# Past seminars

**March 31 (Tu) 4 pm, Zoom Meeting ID TBA**

Sang-hyun Kim (KIAS)

*On Sharkovskii's Theorem*

Sharkovskii's theorem asserts that the set of natural numbers carries a total order << so that p>>q if and only if every continuous interval map having a point with period p must admit a point with period q. We present a proof of this theorem relying on a purely combinatorial lemma and possible applications of the proof. (expository)

**April 7 (Tu)** 4 - 5:30 pm, Zoom Meeting ID 427-852-712

Hyungryul Baik (KAIST)

*Introduction to pseudo-Anosov homeomorphisms *

We will give a brief introduction on pseudo-Anosov surface homeomorphisms, and discuss the major open problems related to it.

**April 15 (W)** 4 - 5:30 pm, Zoom 739-035-2844

National Election Day (no talks)

**April 22 (W)** 4 - 5:30 pm, Zoom 739-035-2844

Dongryul Kim (KAIST)

*Topological Entropy of pseudo-Anosov mapping classes obtained from typical Thurston Construction*

We will talk about probabilistic methods in geometric topology, in particular our recent work joint with Prof. H.Baik and I.Choi, dealing with pseudo-Anosov mapping classes arising from Thurston construction.

**April 29 (W)** 4 - 5:30 pm, Zoom 739-035-2844

Youngjin Bae (KIAS)

*What is Legendrian?*

I will explain the role of Legendrians in the study of symplectic geometry, contact topology, and knot theory. Several Legendrian invariants will be introduced including homotopy type invariants, pseudo-holomorphic curve counting method, and generating families. If time permits, I will also talk about how the Legendrian theory interacts with Lagrangian- and microlocal sheaf theory.

**May 6 (W)** 4 - 5:30 pm, Zoom 739-035-2844

Changsub Kim (KAIST)

*On Translation Lengths of Anosov Maps on Curve Graph of Torus*

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map.

**May 13 (W)** 4 - 5:30 pm, Zoom 739-035-2844

4 pm Korea Standard Time (GST+9) = 9 am French Standard Time (GST+2)

Federica Fanoni (CNRS)

*Big mapping class groups acting on homology*

To try and understand the group of symmetries of a surface, its mapping class group, it is useful to look at its action on the first homology of the surface. For finite-type surfaces this action is fairly well understood. I will recall what happens in this case, introduce infinite-type surfaces (surfaces whose fundamental group is not finitely generated) and discuss joint work with Sebastian Hensel and Nick Vlamis in which we describe the action on homology for these surfaces.

**May 20 (W) 9 - 10:30 am (Note the date and time!)**, Zoom 739-035-2844

9 am Korea Standard Time (UTC+9) May 20 (W) = 5 pm Pacific Daylight Time May 19 (Tu)

Lei Chen (Caltech)

*Actions of Homeo and Diffeo groups on manifolds*

In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is a joint work with Kathryn Mann.

**May 27 (W) 9 - 10:30 am (Note the time!)**, Zoom 739-035-2844

9 am Korea Standard Time (UTC+9) May 27 (W) = 8 pm Toronto Time May 26 (Tu)

Giulio Tiozzo (University of Toronto)

*Central limit theorems for counting measures in coarse negative curvature*

We establish general central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem, and we proved this conjecture in 2018. In our new work, we remove the assumptions of properness and smoothness of the space, or cocompactness of the action, thus proving a general central limit theorem for group actions on hyperbolic spaces. We will see how our techniques replace the classical thermodynamic formalism and allow us to provide new applications, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds. Joint work with I. Gekhtman and S. Taylor.

**June 3 (W) 10 - 11:30 am (Note the time!)**, Zoom 739-035-2844

10 am Korea Standard Time (UTC+9) June 3 (W) = 9 pm Eastern Daylight Time June 2 (Tu)

Sebastian Hurtado (University of Chicago)

*Left orderability of lattices of *SL_{n}(*ℝ*)* . (Joint work with Bertrand Deroin).*

A countable group is said to be left-orderable if it embeds in the group of homeomorphisms of the line. We study the left orderability of lattices in Lie groups. Our main result is that a lattice in a real semi-simple Lie group of higher rank is left orderable if and only if a factor of G is the universal covering of SL_{2}(*ℝ*). In particular every lattice of SL_{n}(*ℝ*)** (if n > 2) is not left orderable. This solves a conjecture of Witte-Morris from the late 90's.

