# Past seminars

# 2020

**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.

**December ****9 and 10**** (W, Th),**** 5 pm Korea = 9 am Switzerland**

Yash Lodha (EPFL)

*Spaces of countable groups*

In this mini course I will describe two spaces of groups. The first is the Grigorchuk space of marked groups, and the second is the Polish space of enumerated groups. Both spaces provide a useful framework for the study of countable, discrete groups. After a gentle introduction, I shall describe some recent results in the field, such as the work of Minasyan, Osin and Witzel on quasi-isometric diversity, and the work of Elayavalli and Goldbring on the generic version of the von Neumann-Day problem. I shall also describe how some of my past and also some recent work (in part with coauthors) fits into the picture.

**December 17 (Th) 10 - 11:30 am,** Zoom 739-035-2844

(= December 16 (W) 8 pm Indiana)

David Fisher (Indiana University)

*Superrigidity, Arithmeticity and Totally Geodesic Submanifolds.*

I will talk about recent joint work relating totally geodesic submanifolds to the fundamental group of finite volume locally symmetric manifolds. This work resolves some questions raised by McMullen and Reid and also illuminates a question of Gromov and Piateski-Shapiro. It is joint work with Uri Bader, Nick Miller, and Matthew Stover.

**2021**

**2021**

**January 7, 11, 13 (Th, M, W), 4 pm Korea**

Yash Lodha (KIAS)

*Amenability and paradoxical decompositions.*

The notion of Amenability has its roots in the seminal work of von Neumann from the early 20th century. Since then, it has become a well studied subfield of modern group theory. In this mini course shall describe amenability from a rather algebraic point of view. In the first lecture we shall define amenability via the notion of Folner sets. The second lecture shall be focused on paradoxical decompositions and Tarski numbers. The third lecture shall focus on some recent results in the field, including new solutions to the von Neumann-Day problem.

**January 15 (F), 10 am Korea**

Chenxi Wu (U of Wisconsin Madison)

*Galois conjugate of entropies of interval maps*

If a unimodal interval map sends the critical point to itself under finite iteration, the exponent of its entropy is an algebraic integer due to Perron-Frobenious theorem. When further assuming that the entropy is in a given interval, we are able to provide an algorithm characterizing the closure of all Galois conjugates of such algebraic integers. In an ongoing work we also generalized it to the setting of core entropy on quadratic Hubbard trees. This is a collaboration with Kathryn Lindsey, Diana Davis, Harrison Bray and Giulio Tiozzo.

**January 20 (W), 2021**

4:30pm - 6:00pm in Korean time = 8:30am - 10:00am in France

Federica Fanoni (CNRS)

*Isospectral hyperbolic surfaces of infinite type*

Two hyperbolic surfaces are said to be (length) isospectral if they have the same length spectrum (i.e., the same collection of lengths of primitive closed geodesics, counted with multiplicity). 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.

**February 3 (W), 2021**

10:00am - 11:30pm in Korean time

Inhyeok Choi (KAIST)

*Simple length spectra of a generic hyperbolic surface determine its isometry class*

Gel'fand asked in 1962 to which extent a closed hyperbolic surface is determined by its length spectrum. Strikingly, the examples of Vignéras and Sunada show that not all surfaces are determined by their length spectra. However, Wolpert proved that 'generic' closed hyperbolic surfaces are determined. Meanwhile, the analogous question with the simple length spectra has not been answered so far. Motivated by the technique of McShane and Parlier, we prove a variant of Wolpert's theorem involving the simple length spectra. To achieve this, we observe a rigidity of length identities over the Teichmüller space, which reads off subsurfaces of low complexity. In this talk, we explain this rigidity and discuss why our argument also applies to infinite-type surfaces. This is joint work with Hyungryul Baik and Dongryul M. Kim.

