pretty — I'd like to spend most of the talk focusing on it.

• Wednesday March 20, 2024 at 14:30, Wachman 617

  Satellite actions on knot concordance

  Allison Miller, Swarthmore College

  Abstract: The collection of knots in the 3-sphere modulo a 4-dimensionally defined equivalence relation called "concordance", is not just a set but a group and a metric space as well. Via the satellite construction, every knot in a solid torus induces a self-map of the concordance set/ group/ metric space. In this talk, we'll survey what is known about these functions: When are they injective/ surjective/ bijective? When are they group homomorphisms? How do they interact with the metric space structure? We will end by discussing recent joint work of mine with two Swarthmore students, Randall Johanningsmeier and Hillary Kim, that unexpectedly provided significant progress towards answering one of these questions.

• Friday March 22, 2024 at 14:30, DRL room A4, University of Pennsylvania

  Measuring fundamental groups using cochains and Hopf invariants

  Nir Gadish, University of Michigan
  
  PATCH Seminar, join with Bryn Mawr, Haverford, Penn, and Swarthmore
  
  Abstract: The classical Hopf invariant uses the linking of two generic fibers to detect elements in higher-homotopy groups of spheres. Sinha-Walter generalized this idea and used "higher linking" to completely characterize elements in the (rational) homotopy groups of any simply connected space. By extending this setup to measure fundamental groups, we arrive at a new invariant theory for groups, which we have termed letter braiding. This is effectively a 0-dimensional linking theory for letters in words, and it realizes every finite-type invariant of any group. We will discuss the topological origins of this theory, its connection to loop spaces, and will explore an application to mapping class groups of surfaces.
  
  In the background talk (10am in room A8), I will discuss studying topological spaces using invariants of words and groups.

• Friday March 22, 2024 at 16:00, DRL room A4, University of Pennsylvania

  Interpolation method in mean curvature flow

  Ao Sun, Lehigh University

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

  Abstract: The interpolation method is a very powerful tool to construct special solutions in geometric analysis. I will present two applications in mean curvature flow: one is constructing a new genus one self-shrinking mean curvature flow, and another one is constructing immortal mean curvature flow with higher multiplicity convergence. The talk is based on joint work with Adrian Chu (UChicago) and joint work with Jingwen Chen (UPenn).

  In the morning background talk (11:30-12:30 in room A8) I will discuss singularities and solitons of mean curvature flow.

• Wednesday March 27, 2024 at 14:30, Wachman 617

  Convergence of unitary representations and spectral gaps

  Michael Magee, Durham University

  Abstract: Let $G$ be an infinite discrete group e.g. hyperbolic 3-manifold group. Finite dimensional unitary representations of $G$ in fixed dimension are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of $G$ alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps.

  The talk is a discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas.

• Tuesday April 2, 2024 at 11:00, Wachman 527

  Two-generator subgroups of free-by-cyclic groups

  Edgar Bering, San Jose State University

  Abstract:  In general, the classification of finitely generated subgroups of a given group is intractable. Restricting to two-generator subgroups in a geometric setting is an exception. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and Shalen proved that a two-generator subgroup of the fundamental group of an orientable atoroidal irreducible 3-manifold is either free, free-abelian, or finite-index. In this talk I will present recent work proving a similar classification theorem for two generator mapping-torus groups of free group endomorphisms: every two generator subgroup is either free or conjugate to a sub-mapping-torus group. As an application we obtain an analog of the Jaco-Shalen result for free-by-cyclic groups with fully irreducible atoroidal monodromy. This is joint work with Naomi Andrew and Ilya Kapovich.

• Friday April 12, 2024 at 10:00, Wachman 617

  PATCH: An introduction to algebraic K-theory

  Anna Marie Bohmann (Vanderbilt)

  Algebraic K-theory is an important invariant of rings (and other related mathematical things). In this talk, we'll give a little bit of background on the classical story of algebraic K-theory and then talk about some of the many ways it's developed in recent years.

• Friday April 12, 2024 at 11:30, Wachman 617

  PATCH: The coarse geometry of Teichm\"uller space and a new notion of complexity length

  Spencer Dowdall (Vanderbilt)

  I will review the key features of Teichmuller space that are relevant to counting lattice points for the action of the mapping class group. Specifically, after introducing Teichmuller space I will discuss the thin regions and their inherent product structure. This will lead us to subsurface projections and Rafi's distance formula and associated combinatorial description of how Teichmuller geodesics pass through thin regions of subsurfaces. I will then introduce a new notion of "complexity length" in Teichmuller space that aims to carefully account for this motion of geodesics through product regions in a way that gives better control on the multiplicative errors and lends itself to counting problems.

• Friday April 12, 2024 at 14:30, Wachman 617

  PATCH: Scissors congruence and the K-theory of covers

  Anna Marie Bohmann (Vanderbilt)

  Scissors congruence, the subject of Hilbert's Third Problem, asks for invariants of polytopes under cutting and pasting operations. One such invariant is obvious: two polytopes that are scissors congruent must have the same volume, but Dehn showed in 1901 that volume is not a complete invariant. Trying to understand these invariants leads to the notion of the scissors congruence group of polytopes, first defined the 1970s. Elegant recent work of Zakharevich allows us to view this as the zeroth level of a series of higher scissors congruence groups.

  In this talk, I'll discuss some of the classical story of scissors congruence and then describe a way to build the higher scissors congruence groups via K-theory of covers, a new framework for such constructions. We'll also see how to relate coinvariants and K-theory to produce concrete nontrivial elements in the higher scissors congruence groups. This work is joint with Gerhardt, Malkiewich, Merling and Zakharevich.

• Friday April 12, 2024 at 16:00, Wachman 617

  PATCH: Counting mapping classes by Nielsen-Thurston type

  Spencer Dowdall (Vanderbilt)

  I will discuss the growth rate of the number of elements of the mapping class group of each Nielsen-Thurston type, that is, either finite-order, reducible, or pseudo-Anosov, measured via the number of lattice points in a ball of radius $R$ in Teichm\"uller space. For the whole mapping class group of the closed genus $g$ surface, Athreya, Bufetov, Eskin, and Mirzakhani have shown this quantity is asymptotic to $e^{(6g-6)R}$ as $R$ tends to infinity. Maher has obtained the same asymptotics for those orbit points that are translates by pseudo-Anosov elements. Obtaining a count for the finite-order or reducible elements is significantly more challenging due to the fact these non-generic subsets are not perceptible to the standard dynamical techniques. I will explain a naive heuristic for why the finite-order elements should grow at the rate of $e^{(3g-3)R}$, that is, with half the exponent. While this approach presents several obstacles, our new notion of complexity length provides the tools needed to make the argument work. Time permitting, I will also explain why the reducible elements grow coarsely at the rate of $e^{(6g-7)R}$. Joint work with Howard Masur.

• Wednesday April 17, 2024 at 14:30, Wachman 617

  NonLERFness of arithmetic hyperbolic manifold groups

  Hongbin Sun, Rutgers University

  Abstract: We show that any arithmetric lattice $\Gamma<\text{Isom}_+(\mathbb{H}^n)$ with $n\geq 4$ is not LERF (locally extended residually finite), including type III lattices in dimension 7. One key ingredient in the proof is the existence of totally geodesic 3-dim submanifolds, which follows from the definition if $\Gamma$ is in type I or II, but is much harder to prove if $\Gamma$ is in type III. This is a joint work with Bogachev and Slavich.

• Wednesday September 4, 2024 at 14:30, Wachman 617

  Solving the word problem in the mapping class group in quasi-linear time

  Saul Schleimer, University of Warwick

  Abstract: Mapping class groups of surfaces are of fundamental importance in dynamics, geometric group theory, and low-dimensional topology.  The word problem for groups in general, the definition of the mapping class group, its finite generation by twists, and the solution to its word problem were all set out by Dehn [1911, 1922, 1938].  Some of this material was rediscovered by Lickorish [1960's] and then by Thurston [1970-80's] -- they gave important applications of the mapping class group to the topology and geometry of three-manifolds.  In the past fifty years, various mathematicians (including Penner, Mosher, Hamidi-Tehrani, D.Thurston, Dynnikov) have given solutions to the word problem in the mapping class group, using a variety of techniques.  All of these algorithms are quadratic-time.

  We give an algorithm requiring only \(O(n log^3(n))\) time.  We do this by combining Dynnikov's approach to curves on surfaces, M\"oller's version of the half-GCD algorithm, and a delicate error analysis in interval arithmetic.

  This is joint work with Mark Bell.

• Wednesday September 11, 2024 at 14:30, Wachman 617

  A metric boundary theory for Carnot groups

  Nate Fisher, Swarthmore College

  Abstract: In this talk, I will try to motivate the use of horofunction boundaries to study nilpotent groups. In particular, for Carnot groups, i.e., stratified nilpotent Lie groups equipped with certain left-invariant homogeneous metrics, all horofunctions are piecewise-defined using Pansu derivatives. For higher Heisenberg groups and filiform Lie groups, two families which generalize the standard 3-dimensional real Heisenberg group, we will discuss the dimensions and topologies of their horofunction boundaries. In doing so, we find that filiform Lie groups of dimension greater than or equal to 8 provide the first-known examples of Carnot groups whose horofunction boundaries are not full-dimension, i.e., of codimension 1.

• Wednesday September 25, 2024 at 14:30, Wachman 617

  The admissible curve graph is not hyperbolic

  Jacob Russell, Swarthmore College

  Abstract: The mapping class group and its subgroups are often illuminated by actions on graphs built from curves on the surface. These actions allow for a variety of questions about the group to be translated into either combinatorial or geometric information about these graphs. We will examine this approach in the case of a stabilizer of a vector field on the surface. These are subgroups that Calderon and Salter have shown are important on the algebraic geometry of Moduli space. This work also suggests that the appropriate graph for these subgroups to act on is the graph of curves with winding number zero. We show the geometry of this graph can be well understood using Masur and Minsky's subsurface projections. As a consequence, we learn that, unlike the traditional curve graph, this admissible curve graph is not hyperbolic.

• Friday October 4, 2024 at 14:30, Wachman 617

  Surface bundles and their coarse geometry

  Chris Leininger, Rice University
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
   

  Abstract: I’ll discuss surface bundles over various spaces with a focus on their monodromy representations and how this influences properties of the fundamental group. In the first talk (9:30am), I will explain Farb and Mosher’s notion of convex cocompactness in the mapping class group, its various incarnations, and its connections to the coarse hyperbolicity of surface bundles. Then I’ll describe more recent work generalizing convex cocompactness to notions of geometric finiteness, examples that illustrate hyperbolicity features, and mention several open problems. In the second talk, I’ll present a new construction of surface bundles over surfaces, providing the first examples of such bundles that are atoroidal.