**June 10 (W) 9 - 10:30 am (Note the time!)**, Zoom 739-035-2844

9 am Korea Standard Time (UTC+9) June 10 (W) = 8 pm Eastern Daylight Time June 9 (Tu)

Henry Segerman (Oklahoma State University)

*From veering triangulations to link spaces and back again*

Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. In unpublished work, Guéritaud and Agol generalise an alternative construction to any closed manifold equipped with a pseudo-Anosov flow without perfect fits.

Schleimer and I build the reverse map. As a first step, we construct the link space for a given veering triangulation. This is a copy of *ℝ*^{2}**, equipped with transverse stable and unstable foliations, from which the Agol--Guéritaud construction recovers the veering triangulation. The link space is analogous to Fenley's orbit space for a pseudo-Anosov flow.

Along the way, we construct a canonical circular ordering of the cusps of the universal cover of a veering triangulation. In work in progress, Manning, Schleimer and I use this to produce Cannon-Thurston maps for all veering triangulations. This gives the first known examples of Cannon-Thurston maps that do not come, even virtually, from surface subgroups.

I will also talk about work with Giannopolous and Schleimer building a census of transverse veering triangulations. The current census lists all transverse veering triangulations with up to 16 tetrahedra, of which there are 87,047.

**June 17 (W) 4 - 5:30 pm**, Zoom 739-035-2844

Marcelo R.R. Alves (Université Libre de Bruxelles)

*Entropy collapse versus entropy rigidity for Reeb and Finsler flows*

The topological entropy of a flow on a compact manifold is a measure of complexity related to many other notions of growth. By celebrated works of Katok and Besson-Courtois-Gallot, the topological entropy of geodesic flows of Riemannian metrics with a fixed volume on a manifold M that carries a metric of negative curvature is uniformly bounded from below by a positive constant depending only on M. We show that this result persists for all (possibly irreversible) Finsler flows, but that on every closed contact manifold there exists a Reeb flow of fixed volume and arbitrarily small entropy. This is joint work with Alberto Abbondandolo, Murat Saglam and Felix Schlenk.

**June 24 (W) 10 - 11:30 am (note the time)**, Zoom 739-035-2844

Nathan Dunfield (University of Illinois Urbana-Champaign)

*Counting incompressible surfaces in 3-manifolds*

Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein.

**July 1 (W) 10 - 11:30 am (note the time)**, Zoom 739-035-2844

Andrés Navas (Universidad de Santiago de Chile)

*Distorted diffeomorphisms and regularity*

The goal is to deal with the following question: for a compact manifold M, does there exist a diffeomorphism that is distorted in the group of *C*^{r}** diffeomorphisms yet undistorted in the group of *C*^{s}** diffeomorphism, where *1 ≤ r < s* ? Although the answer seems to be positive, it seems hard to build explicit examples (these diffeomorphisms necessarily have zero entropy). We will provide such examples for the closed unit interval for r = 1 and s = 2. The distortion part of the proof uses standard techniques on centralizers; the *C*^{2}** part uses recent work with Hélène Eynard on the relation between the Mather invariant and asymptotic distortion of 1-dimensional maps.

**July 8 (W) 9 - 10:30 am (note the time)**, Zoom 739-035-2844

Jing Tao (University of Oklahoma)

Genericity of pseudo-Anosov mapping classes

Let S be a hyperbolic surface of finite type. Thurston’s classification asserts that elements of the mapping class group of S fall into three categories: finite order, reducible, and pseudo-Anosov. There are these three types, but it seems that from any reasonable point of view *most* elements are pseudo-Anosov. In this talk, I will show how to use geodesic currents to establish that pseudo-Anosov mapping classes are generic with respect to several notions of genericity. What these notions have in common is that they arise from functions on the mapping class group that measure the complexity of individual elements seen as mapping classes. This is based on joint work with Viveka Erlandsson and Juan Souto.

**July 15 (W) 10 - 11:30 am (note the time)**, Zoom 739-035-2844

Thomas Koberda (University of Virginia)

*Expanders and right-angled Artin groups*

Right-angled Artin groups provide a fruitful correspondence between group theory and combinatorics. In this talk, I will discuss a characterization of expander graphs via the group theoretic properties of right-angled Artin groups. In the process, I will define a more general notion of vector space expanders, and connect all these concepts to related objects such as dimension expanders and higher dimensional expanders. This is joint work with R. Flores and D. Kahrobaei.