**March**** ****8****, ****9****, 1****0, 11**** (M****, T, W, ****Th), ****10**** pm Korea (****= 8 am EST)**

Sergio Fenley (IAS/Florida State University)

*Partial hyperbolic dynamics in dimension 3.*

This will be a 4 part minicourse. We will first cover some generalities on PH (partially hyperbolic diffeomorphisms). We then restrict to dimension 3 for the ambient manifold, and discuss Pujals' conjecture, and branching foliations, which in some sense are the main technical tool to analyze certain questions about PH in dimension 3. Then we will discuss PH homotopic to the identity. This is very involved and we obtain an enormous amount of structure for such in hyperbolic manifolds and in Seifert manifolds. Any homeomorphism in a closed hyperbolic 3-manifold has a finite power homotopic to the identity. We then study more PH in hyperbolic 3-manifolds and prove that if there is a PH in such a manifold, then there is also an Anosov flow in the manifold. If there is additional time, one possible topic to cover is collapsed Anosov flows.

[Note 1], [Note 2], [Note 3], [Note 4]

**March 24 (W) 4 pm Korea **

Anthony Genevois (CNRS)

*Asymptotically rigid mapping class groups*

In this talk, I will describe asymptotically rigid mapping class groups of some surfaces and explain how they can be used to construct "braided" versions of classical groups, including Thompson's and Houghton's groups. Next, I will explain how to make our asymptotically rigid mapping class groups act on contractible cube complexes with stabilisers isomorphic to finite extensions of braid groups. The rest of the talk will be dedicated to various applications of this construction, including proofs of Funar-Kapoudjian's and Degenhardt's conjectures regarding finiteness properties of braided Ptolemy-Thompons's and Houghton's groups. (Joint work with A. Lonjou and C. Urech.)

**April 21 (W) 4 pm Korea **

Yi Liu (BICMR)

*Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups*

In this talk, I will outline a proof for showing that the profinite completion of the fundamental group determines finite-volume hyperbolic 3-manifolds up to finitely many possibilities. As one of the main steps, I will explain why the Thurston norm on the first (real) cohomology is determined by the completion.

**May 13 (Th) 4 pm Korea**

Masato Mimura (Tohoku U)

*The Green--Tao theorem for number fields*

Joint work with colleagues in Tohoku University: Wataru Kai, Shin-ichiro Seki, Akihiro Munemasa and Kiyoto Yoshino. Ben Green and Terence Tao have proved that the set of rational primes contains arbitrarily long arithmetic progressions. We establish generalizations of this theorem in the setting of (arbitrary) number fields. No serious background on number theory is assumed for this talk. For the preprint, see https://arxiv.org/abs/2012.15669

**May 27 (Th) 10 am Korea (= May 26 9 pm EST)**

Tarik Aougab (Haverford College)

*Simple length rigidity for covers*

Suppose X and Y are finite covers of a fixed hyperbolic surface S. We first show that if for all closed curves gamma on S, gamma admits a simple lift to X if and only if it does to Y, then X and Y are equivalent covers. Using similar ideas, we address the question of when two covers of a fixed hyperbolic surface are isometric when their unmarked simple length spectra agree. We outline some sufficient criteria on the covers for this and generate families of examples. This represents joint work with Max Lahn, Marissa Loving, and Nick Miller.

**July 7 (W) 4 pm Korea**

InSung Park (ICERM)

*Julia sets with conformal dimension one*

Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a post-critically finite rational map f, the Julia set $J_f$ is a fractal defined as the repeller of the dynamics of f. As a fractal embedded in the Riemann sphere, the Julia set of a post-critically finite rational map has conformal dimension between 1 and 2. The Julia set has conformal dimension 2 if and only if it is the whole sphere. However, the other extreme case, when conformal dimension=1, contains diverse Julia sets, including Julia sets of post-critically finite polynomials and Newton maps. In this talk, we show that a Julia set $J_f$ has conformal dimension one if and only if there is an f-invariant graph that has topological entropy zero. In the spirit of Sullivan’s dictionary, we also compare this result with Carrasco and Mackay's work on Gromov hyperbolic groups whose boundaries have conformal dimension one.

**July 14 (W) 4 pm Korea**

InSung Park (ICERM)

*Julia sets with conformal dimension one (continued)*

In this talk, we discuss ideas used to prove the main theorem of the last talk: the equivalence between Julia sets having conformal dimension one and crochet-type Julia sets. The main technique that we use is the conformal energy of maps between graphs, introduced by D. Thurston. The motivation of the conformal energy is to provide a "positive" characterization of post-critically finite rational maps; a "negative" characterization was given by W. Thruston in the 1980s. We will start with historical backgrounds on the characterization problem and see how the conformal energy can be used to estimate the conformal dimensions of Julia sets.