  The surface bundle construction in the second talk represents recent joint work with Autumn Kent. The work on convex cocompactness is also joint with Kent, as well as with Bestvina, Bromberg, Dowdall, Russell, and Schleimer in various combinations. The work on geometric finiteness is joint with Dowdall, Durham, and Sisto.

   

• Friday October 4, 2024 at 16:00, Wachman 617

  Cosmetic surgeries, knot complements, and Chern-Simons invariants

  Tye Lidman, NC State

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

  Abstract: Dehn surgery is an important construction in low-dimensional topology which turns a knot into a new three-manifold. This is deeply tied to the study of knots through their complements. The Cosmetic Surgery Conjecture predicts two different Dehn surgeries on the same knot in the three-sphere always gives different three-manifolds. We show how tools from gauge theory can help to approach this problem, settling the conjecture for almost all knots in the three-sphere. If there is time, we will also discuss how these techniques can prove some generalizations of the knot complement theorem, which says that knots are determined by their complements. This is joint work with Ali Daemi and Mike Miller Eismeier.

  In the morning background talk (at 11:30am), we will learn about instanton Floer homology, an invariant associated to three-manifolds coming from gauge theory which has had a reawakening as a trendy subject over the past 15 years. We will discuss the formal structure and see how it can be applied to topological problems without getting into any of the technical gauge theory.

• Wednesday October 9, 2024 at 14:30, Wachman 617

  Arithmeticity and commensurability of links in thickened surfaces

  Rose Kaplan-Kelly, George Mason University

  Abstract: In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. In particular, a family of generalized alternating links which each correspond to a Euclidean or hyperbolic tiling and admit a right-angled complete hyperbolic structure on their complement. We will give a complete characterization of which right-angled tiling links are arithmetic, and which are pairwise commensurable.  This is joint work with David Futer.

• Wednesday October 16, 2024 at 14:30, Wachman 617

  Profinite properties of clean graphs of groups

  Kasia Jankiewicz, UC Santa Cruz and IAS

  Abstract: A graph of groups is algebraically clean if the edge groups embedding in vertex groups are inclusions of free factors. I will discuss some profinite properties of such groups, and applications to Artin groups. This will include joint work with Kevin Schreve.

• Wednesday October 23, 2024 at 14:30, Wachman 617

  Non-recognizing spaces for stable subgroups

  Sahana Balasubramanya (Lafayette College)

  We say an action of a group  on a space  recognizes all stable subgroups if every stable subgroup of G is quasi-isometrically embedded in the action on . The problem of constructing or identifying such spaces has been extensively studied for many groups, including mapping class groups and right angled Artin groups- these are well known examples of acylindrically hyperbolic groups. In these cases, the recognizing spaces are the largest acylindrical actions for the group. One can therefore ask the question if a largest acylindrical action of an acylindrically hyperbolic group (if it exists) is a recognizing space for stable subgroups in general. We answer this question in the negative by producing an example of a relatively hyperbolic group whose largest acylindrical action fails to recognize all stable subgroups. This is joint work with Marissa Chesser, Alice Kerr, Johanna Mangahas and Marie Trin.

• Friday November 1, 2024 at 14:00, Room DRL 2C6, 209 S 33rd St, Philadelphia, PA 19104, USA

  Geometric finiteness in the mapping class group

  Jacob Russell, Swarthmore College

  PATCH seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

  Abstract: Mosher proposed that the analogy between convex cocompactness in the isometries of hyperbolic space and the mapping class group should extend to the geometrically finite groups of isometries of the hyperbolic space. While no consensus definition of geometric finiteness in the mapping class group has emerged, there are several classes of subgroups that ought to be geometrically finite from several different points of view. We will survey these subgroups with a focus on the stabilizers of multicurves on the surface.

  In the morning background talk, at 10am in room DRL A2, I will introduce the idea of convex cocompactness, in contexts ranging from Kleinian groups to surface bundles.

• Friday November 1, 2024 at 15:45, David Rittenhouse Laboratory, 209 S 33rd St, Philadelphia, PA 19104, USA

  Symplectic annular Khovanov homology and knot symmetry

  Kristen Hendricks, Rutgers University
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Temple)
  
  Abstract: Khovanov homology is a combinatorially-defined invariant which has proved to contain a wealth of geometric information. In 2006 Seidel and Smith introduced a candidate analog of the theory in Lagrangian Floer analog cohomology, which has been shown by Abouzaid and Smith to be isomorphic to the original theory over fields of characteristic zero. The relationship between the theories is still unknown over other fields. In 2010 Seidel and Smith showed there is a spectral sequence relating the symplectic Khovanov homology of a two-periodic knot to the homology of its quotient; in contrast, in 2018 Stoffregen and Zhang used the homotopy type to show that there is a spectral sequence from the combinatorial homology of a two-periodic knot to the annular Khovanov homology of its quotient. (An alternate proof of this result was subsequently given by Borodzik, Politarczyk, and Silvero.) These results necessarily use coefficients in the field of two elements. This inspired investigations of Mak and Seidel into an annular version of symplectic Khovanov homology, which they defined over characteristic zero. In this talk we introduce a new, conceptually straightforward, formulation of symplectic annular Khovanov homology, defined over any field. Using this theory, we show how to recover the Stoffregen-Zhang spectral sequence on the symplectic side. This is joint work with Cheuk Yu Mak and Sriram Raghunath.
   

  In the morning background talk, at 11am in DRL A2, I will introduce equivariant cohomology and Lagrangian Floer cohomology.

   

• Wednesday November 6, 2024 at 14:30, Wachman 617

  Periodic points of endperiodic maps

  Ellis Buckminster (Penn)

  Endperiodic maps are a class of homeomorphisms of infinite-type surfaces whose compactified mapping tori have a natural depth-one foliation. By work of Landry-Minsky-Taylor, every atoroidal endperiodic map is homotopic to a type of map called a spun pseudo-Anosov. Spun pseudo-Anosovs share certain dynamical features with the more familiar pseudo-Anosov maps on finite-type surfaces. A theorem of Thurston states that pseudo-Anosovs minimize the number of periodic points of any given period among all maps in their homotopy class. We prove a similar result for spun pseudo-Anosovs, strengthening a result of Landry-Minsky-Taylor.

• Wednesday November 20, 2024 at 14:30, Wachman 617

  Uniform waist inequalities

  Uri Bader, University of Maryland and Weizmann Institute

  Abstract: Gromov’s waist inequality for the $n$-dimensional sphere $S^n$ is a fundamental result in geometry. It says that the maximal volume of a fiber of a (generic) map from $S^n$ to the $d$-dimensional Euclidean space is at least the $(n-d)$-dimensional volume of an equator sphere $S^{n−d}$, which is a constant times the volume of $S^n$. This constant is the "waist constant".

  A question arises: is there an infinite family of $n$-dimensional compact manifolds satisfying a uniform waist inequality, that is a similar inequality with a uniform waist constant, for a given dimension $d$?

  It is natural to consider the family of all finite covers of a given compact manifold $M$. A positive answer to this question in the case $d=1$ is provided by the Cheeger-Buser inequality, relating the waist constant  with the spectrum of the Laplacian of $M$.    In my talk I will survey gently all of the above and explain a recent solution to the case $d=2$, using a fixed point property for groups acting on $L^1$-spaces. Based on a joint work with Roman Sauer.

• Friday December 6, 2024 at 14:00, Stokes 014, Haverford College

  Local equivalence of Khovanov homology

  Robert Lipshitz, University of Oregon and IAS

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

  Abstract: The notion of local equivalence has been a key tool in recent work on concordance and homology cobordism groups. In this talk, I will describe a variant of this idea that uses a combination of even and odd Khovanov homology. The result is a group whose elements correspond to equivalence classes of certain simple algebraic structure, and which receives a homomorphism from the smooth concordance group. I will sketch some structures on this group and concrete invariants it defines, and perhaps speculate wildly about other places this strategy might be useful. This is joint work with Nathan Dunfield and Dirk Schütz.

  In the morning background talk (9:15 in room KINSC L205), I will construct (Khovanov’s) even Khovanov homology and (Ozsváth-Rasmussen-Szabó’s) odd Khovanov homology and talk about some of their similarities and differences. Most of the talk will assume just an understanding of tensor products of abelian groups / vector spaces.

• Friday December 6, 2024 at 15:30, Stokes 014, Haverford College

  Spinal open books and symplectic fillings with exotic fibers

  Luya Wang, Institute for Advanced Study

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

  Abstract: Pseudoholomorphic curves are pivotal in establishing uniqueness and finiteness results in the classification of symplectic manifolds. In a series of works, Wendl used punctured pseudoholomorphic foliations to classify symplectic fillings of contact three-manifolds supported by planar open books, turning it into a problem about monodromy factorizations. In a joint work with Hyunki Min and Agniva Roy, we build on the works of Lisi--Van Horn-Morris--Wendl in using spinal open books to further delve into the classification problem of symplectic fillings of higher genus open books. In particular, we provide the local model of the mysterious "exotic fibers" in a generalized version of Lefschetz fibrations, which captures a natural type of singularity at infinity. We will give some applications to classifying symplectic fillings via this new phenomenon.

  In the morning background talk (10:30am in room KINSC L205), I will survey some previous results on using planar open books to classify symplectic fillings. I will give some background on open books, contact structures and bordered Lefschetz fibrations. Then I will give an overview of Wendl's proof of his influential theorem: any symplectic filling of a contact 3-manifold supported by a planar open book is deformation equivalent to a bordered Lefschetz fibration.

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

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

• Wednesday January 25, 2023 at 15:30, Wachman 617

  Residually finite central extensions of lattices

  Matthew Stover (Temple)

  The preimage of PSL(2, Z) in any connected cover of PSL(2, R) is residually finite, and one can prove this very explicitly using nilpotent quotients. For n ≥ 2, Deligne famously proved using the congruence subgroup property that the central extension of Sp(2n, Z) by Z determined by its preimage in the universal cover of Sp(2n, R) is not residually finite. I will describe joint work with Domingo Toledo that develops methods, generalizing one interpretation of the argument for PSL(2, Z), to prove residual finiteness (in fact, linearity) of cyclic central extensions of fundamental groups of aspherical manifolds with residually finite fundamental group. I will then describe how this generalization applies to prove residual finiteness of cyclic central extensions of certain arithmetic lattices in PU(n, 1).

