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 $\mathbb{H}$.