**September 1, 2, 3 (W, Th, F) 4 pm Korea**

Javier de la Nuez González (KIAS)

*An introduction to Sela's work on the first order theory of free groups*

The question of whether two non-abelian free groups of different finite ranks satisfy the same collection of first order sentences was originally formulated by Tarski in the 1950s and remained open until 2001, when it was answered in the affirmative by Z. Sela in a celebrated series of papers relying heavily on geometric group theory techniques (an independent solution is due to Kharlampovich and Myasnikov). In this series of talks I will present some of the core ideas in Sela's proof. In particular, I will give an outline of the shortening argument and two of its applications: the construction of Makanin-Razborov diagrams for the analysis of solution sets of equations over free groups and the existence of formal solutions witnessing the validity of sentences. I will assume a certain degree of familiarity with Bass-Serre theory.

**September 29 (W) 4 pm Korea**

Kiho Park (KIAS)

*Transfer operators and limit laws for typical cocycles*

We show that generic matrix cocycles (those called “typical" cocycles) over irreducible subshifts of finite type obey several limit laws with respect to the unique equilibrium states for H\”older potentials. These include the central limit theorem and the large deviation principle. The transfer operator and its spectral properties play key roles in establishing these limit laws. If time permits, we will also discuss the analytic dependence of the top Lyapunov exponent on the underlying equilibrium state. This is joint work with Mark Piraino.

**October 13 (W) 4 - 5 pm Korea**

Javier de la Nuez González (KIAS)

**Title**: Formal solutions and their role in the study of the first order theory of free groups

**Abstract**: I will discuss the existence of formal solutions for positive and related sentences valid in free groups. I will also hint at their role in Sela's solution to the Tarski problem, together with that of other tools.

**November 3 (W) 10 - 11 am Korea**

Hongbin Sun (Rutgers University)

*All finitely generated 3-manifold groups are Grothendieck rigid*

A finitely generated residually finite group G is said to be Grothendieck rigid if for any finitely generated proper subgroup H<G, the inclusion induced homomorphism \hat{H}\to \hat{G} on their profinite completions is not an isomorphism. There do exist finitely presented groups that are not Grothendieck rigid. We will prove that, if we restrict to the family of finitely generated 3-manifold groups, then all these groups are Grothendieck rigid. The proof relies on a precise description on non-separable subgroups of 3-manifold groups.

**December 2 (Th) 9 - 10 am Korea**

Minju Lee (IAS)

*Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends.*

This is joint work with Hee Oh. We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb{H}^d$ is a convex cocompact manifold with

Fuchsian ends. For $d = 3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively

homogeneous. Our results imply the following: for any $k\geq 1$,

(1) the closure of any $k$-horosphere in $M$ is a properly immersed submanifold;

(2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold;

(3) an infinite sequence of maximal properly immersed geodesic $(k+1)$-planes intersecting $\mathrm{core} M$ becomes dense in $M$.

**February 17 (Th) 9 am Korea**

Sam Nariman (Purdue)

*On perfectness of PL homeomorphisms of surfaces *

Diffeomorphism groups and PL homeomorphisms have similar algebraic properties. From the point of view of foliation theory, they both support characteristic classes known as Godbillon-Vey classes and also they both exhibit the local indicability property thanks to Thurston and Calegari-Rolfsen respectively. Thurston and Mather showed that Diff^r_0(M), the group of C^r diffeomorphisms of a closed manifold M that are isotopic to the identity, is perfect (in fact simple) for r≠ dim(M)+1. On the other hand, Epstein proved that PL_0(S^1) and PL_c(R) are perfect groups and left the question of perfectness of PL homeomorphisms of higher dimensional PL manifolds open. In this talk, we explain a homotopy theoretic approach to prove that PL homeomorphisms of surfaces are perfect.

