Geometry and Topology Seminar

Abstract: 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.

I will discuss joint work with Lafont, Minemyer, Sorcar, and Wells that provides an iterative procedure for constructing hyperbolic right-angled Coxeter groups that virtually algebraically fiber (that is, have a homomorphism onto Z with finitely generated kernel). One novel aspect of this procedure is that it produces examples in every cohomological dimension n > 1. This procedure combines work of Jankiewicz, Norin, and Wise with a generalization of a construction due to Osajda involving a `simplicial thickening' process.

Abstract: We discuss what a "typical" short curve on a random large genus hyperbolic surface looks like. In particular, for each $L$, there are finitely many curves of length at most $L$. We find length scales at which such a curve chosen at random is highly likely to be non-simple, or fill the whole surface. It is known that, with respect to many commonly studied random models, a typical surface will be expander. That is, it will be "highly connected," in the sense that we get effective mixing of the geodesic flow. We will give results that hold for all expander surfaces, and hence for random surfaces with respect to many different random models. This is joint work with Ben Dozier.

Abstract: In this talk, I will describe a new geometric and combinatorial structure for the space of complex polynomials with a fixed number of roots. In particular, I will define a metric on the space of monic polynomials with d distinct centered roots, and I will introduce a finite cell structure for the metric completion. Each cell in this complex is a product of two Euclidean simplices, and the combinatorial structure comes from the dual presentation for the d-strand braid group. In particular, this provides a concrete connection between two classifying spaces for the braid group. This is joint work with Jon McCammond.

Abstract: We describe a new construction of invariant train tracks for pseudo-Anosov homeomorphisms with irreducible transition matrix. This fills a gap in the literature concerning the existence of such train tracks. The construction uses invariant train tracks associated to the veering triangulation of the mapping torus of the homeomorphism to eliminate branches which obstruct irreducibility.

In 3-dimension topology, the study of foliations, flows and fundamental group actions on 1-manifolds are closely related. Given a closed 3-manifold, one can construct a fundamental group action on a circle from either a taut foliation or a pseudo-Anosov flow in the manifold by different works of Thurston, Calegari-Dunfield and Fenley. When the foliation is depth-one and the pseudo-Anosov flow is transverse to the foliation, we show (with some extra assumptions) that the circle actions from both settings are topologically conjugate. Moreover, the two circles admit extra structures that are compatible in a natural sense.

Ino Loukidou (University of Chicago)

If M is a hyperbolic 3-manifold fibering over the circle, the fundamental group of M acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (eg taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles - i.e. a circle with a faithful pi_1(M) action preserving a pair of invariant laminations - and these play a key role in relating the dynamical structure to the geometry of M. In this talk we introduce the idea of *Zippers*, which give a new and direct way to construct universal circles, streamlining the known construction in some cases, and giving a host of new constructions in others. This is joint work with Danny Calegari.

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

Background talk (10am): Thinking about dimension 4, stuck in dimension 3: Knots, Concordances, and Homology Cobordisms

Abstract: It is a common theme in topology to study n-manifolds based on the n-1-manifolds they bound, or the n+1-manifolds bounded by them. We’ll explore together why this is an interesting and useful thing to do in dimensions 3 and 4! More specifically, we will talk about knots in the 3-sphere which are secretly related in 4-dimensional ways and how this can help us think about 3-manifolds that are similarly mysteriously connected. 
 

Research talk (2pm): 0-Surgeries on Links

Abstract: In work in progress with Ryan Stees, we show that every closed, oriented 3-manifold can be obtained by 0-surgery on a link. Since the 0-surgery of a link can capture the data of many of the typical isotopy and concordance invariants of a link, particularly in the pairwise linking number 0 case, this result gives us a nice lens through which to study both 3-manifolds and links. However, 0-surgery on a link is certainly not a complete link invariant, and we also give multiple constructions for non-isotopic (and even non-concordant) links with homeomorphic 0-surgeries.

