# Geometry, Groups and

Dynamics/GEAR Seminar

The Geometry, Groups and Dynamics/GEAR Seminar, held at the Illinois hub of GEAR, features speakers of interest to network members. The talks are live streamed on the Illinois Department of Mathematics Youtube channel playlist. Seminars postings can also be found on the Illinois Mathematics Department Events Calendar.

Archive of past lectures.

##### Spring 2019

**12:00 pm, Tuesday, April 30, 2019, 243 Altgeld Hall
Richard Canary (University of Michigan)
Topological Restrictions on Anosov Representations**

Abstract: The theory of Anosov representations was introduced by Francois Labourie in his study of Hitchin representations. They have emerged as the natural analogue, for higher rank Lie groups, of Fuchsian representations, or more generally convex compact representations into rank one Lie groups. We will give a gentle introduction to Anosov representations, followed by a discussion of topological restrictions on the groups which admit Anosov representations into SL(d,R). For example, we will see characterizations of groups admitting Anosov representations into SL(3,R) and SL(4,R) and restrictions on the cohomological dimension for all values of d. (This is joint work with Kostas Tsouvalas.)

**12:00 pm, Thursday, April 25, 2019, 243 Altgeld Hall
Viveka Erlandsson (University of Bristol)
Mirzakhani's curve counting**

Abstract: Mirzakhani proved two theorems about the asymptotic growth of the number of curves in a mapping class group orbit on a surface: one for simple curves and another for general curves. In this talk I will explain how to derive her second theorem from the one about simple curves. Time permitting, I will explain why similar methods can be used to also give a proof for the theorem about simple curves, hence giving a new (and very different) proof of both theorems.

**12:00 pm, Thursday, April 18, 2019, 243 Altgeld Hall
Jean-Pierre Mutanguha (University of Arkansas)
Immersions and Laminations on Free Groups**

Abstract: Using pullbacks, we proved that mapping tori of graph immersions have word-hyperbolic fundamental groups if and only if they have no Baumslag-Solitar subgroups. We will then use laminations to describe an efficient algorithm that determines whether such groups are word-hyperbolic.

**12:00 pm, Tuesday, April 9, 2019, 243 Altgeld Hall
Haizi Yu (University of Illinois)
Geometry Group Theory in Music AI**

Abstract: Is it possible to invent an AI to learn important music concepts directly from sheet music? Is it possible to do this in a human-interpretable form that resembles known music theory and also suggests new theory? We apply our generally developed automatic concept learning model to the domain of music, so as to tackle the above questions and the like. Starting from a connection between existing music concepts and their group-theoretic interpretations, we propose a formal representation of music objects as well as their abstractions and probabilistic patterns. This proposed representation not only reveals internal music structures as mathematical symmetries, but more importantly, are also operational in computational models. As a result, this further yields a learning algorithm that couples knowledge from geometric group theory and statistical inference to automatically discover music concepts without human intervention. Lastly, we briefly demonstrate an ongoing project, called MUS-ROVER, which builds a real web application that delivers to people automatically discovered music rules and concepts, teaching us music composition in a designated style.

**12:00 pm, Tuesday, April 2, 2019, 243 Altgeld Hall
Tarik Aougab (Brown University)
Weil-Petersson analogs for graphs**

Abstract: The Weil-Petersson metric is a Riemannian metric on the Teichmuller space which is natural in the sense that it comes from and reflects the geometry of the hyperbolic metrics on the underlying surface. Motivated by foundational work of McMullen, Pollicott-Sharp (and later Kao) proposed an analogous metric for the moduli space of metrics on a fixed graph. We study this metric and completely characterize its completion in the case of a rose. In this talk, we’ll introduce the Weil-Petersson metric and do our best to motivate the definitions so that no advanced prior knowledge of the subject will be necessary. This represents joint work with Matt Clay and Yo’av Rieck.

**12:00 pm, Thursday, March 28, 2019, 243 Altgeld Hall
Sunny Xiao (Brown University)
Classifying incompressible surfaces in hyperbolic mapping tori
**Abstract: One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path which satisfies certain combinatorial properties in the arc complex of the 4-punctured sphere. This extends and generalizes results of Floyd, Hatcher, and Thurston.

**12:00 pm, Thursday, March 14, 2019, 243 Altgeld Hall
Pierre Will (Institut Fourier)
SL(3,C), SU(2,1) and the Whitehead link complement
**Abstract: In this talk, I will explain how it is possible to construct interesting geometric structures modeled on the boundary at infinity of the complex hyperbolic 2-space. In particular, I will describe examples of hyperbolic 3-manifolds that appear this way. This talk is based on joint works with Antonin Guilloux, and John Parker.

**12:00 pm,Thursday, March 7, 2019, 243 Altgeld Hall
Caglar Uyanik (Yale)
Atoroidal dynamics of subgroups of Out(FN)
**Abstract: I will discuss several examples to illustrate how the dynamics of the Out(FN) action on various spaces reflects on the algebraic structure of the Out(FN)itself. In particular, I will talk about a new subgroup classification theorem for Out(FN) which is joint work with Matt Clay.

**12:00 pm,Tuesday, February 26, 2019, 243 Altgeld Hall
Autumn Kent (University of Wisconsin)
Congruence subgroups in genus one
**Abstract: I’ll discuss a proof of Asada’s theorem that mapping class groups of punctured tori have the congruence subgroup property.