• Wednesday February 1, 2023 at 14:30, Wachman 617

  Classifying arithmetic fully augmented links

  Will Worden (Holy Family University)
  
  
  Fully augmented links (FALs) are a class of hyperbolic links having especially nice geometric and combinatorial structures. A large subclass of these links, called octahedral FALs, have complements that are arithmetic manifolds with trace field $\mathbb{Q}(\sqrt{-1})$. Apart from these, the only other known example of an arithmetic FAL is the minimally twisted 8-chain link, shown to have trace field $\mathbb{Q}(\sqrt{-2})$ by Meyer—Millichap—Trapp. We’ll discuss work joint with Neil Hoffman that shows that these are in fact the only two possible trace fields for arthmetic FALs.

• Friday February 10, 2023 at 14:30, Room 338, Park Science Building, Bryn Mawr College

  Homeomorphism groups of (weakly) self-similar 2-manifolds

  Nicholas Vlamis, CUNY Queens College
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: The study of homeomorphism groups and mapping class groups of infinite-type 2-manifolds is in its infancy, and the deeper we dive into their structure we see that the class of infinite-type surfaces cannot be studied all at once.  Recently, Mann and Rafi have introduced a very useful way to partition this class into several nice subclasses whose homeomorphism groups/mapping class groups often share many properties.  In this talk, we will discuss recent results regarding the structure of homeomorphism groups of the class of weakly self-similar 2-manifolds, and how these can be viewed as natural extensions of results regarding the homeomorphism groups of the 2-sphere, the plane, and the open annulus. 
  
  
  In the morning background talk (at 10:00am), I will give an overview of some fundamental results and tools regarding the algebraic and topological structure of homeomorphism groups of compact (two-)manifolds (with a focus on the sphere).  We will touch on work of Anderson from the 50s, Fisher from the 60s, Kirby from the 60s/70s, and Calegari--Freedman from the aughts. 

• Friday February 10, 2023 at 16:00, Room 338, Park Science Building, Bryn Mawr College

  From embedded contact homology to surface dynamics

  Jo Nelson, Rice University and IAS
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: I will discuss work in progress with Morgan Weiler on knot filtered embedded contact homology (ECH) of open book decompositions of \(S^3\) along \(T(2,q)\) torus knots to deduce information about the dynamics of symplectomorphisms of the genus \((q-1)/2\) pages which are freely isotopic to rotation by \(1/(2q)\) along the boundary. I will explain the interplay between the topology of the open book, its presentation as an orbi-bundle, and our computation of the knot filtered ECH chain complex.  I will describe how knot filtered ECH realizes the relationship between the action and linking of Reeb orbits and its application to the study of the Calabi invariant and periodic orbits of symplectomorphisms of the pages.  
  
  In the morning background talk (at 11:30 am), I will give an introduction to Floer theories and Reeb dynamics for contact manifolds. I will give some background on this subject, including motivation from classical mechanics. I will then explain how to construct Floer theoretic contact invariants, illustrated by numerous graphics.

• Friday March 3, 2023 at 14:30, Swarthmore College, Science Center 158

  A stable homotopy invariant for filled Legendrian submanifolds

  Lisa Traynor, Bryn Mawr College
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: In low-dimensional topology, cobordisms are common and rich objects of study. In symplectic and contact topology, we study topological cobordisms that satisfy additional conditions imposed by symplectic and contact geometry.  These so-called Lagrangian cobordisms between Legendrian submanifolds have proved to be quite interesting: sometimes they have “flexible” phenomena like in the topological world and other times to exhibit “rigidity” that is special to the symplectic and contact world.   

   

  For any Legendrian submanifold that admits a linear-at-infinity generating family there are invariant generating family homology groups.  Sabloff and I established that when the Legendrian submanifold can be filled by a Lagrangian submanifold in such a way that the filling has a generating family that extends the generating family for the Legendrian boundary, then the generating family homology groups of the Legendrian boundary record the topologically invariant singular homology groups of the filling.  I will explain how the generating family homology groups have a spectral lift: there is a generating family spectrum for a Legendrian submanifold whose homology groups agree with our previously defined generating family homology groups.  Moreover, for a Legendrian submanifold that can be filled with a Lagrangian as described above, the generating family spectrum of the Legendrian boundary is equivalent to the suspension spectrum of the filling.  This is joint work with Hiro Lee Tanaka.

  In the morning background talk (10:00am in room 149), I will provide some background on lagrangian cobordisms and higher homotopy theory.

   

• Friday March 3, 2023 at 16:00, Swarthmore College, Science Center 158

  Applications of group actions on the quasi-trees known as projection complexes

  Johanna Mangahas, University of Buffalo
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: Mapping class groups of surfaces have a marvelous library of actions on interesting spaces, in keeping with their central place in geometric group theory. In these talks I will highlight applications of their actions on the projection complexes defined by Bestvina, Bromberg, and Fujiwara, and how these are a special case of a more general picture.  In particular I hope to motivate results and questions growing out of joint work with Matt Clay and Dan Margalit.
  
  In the morning background talk (11:30am in room 149), I will describe some “what/why/how"s around projection complexes.

• Wednesday March 22, 2023 at 14:30, Wachman 617

  Fibered face theory for free-by-cyclic groups and the boundary of the Fried cone

  Ruth Meadow-MacLeod (Temple)

   In this talk I will explain some theorems of Dowdall, Kapovich and Leininger’s work relating to stretch factors of expanding irreducible train track maps on finite graphs with no valence 1 vertices, and the Fried cone of the resulting mapping torus. Then I will discuss the boundary of the Fried cone, and how and when it can be represented by a graph in the mapping torus. There will be plenty of pictures.

• Friday March 31, 2023 at 14:00, DRL room A8, University of Pennsylvania

  Doubles of Gluck twists

  Patrick Naylor, Princeton University
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  
  Abstract: The Gluck twist of an embedded 2-sphere in the 4-sphere is a 4-manifold that is homeomorphic, but not obviously diffeomorphic to the 4-sphere. Despite considerable study, these homotopy spheres have resisted standardization except in special cases. In this talk, I will discuss some conditions that imply the double of a Gluck twist is standard, i.e., is diffeomorphic to the 4-sphere. This is based on joint work with Dave Gabai and Hannah Schwartz. 
  
  
  In the morning background talk (9:30am in room A5), I’ll introduce some of the main ideas, along with some basic constructions of knotted 2-spheres. 

• Friday March 31, 2023 at 15:30, DRL room A8, University of Pennsylvania

  Stabilizations of Heegaard splittings and minimal surfaces

  Daniel Ketover, Rutgers University

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

  Abstract:  In the 1930s, Reidemeister and Singer showed that any two Heegaard surfaces in a three-manifold become isotopic after adding sufficiently many trivial handles. I will show how this topological result gives rise to minimal surfaces of Morse index 2 in many ambient geometries.  In particular, applied to most lens spaces we obtain genus 2 minimal surfaces.  I’ll show using this that the number of distinct genus g minimal surfaces in the round sphere tends to infinity as g does (previously the lower bound for all large genera was two).

  There will be a background talk introducing these ideas, at 11:00am in room A5.

• Wednesday April 12, 2023 at 14:30, Wachman 617

  Upper bound for distance in the pants graph

  Mehdi Yazdi, Kings College London
  
  Abstract: A pants decomposition of a compact orientable surface S is a maximal collection of disjoint non-parallel simple closed curves that cut S into pairs of pants. The pants graph of S is an infinite graph whose vertices are pants decompositions of S, and where two pants decompositions are connected by an edge if they differ by a certain move that exchanges exactly one curve in the pants decomposition. One motivation for studying this graph is a celebrated result of Brock stating that the pants graph is quasi-isometric to the Teichmuller space equipped with the Weil-Petersson metric. Given two pants decompositions, we give an upper bound for their distance in the pants graph as a polynomial function of the Euler characteristic of S and the logarithm of their intersection number. The proof relies on using pre-triangulations, train tracks, and a robust algorithm of Agol, Hass, and Thurston. This is joint work with Marc Lackenby.

• Wednesday April 19, 2023 at 14:30, Wachman 617

  Short curves of end-periodic mapping tori

  Brandis Whitfield (Temple)

  Abstract: Let \(S\) be an infinite-type surface with finitely many ends, all accumulated by genus, and consider an end-periodic homeomorphism \(f\) of \(S\). The end-periodicity of \(f\) ensures that \(M_f\), its associated mapping torus, has a compactification as a \(3\)-manifold with boundary; and further, if \(f\) is atoroidal, then \(M_f\) admits a hyperbolic metric. In ongoing work, we show that given a subsurface \(Y \subset S\), the subsurface projections between a pair of ``positive" and ``negative" \(f\)-invariant multicurves provide bounds for the geodesic length of the boundary of \(Y\) as it resides in \(M_f\).

  In this talk, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic \(3\)-manifolds, and how these techniques may be used in the infinite-type setting.

• Wednesday April 26, 2023 at 14:30, Wachman 617

  Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold

  Fernando Al Assal (Yale)

  Let M be a closed hyperbolic 3-manifold and let Gr(M) be its 2-plane Grassmann bundle. We will discuss the following result: the weak-* limits of the probability area measures on Gr(M) of pleated or minimal closed connected essential K-quasifuchsian surfaces as K goes to 1 are all convex combinations of the probability area measures of the immersed closed totally geodesic surfaces of M and the probability volume (Haar) measure of Gr(M).

• Wednesday September 6, 2023 at 14:30, Wachman 617

  Character varieties, Azumaya algebras, and once-punctured tori

  Yi Wang, University of Pennsylvania

  Abstract: The $SL_2(\mathbb{C})$ character variety is an important tool in studying low-dimensional manifolds. In particular, Culler-Shalen theory connects ideal points of the projectivization of the character variety to essential surfaces in hyperbolic 3-manifolds. Results of Tillmann, Paoluzzi-Porti, and others have related the algebra at these ideal points to the topology of these essential surfaces. In this talk, we will show that certain families of essential once-punctured tori in hyperbolic 3-manifolds are detected by ideal points in character varieties, and discuss how all of this work relates to refining arithmetic invariants of Chinburg-Reid-Stover.

• Wednesday September 13, 2023 at 14:30, Wachman 617

  Faces of the Thurston norm ball dynamically represented by multiple distinct flows

  Anna Parlak (UC Davis)
  
  
  A pseudo-Anosov flow on a closed 3-manifold dynamically represents a face F of the Thurston norm ball if the cone on F is dual to the cone spanned by homology classes of closed orbits of the flow. Fried showed that for every fibered face of the Thurston norm ball there is a unique, up to isotopy and reparameterization, flow which dynamically represents the face.  Mosher found sufficient conditions on a non-circular flow to dynamically represent a non-fibered face, but the problem of the existence and uniqueness of the flow for every non-fibered face was unresolved.