**July 22 (W) 9 - 10:30 am (note the time)**, Zoom 739-035-2844

Alan Reid (Rice University)

*Distinguishing certain triangle groups by their finite quotients.*

We prove that certain arithmetic Fuchsian triangle groups are profinitely rigid in the sense that they are determined by their set of finite quotients amongst all finitely generated residually finite groups. Amongst the examples are the (2,3,8) triangle group.

**July 29 (W) 9 - 10:30 am**, Zoom 739-035-2844

Dan Margalit (Georgia Institute of Technology)

Mapping class groups in complex dynamics

In joint work with James Belk, Justin Lanier and Becca Winarski, we give a simple geometric algorithm that can be used to determine whether or not a post-critically finite topological polynomial is Thurston equivalent to a polynomial. Our methods are rooted in geometric group theory: we consider a complex of isotopy classes of trees and a simplicial map of this complex to itself that we call the lifting map. Our work extends previous work of Nekrashevych. Similar work has been announced by Ishii-Smillie. We will give several applications of our methods, including a solution to Pilgrim's finite global attractor problem in the case of topological polynomials, the solution to a generalization of Hubbard’s twisted rabbit problem (originally solved by Bartholdi–Nekrashevych), and a new proof of Thurston's theorem for topological polynomials.

**August 5 (W) 4 - 5:30 pm,** Zoom 739-035-2844

Mahan Mj (Tata Institute of Fundamental Research)

*Percolation on Hyperbolic groups*

We study first passage percolation (FPP) in a Gromov-hyperbolic group G with boundary equipped with the Patterson-Sullivan measure. We associate an i.i.d. collection of random passage times to each edge of a Cayley graph of G, and investigate classical questions about asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the 'velocity' exists in almost every direction, and is almost surely constant by ergodicity of the G-action on the boundary.

For every point on the boundary, we also show almost sure coalescence of any two geodesic rays directed towards the point. Finally, we show that the variance of the first passage time grows linearly with word distance along word geodesic rays in every fixed boundary direction.

This is joint work with Riddhipratim Basu.

**August 12 (W) No Talk**

**August 19, 21, 24, 25 (W F M Tu, mini-course) 4 - 5:30 pm**, Zoom 739-035-2844

Bram Petri (Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université)

*Extremal problems and probabilistic methods in hyperbolic geometry*

Even if we know many things about hyperbolic manifolds, there are many open extremal problems on them. To name a few:

- How does the maximal systole among closed hyperbolic n-manifolds of volume at most V grow as a function of V?

- How does the minimal diameter among closed hyperbolic n-manifolds of volume at least V grow as a function of V?

- Are there closed hyperbolic n-manifolds of arbitrarily large volume whose spectral gap is larger than that of hyperbolic n-space?

Even for surfaces (i.e n=2), many of these extremal problems are open. In this case, answers to these questions also provide insight into the shape of deformation spaces of hyperbolic surfaces.

In these lectures, I will discuss some of these problems. I will talk about what is known about them and how random constructions of hyperbolic manifolds sometimes provide answers.

**August 26 (W) 10 - 11:30 am,** Zoom 739-035-2844

Andrew Putman (University of Notre Dame)

*The topology of the moduli space of curves*

I will discuss the topology of the moduli space of curves and its finite covers. This is a meeting ground for many different parts of mathematics, and I will try to make this accessible to a broad audience of geometers and topologists.

**September 2 (W) 4 - 5:30 pm,** Zoom 739-035-2844

Michal Ferov (The University of Newcastle)

*Density of metric small cancellation in finitely presented groups*

Small cancellation groups form an interesting class with many desirable properties. It is a well-known fact that small cancellation groups are generic; however, all previously known results of their genericity are asymptotic and provide no information about "small" group presentations. In this talk, I will show how can one obtain closed-form formulas for both lower and upper bounds on the density of small cancellation presentations, and compare the results with experimental data.

**September 9 (W) 4 pm,** Zoom 739-035-2844

Tengren Zhang (National University of Singapore)

*Weakly positive representations*

Given a free group acting on hyperbolic space, the classical ping-pong lemma gives sufficient conditions under which the free group action is convex-cocompact. Similarly, in the setting of higher rank Riemannian symmetric spaces, a generalization of the ping-pong lemma (by Dey-Kapovich-Liu) gives sufficient conditions under which a free group action is induced by an Anosov representation. In this talk, I will explain an analog of the generalized Ping-pong lemma that is ``direction specific”. This gives rise to the notion of a weakly positive representation. As a consequence, we find a new method to construct families of primitive stable representations from rank 2 free groups into higher rank Lie groups.