**June 29 (W). 4 pm Korea (= 3 pm Singapore)**

Ser Peow Tan (National University of Singapore)

*Identities on the hyperbolic thrice punctured sphere.*

Classical identities like the McShane identity, Basmajian identity or Bridgeman identity either are trivial or do not hold on the thrice punctured sphere. A recent result by Basmajian, Parlier and speaker shows that by putting a grading on the cusps of the thrice punctured sphere, one contains infinitely many non-trivial identities. We discuss some interesting aspects of these identities.

**March 10 (Th) 9 -10am Korea**

Tao Li (Boston College)

Taut foliations of 3-manifolds with Heegaard genus two

Let M be a closed, orientable, and irreducible 3-manifold with Heegaard genus two. We prove that if the fundamental group of M is left-orderable then M admits a co-orientable taut foliation.

**Mar 3 (Th) 10 -11 am Korea**

**Mar 4 (F) 9:30 - 10:30 am Korea**

Lei Chen (University of Maryland, College Park)

Talk 1: Nielsen realization problem.

For the natural projection Diff(S_g) ->MCG(S_g), when does this map have a section? MCG(S_g) := \pi_0(Diff(S_g)).

Talk 2: Homomorphism between MCG(S_g) and MCG(S_h), and between different braid groups.

Even more, how can MCG(S_g) acts on spheres?

**July 4 (M), July 6 (W), 10 am Korea.**

**July 12 (Tu), 4 pm Korea**

Thomas Koberda (University of Virginia)

*Mapping class groups, curve graphs, and model theory*

In this series of talks, I will give an introduction to the model theory of the curve graph.

Lecture 1. The curve graph and its automorphism. In the first talk, I will discuss some of the combinatorial and geometric properties of the curve graph of an orientable surface of finite type, and how its structure relates to the study of the mapping class group of the surface. I will concentrate on automorphisms of curve graphs, and discuss the context surrounding Ivanov's metaconjecture concerning "natural" graphs associated to surfaces and their automorphisms.

Lecture 2. An introduction to model theory. I will survey some of the basic notions of model theory, with examples. The main points will be the notion of interpretability, quantifier elimination, stability, and Morley rank. Finally, I will formulate Ivanov's metaconjecture as a precise model-theoretic statement.

Lecture 3. The model theory of the curve graph. I will survey joint work with V. Disarlo and J. de la Nuez Gonzalez about the model theory of the curve graph. Topics will include quantifier elimination, $\omega$-stability, non-definability of certain natural subsets of the curve graph, lack of mutual interpretability between certain geometric graphs, and interpretation rigidity between curve graphs.

**August 12 (F), 10 am Korea**** (hybrid)**

**Note the date/location change!**

**In-Person : KIAS 8101**

**Online : ****http://cayley.kr/vz**

Mark Pengitore (UVa)

*Residual finiteness of the mapping class group with respect to solvable covers*

*In this talk, we will quantify residual finiteness of the mapping class group of finite type with respect to congruence quotients coming from characteristic finite index covers of the underlying surface with solvable deck group. We refer to these quoteints as congruence quotients of solvable type. In particular, we construct an infinite sequence of mapping classes where the minimal congruence quotient of solvable type that detects one of these mapping classes has size which is super polynomial in word length.*

**August 17, 18, 19 (W/Th/F), 10 am Korea**** (hybrid)**

**In-Person : KIAS 1503**

**Online : http://cayley.kr/vz**

Damian Osajda (Universitoy of Wrocław, IMPAN)

*Locally elliptic actions and nonpostitive curvature*

The talks are based on joint works with Karol Duda, Thomas Haettel, Sergey Norin, and Piotr Przytycki.

There are numerous questions concerning actions of torsion groups on nonpositivelycurved spaces. One example is a well-known conjecture stating that subgroups of CAT(0) groups, that is, of groups acting geometrically on CAT(0)spaces, do notc ontain infinite torsion (every element has finite order) subgroups.Proving this conjecture might be seen as a first step towards establishing the Tits Alternative for CAT(0) groups. Notethat for torsiongroups every element acts elliptically on the CAT(0) space, that is, fixes a point. This leads to a concept of locally elliptic actions, that is, actions in which the fixed point set of each element is nonempty.