  I will outline how to show that a non-fibered face can be in fact dynamically represented by multiple topologically inequivalent flows, and discuss how two distinct flows representing the same face may be related. 

   

• Friday September 29, 2023 at 14:30, Park Science Center 336, Bryn Mawr College

  A stack of broken lines: Like BG, but for Morse Theory

  Hiro Lee Tanaka, Texas State University
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Temple)
  
  
   

  Abstract: A $G$--bundle over $X$ is a family of copies of $G$, with one copy for every element of $X$. Families like this arise when studying nice functions on manifolds (i.e., in Morse theory) -- where instead of families of groups, families of broken lines live on moduli of gradient trajectories. And just like $G$--bundles are classified by an object called $BG$, it turns out you can write down the object that classifies families of broken lines -- this object is the stack of broken lines. The amazing fact is that this (geometric) object has an incredibly deep connection to the (algebraic) idea of associativity, and I'll try to explain why this is true. If time allows (which it might not) I'll try to explain why this object is expected to play a central role in enriching Morse theory and various Floer theories over stable homotopy theory. This is joint work with Jacob Lurie.

  In the morning background talk (10am in room 278), I will discuss some needed background for the afternoon.

   

• Friday September 29, 2023 at 16:00, Park Science Center 336, Bryn Mawr College

  Toward a dynamical theory of Thurston's norm

  Michael Landry, Saint Louis University
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
   

  Background talk (11:30am in room 278): Objects associated with 3-manifolds fibering over the circle

  Abstract: A fundamental example in low-dimensional topology is a closed oriented 3-manifold fibering over the circle. Thurston's study of this example led to the celebrated Nielsen-Thurston classification of surface homeomorphisms and the Thurston norm on homology. I will introduce these concepts before further developing some of the rich structure present in the example, touching on flows, foliations, and homeomorphisms of surfaces with infinitely generated fundamental group. I will mention joint work with Minsky and Taylor that fits into the story.

   

  Research talk (4:00pm in room 336): Toward a dynamical theory of Thurston's norm

  Abstract: One might hope to generalize the picture described in the previous talk to the setting of 3-manifolds that do not necessarily fiber over the circle. I will give some of the history of this endeavor, mentioning three conjectures of Mosher from the 1990s. Then I will describe joint work with Tsang that aims to make progress on these conjectures using modern objects called veering branched surfaces.

   

• Wednesday October 11, 2023 at 14:30, Wachman 617

  Large volume fibered knots in 3-manifolds

  Rob Oakley, Temple University

  Abstract: Let $M$ be a closed, connected, oriented, 3-manifold. Alexander proved that every such $M$ contains a fibered link. In this talk I will describe work that uses this idea to show that for hyperbolic fibered knots in $M$, the volume and genus are unrelated. I will also discuss a connection to a question of Hirose, Kalfagianni, and Kin about volumes of hyperbolic fibered 3-manifolds that are double branched covers.

• Wednesday October 25, 2023 at 14:30, Wachman 617

  Counting fixed points of pseudo-Anosov maps

  Dave Futer, Temple University

  Abstract: Let $S$ be a hyperbolic surface and $f$ a pseudo-Anosov map on $S$. I will describe a result that predicts the number of fixed points of $f$, up to constants that depend only on the surface $S$. If $f$ satisfies a mild condition called “strongly irreducible,” then the logarithm of the number of fixed points of $f$ is coarsely equal to its translation length on the Teichmuller space of $S$. Without this mild condition, there is still a coarse formula.

  This result and its proof has some applications to the search for surface subgroups of mapping class groups, and relations between the hyperbolic volume and the knot Floer invariants of fibered hyperbolic knots. This is joint work with Tarik Aougab and Sam Taylor.

• Friday November 3, 2023 at 09:30, Wachman 617

  Using finite groups to approximate infinite groups

  Tam Cheetham-West, Yale University

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

  Abstract: Some infinite groups, like the group of integers with addition, have lots of finite quotients. What can we use these finite quotients to do? What do these collections of finite quotients remember about the groups that produce them? This is the background talk for the research lecture at 2:30pm.

• Friday November 3, 2023 at 11:30, Wachman 617

  Fibered knots: what, why and how

  Siddhi Krishna, Columbia University

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

  Abstract: Fibered knots show up all over low-dimensional topology, as they provide a robust way to investigate interactions between phenomena of different dimensions. In this talk, I'll survey what they are, why you should care, and how to identify them. Then, as time permits, I'll also sketch a proof that positive braid knots are fibered. I will assume very little background for this talk -- all are welcome!

• Friday November 3, 2023 at 14:30, Wachman 617

  Finite quotients of fibered, hyperbolic 3-manifold groups

  Tam Cheetham-West, Yale University

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

  Abstract: The finite quotients of the fundamental group of a 3-manifold are the deck groups of its finite regular covers. We often pass to these finite-sheeted covers for different reasons, and these deck groups are organized into a topological group called the profinite completion of a 3-manifold group. In this talk, we will discuss how to leverage certain properties of mapping class groups of finite-type surfaces to study the profinite completions of the fundamental groups of fibered hyperbolic 3-manifolds of finite volume.

• Friday November 3, 2023 at 16:00, Wachman 617

  Braid positivity, taut foliations, and unknot detection

  Siddhi Krishna, Columbia University

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

  Abstract: The L-space conjecture predicts that three seemingly different ways to measure the "size" of a 3-manifold are equivalent. In particular, it predicts that a manifold with the "extra" geometric structure of a taut foliation also has "extra" Heegaard Floer homology. In this talk, I'll discuss the motivation for this conjecture, and describe some new results which produce taut foliations by leveraging special properties of positive braid knots. Along the way, we will produce some novel obstructions to braid positivity. I will not assume any background knowledge in Floer or foliation theories; all are welcome!

• Wednesday November 8, 2023 at 14:30, Wachman 617

  Currents with corners

  Tarik Aougab, Haverford College

  Abstract: We introduce the notion of a geodesic current with corners, a generalization of a geodesic current in which there are singularities (the “corners”) at which invariance under the geodesic flow can be violated. Recall that the set of closed geodesics is, in the appropriate sense, dense in the space of geodesic currents; the motivation behind currents with corners is to construct a space in which graphs on S play the role of closed curves. Another fruitful perspective is that geodesic currents reside “at infinity” in the space of currents with corners, in the sense that their (non-existent) corners have been pushed out to infinity. As an application, we count (weighted) triangulations in a mapping class group orbit with respect to (weighted) length, and we obtain asymptotics that parallel results of Mirzakhani, Erlandsson-Souto, and Rafi-Souto for curves. This represents joint work with Jayadev Athreya.

• Friday December 8, 2023 at 14:30, DRL room A8, University of Pennsylvania

  Scissors congruences and algebraic K-theory

  Cary Malkiewich, Binghamton University
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: Scissors congruence is the study of polytopes, up to relations that cut into pieces and rearrange the pieces to form a new polytope. One fundamental goal is to give effective invariants that determine when two polytopes can be related in this way. This has been done for low- dimensional geometries, but is open in dimensions 5 and greater.
   

  In the first talk (at 10:00am), we'll describe classical work on this problem, including Hilbert's Third Problem and its soluFon. This can be phrased as the computaFon of a certain abelian group, the 0th scissors congruence group. We'll then introduce the higher scissors congruence groups, defined by Zakharevich using algebraic K-theory.

  In the second talk (at 2:30pm), we'll describe recent results on higher scissors congruence groups. The main result is an analogue of the Madsen-Weiss theorem that computed the stable cohomology of mapping class groups. In this context, it gives us the higher scissors congruence groups for all one-dimensional geometries. We also explain ongoing work that simplifies the definiFon of the higher scissors congruence groups, relaFng our calculaFons back to the homology of the group of interval exchange transformaFons.

  Much of this is based on joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich, and separately with Alexander Kupers, Ezekiel Lemann, Jeremy Miller, and Robin Sroka.

   

• Friday December 8, 2023 at 16:00, DRL room A8, University of Pennsylvania

  Composing and decomposing surfaces and functions in $R^n$ and $H^n$

  Robert Young, Courant Institute
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: In these two talks, we'll discuss the role of complexity in geometry and topology - how to build complicated objects, how to break them up into simple pieces, and how to use these decompositions to study problems in geometric measure theory and metric geometry.
   

  In the first talk (11:30am), we'll consider surfaces in $\mathbb{R}^n$ , discuss how to quantify the nonorientability of a surface, and explain how this relates to a paradoxical example of L. C. Young.

  In the second talk (4:00pm), we'll consider surfaces in the Heisenberg group, the simplest example of a noncommutative nilpotent Lie group. We'll explore how that noncommutativity affects its geometry, how good embeddings of $\mathbb{H}$ must be bumpy at many scales, and how to study embeddings into $L_1$ by studying surfaces in $\mathbb{H}$.

   

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

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

• Wednesday February 23, 2022 at 14:30, Wachman 617

  Canonical forms for free group automorphisms.

  Jean Pierre Mutanguha , IAS 
   
  The Nielsen-Thurston theory of surface homeomorphism can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar decomposition for free group automorphisms. (Un)Fortunately, free group automorphismscan have arbitrarily complicated behaviour. This forms a significant barrier to translating specific arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

• Wednesday March 9, 2022 at 14:30, Wachman 617

  Effective mapping class group dynamics

  Francisco Arana-Herrera, IAS 
   
  Motivated by counting problems for closed geodesics on hyperbolic surfaces, I will present a family of new results describing the dynamics of mapping class groups on Teichmuller spaces and spaces of closed curves of closed surfaces.
   

• Wednesday March 16, 2022 at 14:30, Wachman 617

  Random Groups Acting on CAT(0) Cube Complexes

  MurphyKate Montee, Carleton College 
  
  Random groups are one way to study "typical" behavior of groups. In the Gromov density model, we often find a threshold density above which a property is satisfied with probability 1, and below which it is satisfied with probability 0. Two properties of random groups that have been well studied are cubulation (or more generally, acting cocompactly on a CAT(0) cube complex without global fixed point) and Property (T). In this setting these are mutually exclusive properties, but the threshold densities are not known. In this talk I'll present a method to show that random groups with density less than 3/14 act on a CAT(0) cube complex, and discuss how this might be extended to densities up to 1/4. This extends results of Ollivier-Wise and Mackay-Przytycki at densities less than 1/5 and 5/24, respectively.

• Wednesday March 23, 2022 at 14:30, Wachman 617

  Pseudo-Anosov flows, foliations, and left-orders

  Jonathan Zhung, Princeton University 
   
  Given a codimension 1 foliation on a 3-manifold, we say that its leaf space has branching if the lift of the foliation to the universal  cover fails to be a product foliation. I'll talk about some examples  of this phenomenon, and explain how a detailed understanding of the branching of some foliations can help us produce left-orderings of  fundamental groups.