**October 14 (W) 4 pm, Korea = 8 am, UK (note the time!)**

Henry Wilton (Cambridge)

*On stable primitivity rank*

The commutator length of an element w of the commutator subgroup of a group G is the minimal number of commutators needed to express w as a product of commutators. A more fruitful definition is obtained by stabilising the definition, yielding the notion of *stable* commutator length.

In the context of free groups, Puder has recently introduced the notion of *primitivity rank*, which can be thought of as a homotopical version of commutator length. In this talk, I’ll propose a stable version of primitivity rank, and state some of its properties.

**October 27, 29, 30 (TThF) 9 am**

Michael Landry (Washington University St. Louis)

*Lecture 1: **The Thurston norm, flows, and the Teichmüller polynomial*

This first talk will be mostly expository. We will introduce the Thurston norm and discuss how the norm interacts with certain pseudo-Anosov flows. We will also introduce the Teichmüller polynomial and talk about how it can be used to compute growth rates of closed orbits of pseudo-Anosov flows.

Note: https://www.math.wustl.edu/~landry/visgat1.pdf

*Lecture 2: ** Veering triangulations, the Thurston norm, and surfaces in 3-manifolds*

We will introduce veering triangulations and explain how a given veering triangulation encodes much information about a face of the Thurston norm unit ball.

Note: https://www.math.wustl.edu/~landry/visgat2.pdf

*Lecture 3: A polynomial invariant of veering triangulations *

We introduce a polynomial invariant of veering triangulations that can be viewed as a generalization of the Teichmüller polynomial. This is joint work with Yair Minsky and Samuel Taylor.

Note: https://www.math.wustl.edu/~landry/visgat3.pdf

**November 11 (W) 4 pm**

Ilya Gekhtman (Technion)

*Martin, Floyd and Bowditch boundaries of relatively hyperbolic groups*

Abstract: Consider a transient random walk on a countable group $G$. The Green distance between two points in the group is defined to be minus the boundary of the probability that a random path starting at the first point ever reaches the second. The Martin compactification of the random walk is a topological space defined to be the horofunction boundary of the Green distance. It is a topological model for the Poisson boundary.

The Martin boundary typically heavily depends on the random walk; it is thus exciting when for some large class of random walks, the Martin boundary is equivariantly homeomorphic to some well known geometric boundary of the group. Ancona showed in 1988 that this is the case for finitely supported random walks on hyperbolic groups: the Martin boundary is identified with the Gromov boundary.

We generalize Ancona's results to relatively hyperbolic groups: the Martin boundary equivariantly continuously surjects onto the Gromov boundary of any hyperbolic space on which the group acts geometrically finitely (called the Bowditch boundary), and the preimage of any conical limit point is a singleton. When the parabolic subgroups are virtually abelian (e.g. for Kleinian groups) we show that the preimage of a parabolic fixed point is a sphere of appropriate dimension, so the Martin boundary can be identified with a Sierpinski carpet.

A major technical tool is a generalization of a deviation inequality due to Ancona saying the Green distance is nearly additive along word geodesics, which has various other applications, including to comparing harmonic and Patterson-Sullivan measures for negatively curved manifolds and to local limit theorems for random walks.

We do all this using an intermediate construction called the Floyd metric obtaining by suitably rescaling the Cayley graph and considering the associated completion called the Floyd compactification. We show that for any finitely supported random walk on a finitely generated group, the Martin boundary surjects to the Floyd boundary, which in turn by work of Gerasimov covers the Bowditch boundary of relatively hyperbolic groups. This is based on several joint works with subsets of Dussaule, Gerasimov, Potyagailo, and Yang.

**November 25 (W) 10 am, Korea (note the time!) **=** **Nov 24 (Tu) 8 pm New York EST

Kathryn Mann (Cornell)

Stability for hyperbolic groups acting on their boundaries.

A hyperbolic group acts naturally by homeomorphisms on its Gromov boundary. The theme of this talk is to say that, in many cases, such an action has very rigid dynamics.

Jonathan Bowden and I studied a special case of this, showing if G is the fundamental group of a compact, negatively curved Riemannian manifold, then the action of G on its boundary is what dynamicists call "topologically stable", meaning that small perturbations contain the same dynamical information as the original action. In new work with Jason Manning, we extend this to hyperbolic groups with sphere boundary, using large-scale geometric techniques. I will give some of the history of this problem and a sketch of the techniques used in the proof.