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

Background talk (11am): Waist inequalities, Property T, and Higher Expansion

Abstract: Suppose we are given two simplicial complexes $X$ and $Y$, where $X$ is "complicated” and $Y$ is lower dimensional than $X$. Then must a map $f: X \to Y$ have at least one "complicated” fiber? This talk will give an overview of a program Gromov initiated to prove quantitative statements of this kind, called waist inequalities. Both this talk and the next will feature examples where what "complicated" means can be either geometric (e.g. volume) or topological (e.g., a measure of the largeness of the fundamental group.) To give a sense for the theory I will first talk about what is known for concrete examples like the sphere and the torus, before moving towards the wilder setting of negative curvature. Along the way, I will as time permits describe connections to scalar curvature, (higher versions of) property $T$, systolic geometry, and various notions of higher expander simplicial complexes and manifolds originating in computer science. This talk will move slowly and not assume prior knowledge in this area.

Research talk (3:45pm): Minimal Submanifolds and Waist Inequalities for Locally Symmetric Spaces

Abstract: This talk will focus on the case of nonpositively curved locally symmetric spaces. In addition to being the most natural non-positively curved spaces to study from the perspective of differential geometry, they also have strong connections to geometric group theory, number theory, and algebraic geometry. I will describe recent joint work with Mikolaj Fraczyk that establishes a number of different kinds of higher expansion properties for families of manifolds in this setting by bringing new tools into the picture from representation theory and minimal surface theory. One goal will be to explain how knowledge of the unitary representations of a semisimple Lie group can be used to study the geometry of the associated locally symmetric spaces. On the minimal surface side, we establish new monotonicity formulas, or volume growth estimates, for minimal submanifolds of low-codimension in nonpositively curved symmetric spaces. I will explain how this can be played against the information coming from representation theory to prove waist inequalities. This talk may not move as slowly as the first talk but it will not assume prior knowledge in this area.

Thomas Hill (Utah)

For a locally finite, connected graph $\Gamma$, let $\operatorname{Map}(\Gamma)$ denote the group of proper homotopy equivalences of $\Gamma$ up to proper homotopy.  Excluding sporadic cases, we show $\operatorname{Aut}(\mathcal{S}(M_\Gamma)) \cong \operatorname{Map}(\Gamma)$, where $\mathcal{S}(M_\Gamma)$ is the sphere complex of the doubled handlebody $M_\Gamma$ associated to $\Gamma$.  We also construct an exhaustion of $\mathcal{S}(M_\Gamma)$ by finite strongly rigid sets when $\Gamma$ has finite rank and finitely many rays, and an appropriate generalization otherwise. This is joint work with Michael Kopreski, Rebecca Rechkin, George Shaji, and Brian Udall.  

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

Abstract: In the morning background talk (10am in Park 337), I will provide an introduction to symplectic embeddings. I will explain the background to the symplectic embedding problem and some basic results about embedding four dimensional ellipsoids.

In the afternoon research talk (2:30pm in Park 300), I will discuss calculations of the embedding capacity function for ellipsoids into balls and related target spaces.

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

In the morning background talk (11:30am in Park 337), I will give some background on the moduli and deformation spaces of quadratic rational maps with fixed portraits and their compactifications. This talk will include definitions and techniques from the theory of rational maps, algebraic geometry, and low-dimensional topology that will be relevant to the second talk.

In the afternoon research talk (4:00pm in Park 300), I will discuss a question of J. Milnor on the moduli spaces of quadratic rational and its extension by A. Epstein. Using some of the techniques introduced in part #1, we will show that Epstein's deformation space is disconnected and give further properties. Part of this work described is joint with S. Koch.

Abstract: Symmetries of a group $G$ are encoded in the automorphism group $Aut(G)$. "Hidden symmetries" are encoded in the abstract commensurator $Comm(G)$. While many classes of finitely generated groups have reasonably well-understood commensurator -- for example, when $G$ is an arithmetic group, $Comm(G)$ is typically a group of matrices with rational entries -- the abstract commensurator of a free group, $Comm(F_2)$, is still somewhat mysterious. I will explain how Edgar A. Bering IV and I fleshed out a topological perspective of commensurations that allowed us to show that every countable locally finite group is a subgroup of $Comm(F_2)$.