• Wednesday March 30, 2022 at 14:30, Wachman Hall 617

  Geometric Combinatorics of Complex Polynomials

  Michael Dougherty, Swarthmore College 
   
  There are two commonly-used presentations for the braid group. In Artin's original presentation, we linearly order the n strands and use n-1 half-twists between adjacent strands to generate the group. The dual presentation, defined by Birman, Ko, and Lee in 1998, introduces additional symmetry by using the larger generating set of all half-twists between any pair of strands. Each presentation has an associated cell complex which is a classifying space for the braid group: the Salvetti complex for the standard presentation and the dual braid complex for the dual presentation. In this talk, I will present a combinatorial perspective for complex polynomials which comes from the dual presentation and describe how this leads to a cell structure for the spaceof complex polynomials which arises from the dual braid complex. This is joint work with Jon McCammond.
   
   

• Friday April 1, 2022 at 14:30, Wachman 617

  Anosov flows, foliated planes, and ideal circles

  Kathryn Mann, Cornell University
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
  
  
  Abstract: 
  From an Anosov flow on a 3-manifold, one can extract an action of the fundamental group of the manifold on a plane preserving a pair of transverse foliations, and on a compactification of the plane by an ideal circle. My talks will give an introduction to this picture and show a recent application, joint with Thomas Barthelme and Steven Frankel on the classification problem for Anosov flows. By proving rigidity results about group actions on planes and circles, we show that transitive (pseudo-)Anosov flows are determined (up to orbit equivalence) by the algebraic data of the set of free homotopy classes of closed orbits.
  
  In the morning background talk (at 9:30am), I will give an introduction to basic examples and structure theory of Anosov flows on 3-manifolds, focusing on the relationship between the geometry and topology of a manifold and the possible examples of flows it admits.  

• Friday April 1, 2022 at 16:00, Wachman 617

  Manifold topology via isovariant homotopy theory

  Inbar Klang, Columbia University
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
  
  Abstract: Homotopy theory has proven to be a robust tool for studying non-homotopical questions about manifolds; for example, surgery theory addresses manifold classification questions using homotopy theory. In joint work with Sarah Yeakel, we are developing a program to study manifold topology via isovariant homotopy theory. I'll explain what isovariant homotopy theory is and how it relates to the study of manifolds via their configuration spaces, and talk about an application to fixed point theory.
  
  In the morning background talk (at 11:00am), I will talk about configuration spaces and about homotopical fixed point theory. These are two relevant examples of how homotopy theory is used in manifold topology.

• Monday April 4, 2022 at 15:30, Wachman Hall 617

  Ph.D. Thesis Examination of Khanh Le

  
  
  Thesis: Left-orderability of Dehn surgeries on knot complements
   

  Date: Monday, April 4, 2022
   

  Time: 3:30 p.m. - 4:30 p.m.

  Place: 617 Wachman Hall, 6th floor

   

   All faculty are invited to attend the exam, ask questions and participate in the follow up discussions. 

   

• Wednesday April 6, 2022 at 14:30,

  Counting graphs on hyperbolic surfaces

  Tarik Aougab, Haverford College 
  
  Mirzakhani's beautiful work allows one to count closed geodesics in a specified orbit of the mapping class group on a hyperbolic surface. Later work of Erlandsson-Souto and Rafi-Soutoreproves these counting results while avoiding some of the most difficult aspects of Mirzakhani's proofs by recasting the problem as a convergence statement for a certain family of measures on the space of geodesic currents.We will follow this approach to count harmonic graphs, graphs that arise as the image of a harmonic map from a weighted graph into a hyperbolic surface. To do this, we define currents with corners, ageneralization of a geodesic current that allows for singularities which we think of as corresponding to the vertices of a graph. This represents joint work with Jayadev Athreya and Ryokichi Tanaka.

• Wednesday April 20, 2022 at 14:30,

  Leighton's Theorem, Kneser Complexes, and Quasi-isometric rigidity

  Daniel Woodhouse

  A spectre is haunting Geometric Group Theory -- the spectre of a generalized Leighton's Graph Covering Theorem. The original theorem states that any two graphs with common universal cover have a common finite cover. Haglund conjectured that this should generalize to all compact special cube complexes. I will talk about recent progress on this, my own contributions alongside others. I will discuss the implications for quasi-isometric rigidity, and for hyperbolic groups in particular. I will give some conjectures and explain why they should be true and very loosely how (other people) will likely one day prove them.

• Friday April 22, 2022 at 14:00, DRL room A2, at Penn

  Isotopy vs. homotopy for disks with a common dual

  Hannah Schwartz, Princeton University

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

  Abstract: Recent work of both Gabai and Schneiderman-Teichner on the smooth isotopy of homotopic surfaces with a common dual has reinvigorated the study of concordance invariants defined by Freedman and Quinn in the 90's, along with homotopy theoretic isotopy invariants of Dax from the 70's. We will outline, give context to, and discuss techniques used to prove these so called "light bulb theorems", and present new light bulb theorems for disks rather than spheres.

  At 9:30am, there will be a background talk on picture-based geometric interpretations of the Freedman-Quinn and Dax invariants.

• Friday April 22, 2022 at 15:30, DRL room A2, at Penn

  Peripheral birationality for 3-dimensional convex co-compact PSL(2, C) varieties

  Franco Vargas Pallete, Yale University

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

  Abstract: It is a consequence of a well-known result of Ahlfors and Bers that the \(PSL(2, C)\) character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding \(PSL(2, C)\) character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.

  At 11:00am, there will be a background talk on volume in hyperbolic geometry, including volume rigidity and the Bonahon-Schlafli formula.

• Wednesday May 4, 2022 at 14:30, Wachman Hall 617

  The guessing geodesics lemma

  Anschel Schaffer-Cohen, University of Pennsylvania 
  
  When proving that geodesic triangles in a given space are delta-thin, the hardest step is often simply defining the geodesics. The guessing geodesics lemma allows us to skip that step entirely, by replacing true geodesics with paths that are "good enough". In this expository talk, I'll give a proof of this lemma and demonstrate its use in a very elegant proof about the curve complex of a surface.

• Wednesday September 7, 2022 at 14:30, Wachman 617

  Model Theory As Applied to Geometric Group Theory

  Paul Rapoport, Temple University
  
  Abstract: 
  We start by recapitulating aspects of model theory in order to explain the concept of a transfer principle, motivating the idea by applying it to \mathbb{C} and cofinite collections of \overline{\mathbb{F}_p}. We explain certain "data structures" from the speaker's preprint making use of these ideas to bring tools from model theory to bear on problems more relevant to geometric group theory, using \Sigma_{G, n} as a motivating example, and finish up by showing the use of these techniques in context.
  
  
   

• Wednesday September 14, 2022 at 14:30, Wachman 617

  Invariant random subgroups and their growth rates

  Ilya Gekhtman, Technion
  
  Abstract: 
  Invariant random subgroups (IRS) are conjugation invariant probability measures on the space of subgroups in a given group G. They arise as point stabilizers of probability measure preserving actions. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices lattices. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting. A more general notion is a stationary random subgroup (SRS) where the measure on the space of subgroups is no longer required to be conjugation invariant, but only stationary with respect to some random walk. SRS are useful in studying IRS which are in themselves useful for studying lattices.
  
  Jointly with Arie Levit, we prove such a result: the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 symmetric space, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. We prove a related bound for SRS, with "half" replaced by entropy divided by drift of the random walk.

• Friday September 23, 2022 at 14:30, Wachman 617

  Unknotting via null-homologous twists and multi-twists

  Samantha Allen (Duquesne University)
  
  PATCH Seminar, joint with Bryn Mawr, Haverford, Swarthmore, and Penn
  
  Abstract: The untwisting number of a knot K is the minimum number of null-homologous full twists required to unknot K. The surgery description number of K can be defined similarly, allowing for multiple full twists in a single twisting region. We can find no examples of knots in the literature where these two invariants are not equal. In this talk, I will provide the first known examples where untwisting number and surgery description number are not equal and discuss challenges to distinguishing these invariants in general.  While these invariants are defined geometrically, our methods will involve several “algebraic” versions of unknotting operations.  In addition, we show the surprising result that the untwisting number of a knot is at most three times its surgery description number.  This work is joint with Kenan Ince, Seungwon Kim, Benjamin Ruppik, and Hannah Turner.
  
  In the morning background talk (at 9:30am), I will discuss the definitions of many different knot invariants, the linking pairing on the homology of the double branched cover of S^3 branched over a knot, and, if time permits, some basics of Kirby calculus that will be useful.

• Friday September 23, 2022 at 16:00, Wachman 617

  Nielsen Realization for 3-manifolds

  Bena Tshishiku (Brown University)
  
  PATCH Seminar, joint with Bryn Mawr, Haverford, Swarthmore, and Penn
  
  Abstract: Given a manifold M, the Nielsen realization problem asks when a subgroup of the mapping class group Mod(M) can be lifted to the diffeomorphism group Diff(M) under the natural projection Diff(M) → Mod(M). In this talk we consider the Nielsen realization problem for 3-manifolds and give a solution for subgroups of Mod(M) generated by sphere twists. This is joint work with Lei Chen. 
  
  In the morning background talk, at 11:30, I will introduce the Nielsen realization problem for group actions on manifolds and explain some of its connections to geometry, topology, and dynamics. 

• Wednesday October 12, 2022 at 14:30, Wachman 617

  Relative Cubulation of Small Cancellation Quotients

  Eduard Einstein (Swarthmore)
  
  
  Daniel Groves and I introduced relatively geometric actions, a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex. Building on results of Martin and Steenbock for properly and cocompactly cubulated groups, Thomas Ng and I proved that C’(1/6)--small cancellation free products of relatively cubulable groups are relatively cubulable. The flexibility of relatively geometric actions allowed us to prove that C’(1/6)--small cancellation free products of residually finite groups are residually finite – without any need to assume that the factors are cubulable. In this talk, I will discuss techniques used to produce relatively geometric cubulations, applications to small cancellation quotients and  potential future applications to random groups and small cancellation quotients of relatively hyperbolic groups. 
   

• Wednesday October 19, 2022 at 14:30, Wachman 617

  Infinitely many virtual geometric triangulations

  Dave Futer (Temple)
  
  
  Since the pioneering work of Thurston, it has been believed that every cusped hyperbolic 3-manifold should admit a decomposition into a union of positively oriented ideal tetrahedra. Somewhat shockingly, the question of whether such a geometric triangulation exists is still open today. Luo, Schleimer, and Tillmann proved that geometric ideal triangulations of this sort exist in some finite cover of every cusped 3-manifold. We extend their result by showing that every cusped hyperbolic 3-manifold has a finite cover admitting an infinite trivalent tree of geometric ideal triangulations. Furthermore, every sufficiently long Dehn filling of this cover also admits infinitely many geometric ideal triangulations.
  