Generalizing the above conjecture and a number of other related open questions we state the following Meta-Conjecture: Every locally elliptic actionof a finitely generated group on a finite dimensional nonpositively curved complex is elliptic, that is, has a global fixed point. Here, `nonpositively curved complex' could mean: CAT(0), Gromov hyperbolic, small cancellation, systolic, or Helly polyhedral complex, and many others.`Finite dimensional' could refer to finite dimensionality as a polyhedral complex.The assumptions in the Meta-Conjecture are important. Every not finitely generated group acts on a tree without a fixed point. Some infinite torsion groups, even some infinite Burnside groups (there is a common bound on orders of elements) act without fixed points on infinite dimensional CAT(0) spaces. Also the combinatorial setting, that is, considering complexes instead of arbitrary metric spaces is essential.

In the talks I will explain in details motivations for the Meta-Conjecture, some of its consequences, and relations to well-known open problems. I will present the actualconjectures being specifications of the Meta-Conjecture and explain state of the art,focusing on recent results of my collaborators and myself, and providing (ideas of) proofs.

**August 24 (W), 10 am Korea (hybrid)**

**In-Person : KIAS 1423**

**Online : http://cayley.kr/vz**

Seung-Yeon Ryoo (Princeton)

*Vertical versus horizontal inequalities on nilpotent Lie groups and groups of polynomial growth*

It is known that simply connected nonabelian nilpotent Lie groups and not virtually abelian groups of polynomial growth fail to embed bilipschitzly into uniformly convex Banach spaces, because these groups have Carnot groups as asymptotic cone and because Carnot groups fail to embed bilipschitzly into uniformly convex Banach spaces by the Pansu--Semmes nonembeddability argument. We quantify this fact by providing a lower bound on the distortion of balls in the aforementioned groups into uniformly convex spaces. In particular, we show that the $L^p$-distortion, $(1<p<\infty)$, of a ball of radius $n\ge 2$ in the aforementioned groups is exactly $(\log n)^{1/\max\{p,2\}}$ up to constants. We achieve this by establishing ``vertical-versus-horizontal Poincaré inequalities'' which are specifically tailored to measuring the bilipschitz distortion of these balls and which demonstrate a collapse of embeddings along central directions. We provide some conjectural bounds on the L^1-distortion of these balls.

**September 6 (Tu) and 8 (Th), 10 am**

**In-person** KIAS 1503

**Online **http://cayley.kr/vz

Ken'ichi Ohshika (Gakushuin University)

*Finsler metrics on Teichmuller space and their rigidity (part I and part II)*

In the first part, I shall talk about recent joint work with Yi Huang and Athanase Papadopoulos on convex geometry of cotangent spaces of Teichmuller space with respect to Thurston’s asymmetric metric. As an application, I shall present the infinitesimal rigidity of Thurston’s metric under the action of the extended mapping class group.

In the second part, I shall talk about joint work in progress with Yi Huang, Huiping Pan and Ahanase Papadopoulos on the earthquake metric on Teichmuller space. The earthquake metric is a new Finsler metric introduced by Thurston, but has not been studied since then. I shall present its definition, basic properties first, and then show its duality to Thurston’s metric, which implies the infinitesimal rigidity of the earthquake metric. I shall also show that the metric is incomplete just like the Weil-Petersson metric.

**October 13 (Th) and 14 (F), 10 am**

**Online **http://cayley.kr/vz

Gromov boundary extended

Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it has been an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce two topological spaces that are natural analogs of the Gromov boundary for a larger class of metric spaces. First we construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space that can be associated to all finitely generated groups. Furthermore, for many groups, the sublinear boundary can be identified with the Poisson boundaries of the associated group, thus providing a QI-invariant model for Poisson boundaries. This result answers the open problems regarding QI-invariant models of CAT(0) groups and the mapping class group. Lastly, for a subset of the metric spaces we define a compactification of the sublinearly Morse boundary and show that in these cases they are naturally identified with the Bowditch boundary. This is a series of joint work with Kasra Rafi and Giulio Tiozzo.