  
  The proof involves a mixture of geometric constructions and subgroup separability tools. One of the separability tools is a new theorem about separating a peripheral subgroup from every conjugate of a coset. I will try to give a glimpse into both the geometry and the separability. This is joint work with Emily Hamilton and Neil Hoffman.

• Friday October 21, 2022 at 14:30, KINSC room Hilles 011, Haverford College

  Fibered knots and train track maps

  Braeden Reinoso, Boston College
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract:  Knot Floer homology is a knot invariant which comes in the form of a graded vector space. The algebra of this vector space determines much of the geometry of the knot, including its genus and whether it is fibered. For fibered knots, knot Floer homology also provides a great deal of geometric information about the monodromy of the fibration. The goal of this talk will be to describe some applications of this theory to knot detection problems. The strategy relies heavily on the theory of train tracks for surface automorphisms, which I will describe in the intro talk (at 9:15am).
   

• Friday October 21, 2022 at 16:00, KINSC room Hilles 011, Haverford College

  Geometric combinatorics of complex polynomials

  Michael Dougherty, Lafayette College
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract:  There are two commonly-used presentations for the braid group. In Artin's original presentation, we linearly order the n strands and use $n-1$ half-twists between adjacent strands to generate the group. The dual presentation, defined by Birman, Ko, and Lee in 1998, introduces additional symmetry by using the larger generating set of all half-twists between any pair of strands. Each presentation has an associated cell complex which is a classifying space for the braid group: the Salvetti complex for the standard presentation and the dual braid complex for the dual presentation. In this talk, I will present a combinatorial perspective for complex polynomials which comes from the dual presentation and describe how this leads to a new cell structure for the space of complex polynomials with distinct roots. This is joint work with Jon McCammond.
  
  
  
  In the introductory talk (at 11:15am), I will discuss braids, partitions, and configuration spaces.

• Wednesday October 26, 2022 at 14:30, Wachman 617

  Normal generation vs small translation length

  Harry Baik, KAIST
  
  Abstract: I will talk about mapping class group action on the curve complex with the following  conjecture in mind: an element with small stable translation length is a normal generator. This conjecture is motivated by a similar statement in the case of the action on the Teichmüller space proved by Lanier-Margalit. I will discuss various partial results (based on joint works with various subsets of {Dongryul Kim, Hyunshik Shin, Philippe Tranchida, Chenxi Wu}). 
   

• Wednesday November 2, 2022 at 14:30, Wachman 617

  Asymmetry of a typical outer automorphism

  Inyeok Choi, KAIST
  
  Abstract: The outer automorphism group Out(F_N) and the mapping class group share numerous analogies, yet they also have some striking contrasts. Among them is the innate asymmetry of some fully irreducible automorphisms. A decade ago, Handel and Mosher showed that the forward and the backward expansion factors of a parageometric fully irreducible automorphism are distinct. This mismatch is related to the asymmetry of the Lipschitz metric on Outer space. In this talk, we will observe that this phenomenon is generic in rank at least 3. More precisely, I'll explain why a typical outer automorphism arising from a random walk on Out(F_N) is a fully irreducible automorphism with distinct forward and backward expansion factors.

• Wednesday November 9, 2022 at 14:30, Wachman 617

  Gaps Between Slopes of Saddle Connections on the 2n-gon

  Taylor McAdam, Yale
  
  Abstract:  Motivated by questions from number theory, we consider the set of slopes of saddle connections on a fixed translation surface $(X,\omega)$ of length at most $N$. How do the slopes distribute as $N$ tends toward infinity? By a result of Veech we know that the directions of saddle connections equidistribute as $N$ goes to infinity, which suggests that they appear quite “random” or uniformly distributed. However, we may consider finer notions of randomness, such as the distribution of the sizes of (renormalized) gaps between slopes as $N$ tends toward infinity. Athreya-Cheung show that finding this gap distribution for a translation surface can be translated into a problem of computing the return times of the horocycle flow to an appropriate transversal on \(\rm{SL}_2(R)/\rm{SL}(X,\omega)\), where $\rm{SL}(X,\omega)$ is the Veech group of $(X,\omega)$. 
  
  In contrast to the distribution of slopes, the distribution of slope \emph{gaps} of a translation surface appears to be highly non-random, and for any Veech surface (i.e. translation surface with a lattice Veech group) the distribution has no support at 0, quadratic tail, and is continuous and piecewise real-analytic with finitely many points of non-analyticity (Athreya-Chaika, Athreya-Chaika-Lelievre, Uyanik-Work, Kumanduri-Sanchez-Wang). Work of Uyanik-Work and Kumanduri-Sanchez-Wang provides a practical method of computing such distributions, but this has been carried out in only a small number of cases, and many questions about the general behavior of slope gap distributions remain open. In this talk, we will discuss some of the motivation for this area of study, as well as the method of turning the problem of slope gap distributions into a problem in dynamics. Finally, we will discuss the slope gap distributions for the family of translation surfaces given by regular $2n$-gons with opposite sides glued. We provide linear upper- and lower-bounds on the number of points of non-analyticity in terms of $n$, providing the first example of a family of slope gap distributions with unbounded number of points of non-analyticity.

• Friday November 11, 2022 at 13:00, DRL room A4, University of Pennsylvania

  Steklov eigenspaces and free boundary minimal surfaces

  Rob Kusner, UMass
  
  PATCH seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: The coordinate functions on a free boundary minimal surface (FBMS) in the unit
  ball \(B^n\) are Steklov eigenfunctions with eigenvalue 1. For many embedded FBMS in \(B^3\) we show its first Steklov eigenspace  coincides with the span of its coordinate functions, affirming a conjecture of Fraser & Li in an even stronger form.  One corollary is a partial resolution of the Fraser-Schoen conjecture: the critical catenoid is the unique embedded FBM annulus in \(B^3\) with antipodal symmetry.  This is joint work with Peter McGrath.

• Friday November 11, 2022 at 15:00, DRL room A4, University of Pennsylvania

  Khovanov homology and exotic surfaces in the 4-ball

  Isaac Sundberg, Max Planck Institute of Mathematics
  
  PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)
  
  Abstract: In this talk, we discuss recently developed techniques from Khovanov homology used to exhibit pairs of exotic Seifert surfaces in the 4-ball, as well as potential applications of these techniques toward producing an infinite family of exotic slice disks bounding a common knot. This is joint work with Kyle Hayden, Seungwon Kim, Maggie Miller, and JungHwan Park.

• Wednesday November 16, 2022 at 14:30, Wachman 617

  ``Big Out(F_n)" and its Coarse Geometry

  Hanna Hoganson, Maryland
  
  Recently, Algom-Kfir and Bestvina introduced mapping class groups of locally finite graphs as a proposed analog of Out(F_n) in the infinite-type setting. In this talk we will introduce the classification of infinite-type graphs, their mapping class groups, and some important types of elements in these groups.  Using a framework established by Rosendal, we will then discuss the coarse geometry of the pure mapping class groups and related properties, including results on asymptotic dimension. This is joint work with George Domat and Sanghoon Kwak.

• Wednesday November 30, 2022 at 14:30, Wachman 617

  Hidden symmetries and Dehn fillings of all but one cusp of an infinite family of tetrahedral links 

  Priyadip Mondal, Rutgers
  
  
  A hidden symmetry of a finite volume hyperbolic 3-manifold M is an isometry between two of its finite covers which is not a lift of a self-isometry of M. This talk will center around the study of hidden symmetries for hyperbolic knot complements guided by the following question of Neumann and Reid: Is there a hyperbolic knot except for the figure eight knot and the two dodecahedral knots of Aitchison and Rubinstein, whose complement has a hidden symmetry? 
  
  Hyperbolic knot complements that we will consider in our talk originate from hyperbolic link complements that are tetrahedral manifolds, i.e., they have a decomposition into regular ideal tetrahedra. The Fominykh-Garoufalidis-Goerner-Tarkaev-Vesnin census provides abundant examples of tetrahedral manifolds. In this talk, we will concentrate on investigating the existence of hidden symmetries in hyperbolic knot complements obtained from Dehn filling all but one cusp of the members of an infinite family of tetrahedral link complements, all of which cyclically cover the complement of a single tetrahedral link from this census. 

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

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

• Friday January 29, 2021 at 10:40, over Zoom

  Isospectral hyperbolic surfaces of infinite genus

  Federica Fanoni, University of Paris

  Abstract: Two hyperbolic surfaces are said to be (length) isospectral if they have the same collection of lengths of primitive closed geodesics, counted with multiplicity (i.e. if they have the same length spectrum). For closed surfaces, there is an upper bound on the size of isospectral hyperbolic structures depending only on the topology. We will show that the situation is different for infinite-type surfaces, by constructing large families of isospectral hyperbolic structures on surfaces of infinite genus.

• Friday February 12, 2021 at 10:40,

  Constructing examples and non-examples of jigsaw pseudomodular groups 

  Carmen Galaz-Garcia
  University of California, Santa Barbara
   
  Consider a discrete subgroup H of PSL(2,R) and its action on IH^2 the upper half-plane model of the hyperbolic plane. The cusp set of H is the set of points in the boundary at infinity of IH^2 fixed by its parabolic elements. For example, the cusp set of PSL(2,Z) is QU{oo}. A natural question is: how strong is the cusp set as an invariant? More precisely, if H has cusp set  QU{oo} , is it commensurable with PSL(2,Z)? A negative answer was provided by Long and Reid in 2001 by constructing finitely many examples of pseudomodular groups. In 2016 Lou, Tan and Vo produced two infinite families of pseudomodular groups via the jigsaw construction. In this talk we will construct a third family of pseudomodular groups obtained with the jigsaw construction and also show that "many" of the simplest jigsaw groups are not pseudomodular.  
   
   
   

• Friday February 19, 2021 at 10:40, over Zoom

  A polynomial invariant for veering triangulations

   

  Sam Taylor

  Temple University

  Veering triangulations form a rich class of ideal triangulations of cusped hyperbolic manifolds that were introduced by Agol and have connections to hyperbolic geometry, Teichmuller theory, and the curve complex. In this talk, we introduce a polynomial invariant of a veering triangulation that, when the triangulation comes from a fibration, recovers the Teichmuller polynomial introduced by McMullen. We show that in general, the polynomial determines a (typically non-fibered) face of the Thurston norm ball and that the classes contained in the cone over this face have representatives that are carried by the veering triangulation itself.

  This is joint work with Michael Landry and Yair Minsky.
   

   

• Friday February 26, 2021 at 10:40, over Zoom

  Cannon-Thurston maps: to exist or not to exist

   

  Radhika Gupta, Temple University

  
  Consider a hyperbolic 3-manifold which is the mapping torus of a pseudo-Anosov on a closed surface of genus at least 2. Cannon and Thurston showed that the inclusion map from the surface into the 3-manifold extends to a continuous, surjective map between the visual boundaries of the respective universal covers. This gives a surjective map from a circle to a 2-sphere. In this talk, we will explore what happens when we consider the mapping torus of a surface with boundary, which is not hyperbolic but CAT(0) with isolated flats.

   

• Friday March 5, 2021 at 10:40, over Zoom

  Geodesic currents and the smoothing property

   

  Didac Martinez-Granado, UC Davis

  Abstract: Geodesic currents are measures that realize a closure of the space of curves on a closed surface. Bonahon introduced geodesic currents in 1986, showed that geometric intersection number extends to geodesic currents and realized hyperbolic length of a curve as intersection number with a geodesic current associated to the hyperbolic structure. Since then, other functions on curves have been shown to extend to geodesic currents. Some of them extend as intersection numbers, such as negatively curved Riemannian lengths (Otal, 1990) or word length w.r.t. simple generating sets of a surface group (Erlandsson, 2016). Some other functions aren't intersection numbers but extend continuously (Erlandsson-Parlier-Souto, 2016), such as word length w.r.t. non-simple generating sets or extremal length of curves. In this talk we present a criterion for a function on curves to extend continuously to geodesic currents. This is joint work with Dylan Thurston.

   

• Friday March 12, 2021 at 11:00, over Zoom

  RAAGs in MCGs

  Ian Runnels, University of Virginia

  Abstract: Inspired by Ivanov's proof of the Tits alternative for mapping class groups via ping-pong on the space of projective measured laminations, Koberda showed that right-angled Artin subgroups of mapping class groups abound. We will outline an alternate proof of this fact using the hierarchy of curve graphs, which lends itself to effective computations and stronger geometric conclusions. Time permitting, we will also discuss some applications to the study of convex cocompact subgroups of mapping class groups.

• Friday March 19, 2021 at 10:40, over Zoom

  Cubical geometry in big mapping class groups

   

  Anschel Schaffer-Cohen, University of Pennsylvania

  Abstract: Mapping class groups of infinite-type surfaces, also known as big mapping class groups, can be studied geometrically from the perspective of coarsely bounded generating sets. Within this framework, we describe a large family of surfaces--the avenue surfaces without significant genus--and show that the mapping class group of any such surface is quasi-isometric to an infinite-dimensional cube graph. As a consequence, we see that these mapping class groups are all quasi-isometric to each other, and that they are all a-T-menable. Both of these properties are notable in that they are known to fail for mapping class groups of finite-type surfaces.

   

• Friday April 9, 2021 at 13:00, over Zoom

  Covers, curves, and length spectra

  Marissa Loving, Georgia Tech

  Abstract: In this talk, I will share some of my ongoing work with Tarik Aougab, Max Lahn, and Nick Miller in which we explore the simple length spectrum rigidity of hyperbolic metrics arising from Sunada’s construction. Along the way we give a characterization of equivalent covers (not necessarily regular) in terms of simple elevations of curves, generalizing previous work with Aougab, Lahn, and Xiao.

• Friday April 16, 2021 at 10:40, over Zoom

  A polynomial invariant for square triangulations

   

  Aaron Abrams, Washington & Lee University

  Abstract: A celebrated theorem of Monsky from 1970 implies that it is impossible to dissect a square into an odd number of triangles of equal area.
  
  Which begs the question:  which (tuples of) areas can be the areas of the triangles in a dissection of a square?
  
  
  It turns out that for each combinatorial type of triangulation, there is exactly one polynomial relation that is satisfied by the areas of the triangles in every geometric realization of the given triangulation.  I will talk about recent discoveries and current mysteries surrounding this polynomial invariant.

   

• Wednesday September 8, 2021 at 14:30, Wachman 617

  Systoles and cosmetic surgeries

  Dave Futer, Temple University

  Abstract: The cosmetic surgery conjecture, posed by Cameron Gordon in 1990, is a uniqueness statement that (essentially) says a knot in an arbitrary 3-manifold is determined by its complement \(N\). In the past three decades, this conjecture has been extensively studied, especially in the setting where the knot complement \(N\) embeds into the 3-sphere. Many different invariants of knots and 3-manifolds have been applied to this problem.

  After surveying some of this recent work, I will describe a recent result that uses hyperbolic methods, particularly short geodesics, to reduce the cosmetic surgery conjecture for any particular \(N\) to a finite computer search. This is joint work with Jessica Purcell and Saul Schleimer.

• Wednesday September 15, 2021 at 14:30, Wachman 617

  Left orderability for surgeries on [1,1,2,2,2j] two-bridge knots

  Khánh Lê, Temple University

  Abstract: A group is called left-orderable if it admits a total ordering that is invariant under left multiplication. In 3-manifold topology, left orderability is an important concept due to its role in the L-space conjecture. There has been a substantial effort in developing tools to order the fundamental group of rational homology 3-spheres. In a recent work, Xinghua Gao encoded information about hyperbolic \(\widetilde{PSL}_2{\mathbb R}\) representations of a one-cusped 3-manifold \(M\) in the holonomy extension locus and used it to order intervals of Dehn fillings assuming a strong technical condition of the character variety of \(M\). In this talk, we will show how to weaken this condition to a local condition at the non-abelian reducible representation. As an application, we construct left orders on an interval of Dehn fillings on the \([1,1,2,2,2j]\) two-bridge knots.

• Wednesday September 22, 2021 at 14:30, Wachman 617

  Right-angled links in thickened surfaces

  Rose Kaplan-Kelly, Temple University

  Abstract: Traditionally, alternating links, links with a projection diagram that can be given an orientation such that the link's crossings alternate between over- and under-crossings, are studied with alternating diagrams on \(S^2\) in \(S^3\). In this talk, we will consider links which are alternating on higher genus surfaces \(S_g\) in \(S_g x I\). We will define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property is equivalent to the link having two totally geodesic checkerboard surfaces, and equivalent to a set of restrictions on the link's alternating projection diagram. We will then use these diagram restrictions to classify RGCR links according to the polygons in their checkerboard surfaces and provide a bound on the number of RGCR links for a given surface of genus g. Along the way, we will answer a question posed by Champanerkar, Kofman, and Purcell about links with alternating projections on the torus.

• Wednesday September 29, 2021 at 14:30, Wachman 617

  Decompositions of groups and their quotients using graphs and cell complexes

  Thomas Ng, Technion

  Abstract:Bass-Serre theory plays an important role in studying manifolds via decompositions along essential submanifolds by graphically encoding their fundamental groups as a combination of more easily understood subgroups and morphisms between them. Subgroups inherently are related to covers of such graphs of groups, but using these tools to geometrically study quotients is more mysterious. I will discuss how certain classes of quotients admit the generalized structure of developable complex of groups. I will go on to mention joint work with Radhika Gupta and Kasia Jankiewicz demonstrating how to use this structure to prove locally uniform exponential growth for certain Artin groups and the Higman group.

• Friday October 1, 2021 at 15:00, Bryn Mawr College, Park Science Center 159

  Graphically discrete groups and rigidity

  Emily Stark, Wesleyan University

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

  Abstract: Rigidity theorems prove that a group's geometry determines its algebra, typically up to virtual isomorphism. Motivated by interest in rigidity, we study the family of graphically discrete groups. In this talk, we will present rigidity consequences for groups in this family. We will present classic examples as well as new results that imply this property is not a quasi- isometry invariant. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

  In the morning background talk (10-11am), I will discuss coarse geometry and present examples of how groups with similar large-scale geometry often share common algebraic features.

• Friday October 1, 2021 at 16:30, Bryn Mawr College, Park Science Center 159

  Liouville cobordisms

  Emmy Murphy, Princeton University and IAS

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

  Abstract: In this talk we'll discuss some interesting Liouville cobordisms arising in the particular case when the negative boundary is an overtwisted contact manifold. This will center on two independent constructions: concordances in the high-dimensional setting, and cobordisms with high-index (and therefore non-Weinstein) topological type.

  In the morning background talk (11:30am-12:30pm), we'll discuss the basics of Liouville manifolds and Weinstein handles. This is a method by which new symplectic manifolds can be constructed from old, using isotropic/Legendrian submanifolds of contact manifolds.

• Wednesday October 6, 2021 at 14:30, Wachman 617

  Polynomial invariants of free-by-cyclic groups

  Radhika Gupta, Temple University

  Abstract: We will first talk about some polynomial invariants for fibered hyperbolic 3-manifolds, namely the Teichmüller polynomial and Alexander polynomial. We will then develop analogous theory for free-by-cyclic groups and explore the relation between the corresponding polynomials. This is based on joint work with Sam Taylor and Spencer Dowdall.

• Thursday October 21, 2021 at 17:15, Zoom talk

  Topology of the space of contact structures on the 3-sphere 

  Yasha Eliashberg, Stanford University

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

  Abstract: I will prove that the space of positive tight contact structures on the 3-sphere is homotopy equivalent to the real projective plane. The talk is based on a joint work with N. Mishachev.

• Thursday October 21, 2021 at 18:15, Zoom talk

  Dilatation for random point-pushing pseudo-Anosovs 

  Tarik Aougab, Haverford College

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

  Abstract: Choose a homotopically non-trivial curve “at random” on a compact orientable surface- what properties is it likely to have? We address this when, by “choose at random”, one means running a random walk on the Cayley graph for the fundamental group with respect to the standard generating set. In particular, we focus on self-intersection number and the location of the metric in Teichmuller space minimizing the geodesic length of the curve. As an application, we show how to improve bounds (due to Dowdall) on dilatation of a point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of the defining curve, for a “random” point-pushing map. This represents joint work with Jonah Gaster.

• Wednesday October 27, 2021 at 14:30, Wachman 617

  Geometric triangulations of a family of hyperbolic 3-braids

  Barbara Nimershiem, Franklin & Marshall College

  Abstract: After constructing topological triangulations of complements of closures of the braids \( C^2\sigma_1^p\sigma_2^{-1} \), where \(p\geq 1\), I will argue that the constructed triangulations meet a stronger condition: They are actually geometric. The proof follows a procedure outlined by Futer and Gu\'eritaud and uses a theorem of Casson and Rivin. If time allows, I will also discuss possible generalizations of the construction.

• Wednesday November 10, 2021 at 14:30, Wachman 617

  Boundaries of fibered faces and limits of stretch factors

  Sam Taylor, Temple University

  Abstract: Pseudo-Anosov stretch factors are organized by the fibered face theory of Thurston, Fried, and McMullen. In this talk, I’ll explain the precise sense in which this is true, draw lots of pictures that illustrate it, and use this perspective to answer a question of Chris Leininger. Informally, he asks what can limit points of stretch factors coming from a single 3-manifold look like, and we’ll see how they are actually stretch factors of homeomorphisms of "infinite type" surface that are “wrapped-up” inside in the original manifold.

  Anything original is joint work with Landry and Minsky.

• Wednesday November 17, 2021 at 14:30, Wachman 617

  Train track maps and CTs on graphs of groups

  Rylee Lyman, Rutgers University Newark

  Abstract: A homotopy equivalence of a graph is a train track map when it sends vertices to vertices and the restriction of any iterate of the map to an edge yields an immersion. (Relative) train track maps were introduced by Bestvina and Handel in 1992; since then they have become one of the main tools in the study of outer automorphisms of free groups. More recently in 2011, Feighn and Handel introduced a stronger kind of relative train track map called a CT and proved their existence for all outer automorphisms after passing to a power. We extend the theory of relative train track maps to graphs of groups with finitely generated, co-Hopfian edge groups and the theory of CTs to free products (that is, graphs of groups with trivial edge groups).

• Friday December 3, 2021 at 09:30, Wachman 617

  Hyperbolic links in the tangent bundle

  Andrew Yarmola, Princeton University

  PATCH Background talk

  Abstract: Staring with a background on the geometry and topology of 2- and 3-manifolds, we will introduce hyperbolic 3-manifolds that arise as link complements in the projectivized tangent bundle \( PT(S)\) of a surface \(S\). Specifically, we will focus on the case where the link is the canonical lift of a family \( C\) of smooth curves. When \(C\) is filling and in minimal position on \(S\), the resulting 3-manifold \( M_C\) turns out to be finite-volume and hyperbolic and therefore any invariants of \( M_C\) (such as volume, homology, cusp shape and volume, number of tetrahedra in canonical triangulations, etc) are now mapping class group invariants of \( C \). Outside of this connection, these links may be of independent interest as they include all Lorenz links and provide an infinite family Legendrian links for the natural contact structure on \( PT(S)\).

• Friday December 3, 2021 at 11:00, Wachman 617

  Introduction to knot concordance

  Allison Miller, Swarthmore College

  PATCH Background Talk

  We will talk about the basics of knot concordance and answer some of the following questions, depending on audience preferences: What should it mean for a knot to be "simple" from a 4-dimensional perspective? Is there a sensible and interesting definition of when we should think of two knots as being "4-d equivalent". What do classical knot invariants like the Alexander polynomial have to say? What structure can we find or build from a 4-dimensional perspective on knot theory? Can this help us understand the weird world of 4-manifold topology more broadly?

• Friday December 3, 2021 at 15:00, Wachman 617

  Invariants of hyperbolic canonical links in tangent bundles

  Andrew Yarmola, Princeton University

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

  Abstract: Let \( S \) be a surface of negative Euler characteristic and consider a finite filling collection \( C \) of closed curves on \( S \) in minimal position. An observation of Foulon and Hasselblatt shows that \( M_C = PT(S) \setminus \hat{C} \) is a finite-volume hyperbolic 3-manifold, where \( PT(S) \) is the projectivized tangent bundle and \( \hat C \) is the set of tangent lines to \( C \). In particular, any invariant of \( M_C\) is a mapping class group invariant of the collection \( C \). In this talk, we will go over results that explain the behavior and provide coarse bounds on the volume of \( M_C\) in terms of topological and geometric properties of the family \( C\) . For example, when \( C \) is a filling pair of simple closed curves, we show that the volume is coarsely comparable to Weil-Petersson distance between strata in Teichmuller space. Further, we will explain algorithmic methods and tools for building such links and computing invariants.

• Friday December 3, 2021 at 16:30, Wachman 617

  Satellite operators and knot concordance

  Allison Miller, Swarthmore College

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

  Abstract: The classical satellite construction associates to any pattern P in a solid torus and companion knot K in the 3-sphere a satellite knot P(K), the image of P when the solid torus is ‘tied into’ the knot K. This operation descends to a well-defined map on the set of (smooth or topological) concordance classes of knots. Many natural questions about these maps remain open: when are they surjective, injective, or bijective? How do they behave with respect to measures of 4-dimensional complexity? How do they interact with additional group or metric space structure on the concordance set?

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

The Seminar usually takes place on Fridays at 10:40 AM via Zoom (please contact the seminar organizers for the Zoom link), not in Room 617 on the sixth floor of Wachman Hall.

• Wednesday January 22, 2020 at 14:30, Wachman 617

  Detecting Geometric Outer Automorphisms

  Edgar A. Bering IV
  Temple University
  
  Outer automorphisms of a free group are a fundamental example in geometric group theory and low dimensional topology. One approach to their study is by analogy with the mapping class groups of surfaces. This analogy is made concrete by the natural inclusions Mod(S) -> Out(F) that occur whenever S has free fundamental group. Outer automorphisms in the image of these inclusions are called geometric. In 1992, Bestvina and Handel gave an algorithm for deciding when an irreducible outer automorphism is geometric. I will describe current joint work with Yulan Qing and Derrick Wigglesworth to give an algorithm to decide when a general outer automorphism is geometric.

• Wednesday January 29, 2020 at 14:30, Wachman 617

  From Hierarchical to Relative Hyperbolicity

  Jacob Russell, CUNY Graduate Center

  The success of Gromov’s coarsely hyperbolic spaces has inspired a multitude of generalizations. We compare the first of these generalizations, relatively hyperbolic spaces, with the more recently introduced hierarchically hyperbolic spaces. We show that relative hyperbolicity can be detected by examining simple combinatorial data associated to a hierarchically hyperbolic space. As an application, we classify when the separating curve graph of a surface is relatively hyperbolic.

• Wednesday February 5, 2020 at 14:30, Wachman 617

  An Analog to the Curve Complex for FC Type Artin groups

  Rose Morris-Wright Brandeis University

  Artin groups are a generalization of braid groups that provide a rich field of examples and counter-examples for many algebraic, geometric, and topological properties. Any given Artin group contains many subgroups isomorphic to other Artin groups, creating a hierarchical structure similar to that of mapping class groups. I generalize and unify the work of Kim and Koberda on right angled Artin groups and the work of Cumplido, Gonzales-Meneses, Gebhardt, and Wiest on finite type Artin groups, to construct a simplicial complex in analogy to the curve complex. I will define this complex, and discuss some properties that this complex shares with the curve complex of a mapping class group.

• Wednesday February 12, 2020 at 14:30, Wachman 617

  Smooth Fibrations of the 3-Sphere by Simple Closed Curves

  Leandro Lichtenfelz, University of Pennsylvania We show that the moduli space of all smooth fibrations of a 3-sphere by oriented simple closed curves has the homotopy type of a disjoint union of a pair of 2-spheres, which coincides with the homotopy type of the finite-dimensional subspace of Hopf fibrations. In the course of the proof, we present a pair of entangled fiber bundles in which the diffeomorphism group of the 3-sphere is the total space of the first bundle, whose fiber is the total space of the second bundle, whose base space is the diffeomorphism group of the 2-sphere. This is joint work with D. DeTurck, H. Gluck, M. Merling and J. Yang.

• Friday February 28, 2020 at 14:30, Haverford College, room Sharpless S113

  Convergence of normalized Betti numbers in nonpositive curvature 

  Ian Biringer, Boston College

  PATCH Seminar, at Haverford College

  Abstract: We’ll show that if \(X\) is any symmetric space other than 3-dimensional hyperbolic space and \(M\) is any finite volume manifold that is a quotient of \(X\), then the normalized Betti numbers of M are “testable", i.e. one can guess their values by sampling the complex at random points. This is joint with Abert-Bergeron-Gelander, and extends some of our older work with Nikolov, Raimbault and Samet. The content of the recent paper involves a random discretization process that converts the "thick part" of \(M\) into a simplicial complex, together with an analysis of the "thin parts" of \(M\). As a corollary, we can prove that whenever \(X\) is a higher rank irreducible symmetric space and \(M_i\) is any sequence of finite volume quotients of \(X\), the normalized Betti numbers of the \(M_i\) converge to the "\(L^2\)-Betti numbers" of \(X\).

  There will also be a background talk on this topic at 9:30am.

• Friday February 28, 2020 at 16:00, Haverford College, room Sharpless S113

  Right-veering open books and the Upsilon invariant 

  Diana Hubbard, CUNY

  PATCH Seminar, at Haverford College

  Abstract: Fibered knots in a three-manifold \(Y\) can be thought of as the binding of an open book decomposition for \(Y\). A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

  There will also be a background talk on this topic at 11:00am.

• Wednesday March 11, 2020 at 14:30, Wachman 617

  Hyperbolic volume and bounded cohomology

  James Farre, Yale University
  
  Abstract: A natural notion of complexity for a closed manifold \(M\) is the smallest number of top dimensional simplices it takes to triangulate \(M\).  Gromov showed that a variant of this notion called simplicial volume gives a lower bound for the volume of \(M\) with respect to any (normalized) Riemannian metric.  The heart of his proof factors through the dual notion of bounded cohomology.  I will define bounded cohomology of discrete groups illustrated by some examples coming from computing the volumes of geodesic simplices in hyperbolic space.  Although bounded cohomology is often an unwieldy object evading computation, we give some conditions for volume classes to be non-vanishing in low dimensions.  We then ask, ``When do higher dimensional volume classes vanish?’’

• Friday September 11, 2020 at 10:40, over Zoom

  Non-uniquely ergodic trees in the boundary of Outer space

  Radhika Gupta, Temple University

  There exist non-uniquely ergodic arational laminations on a surface of genus \( g \geq 2\). That is, there exists an arational lamination which supports two transverse measures that are not scalar multiples of each other. In analogy, one asks if 'arational' trees in the boundary of Outer space support metrics that are not scalar multiples of each other. In this talk, I will first talk about laminations on surfaces. Then we will see some examples of trees in Outer space and understand what it means for a tree to be non-uniquely ergodic. Time permitting, I will describe the construction of a "non-geometric", arational, non-uniquely ergodic \(F_n\)-tree. This is joint work with Mladen Bestvina and Jing Tao.

• Friday September 18, 2020 at 10:40,

  Anschel Schaffer-Cohen TBA 

  Anschel Schaffer-Cohen, University of Pennsylvania

  Title/abstract TBA