**12:00 pm, Thursday, February 21, 2019, 243 Altgeld Hall
Ying Hu (University of Nebraska-Omaha)
Taut foliations and left-orderability of 3 manifold groups**

Abstract: A group G is called left-orderable if there exists a strict total order on G which is invariant under the left-multiplication. Given an irreducible 3-manifold M, it is conjectured that the fundamental group of the 3-manifold is left-orderable if and only if M admits a co-orientable taut foliation. In this talk, we will discuss the left-orderability of the fundamental groups of 3-manifolds that admit co-orientable taut foliations.

**12:00 pm, Thursday, February 14, 2019, 243 Altgeld Hall
Marissa Loving (Illinois Mathematics)
Spectral Rigidity of q-differential Metrics**

Abstract: When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.

**12:00 pm, Tuesday, February 5, 2019, 243 Altgeld Hall
Anja Randecker (University of Toronto)
Asymptotics of the expected diameter of translation surfaces**

Abstract: For the hyperbolic structure on a Riemann surface, Mirzakhani has proven asymptotics of the expected diameter for large genus surfaces. An abelian differential equips a Riemann surface with a translation structure. In joint work with Howard Masur and Kasra Rafi, we prove asymptotics for large genus translation surfaces of area 1. Unlike in the case of hyperbolic surfaces, the expected diameter goes to zero as the genus goes to infinity.

**12:00 pm, Tuesday, January 29, 2019, 243 Altgeld Hall**

Diaaeldin Taha (University of Washington)

The Farey Sequence Next-Term Algorithm, and the Boca-Cobeli-Zaharescu Map Analogue for Hecke Triangle Groups G_q

Abstract: The Farey sequence is a famous enumeration of the rationals that permeates number theory. In the early 2000s, F. Boca, C. Cobeli, and A. Zaharescu encoded a surprisingly simple algorithm for generating--in increasing order--the elements of each level of the Farey sequence as what grew to be known as the BCZ map, and demonstrated how that map can be used to study the statistics of subsets of the Farey fractions. In this talk, we present a generalization of the BCZ map to all Hekce triangle groups G_q, q \geq 3, with the G_3 = SL(2, \mathbb{Z}) case being the "classical" BCZ map. If time permits, we will present some applications of the G_q-BCZ maps to the statistics of the discrete G_q linear orbits in the plane \mathbb{R}^2 (i.e. the discrete sets \Lambda_q = G_q (1, 0)^T).

Diaaeldin Taha (University of Washington)

The Farey Sequence Next-Term Algorithm, and the Boca-Cobeli-Zaharescu Map Analogue for Hecke Triangle Groups G_q

**12:00 pm, Thursday, January 24, 2019, 243 Altgeld Hall
Joshua Pankau (University of Iowa)
Stretch Factors Coming From Thurston's Construction**

Abstract: Associated to every pseudo-Anosov map is a real number called its stretch factor. Thurston proved that stretch factors are algebraic units, but it is unknown exactly which algebraic units are stretch factors. In this talk I will discuss a construction of pseudo-Anosov maps due to Thurston, and discuss my recent results where I classified (up to power) the stretch factors coming from this construction. We will primarily focus on a specific class of algebraic units known as Salem numbers. This talk is intended to be accessible to everyone.

** 12:00 pm, Thursday, January 17, 2019, 243 Altgeld Hall
Ilya Kapovich (Hunter College)
Index properties of random automorphisms of free groups**

Abstract: For automorphisms of the free group Fr, being "fully irreducible" is the main analog of the property of being a pseudo-Anosov element of the mapping class group. It has been known, because of general results about random walks on groups acting on Gromov-hyperbolic spaces, that a "random" (in the sense of being generated by a long random walk) element ϕ of Out(Fr) is fully irreducible and atoroidal. But finer structural properties of such random fully irreducibles ϕ∈Out(Fr) have not been understood. We prove that for a "random" ϕ∈Out(Fr) (where r≥3), the attracting and repelling ℝ-trees of ϕ are trivalent, that is all of their branch points have valency three, and that these trees are non-geometric (and thus have index <2r−2). The talk is based on a joint paper with Joseph Maher, Samuel Taylor and Catherine Pfaff.

**Fall 2018**

**12:00 pm, Tuesday, November 13, 2018, 243 Altgeld Hall
Margaret Nichols (University of Chicago)
Taut sutured handlebodies as twisted homology products**

Abstract: A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’. One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such? We explore some classes of relatively simple sutured manifolds, and see one class is always a rational homology product, but that the next natural class contains examples which require twisting. We also find examples that require twisting by a representation which cannot be ‘too simple’.

**12:00 pm, 243 Altgeld Hall,Tuesday, October 2, 2018
Alex Zupan (University of Nebraska-Lincoln)
Generalized square knots and the 4-dimensional Poincare Conjecture**

Abstract: The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. In this talk, we consider homotopy 4-spheres X built with two 2-handles and two 3-handles, which are uniquely determined by the attaching link L for the 2-handles in the 3-sphere. We prove that if one of the components of L is the connected sum of a torus knot T(p,2) and its mirror (a generalized square knot), then X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.

**12:00 pm, 243 Altgeld Hall,Tuesday, September 25, 2018
Justin Lanier (Georgia Tech Math)
Normal generators for mapping class groups are abundant**

Abstract: For mapping class groups of surfaces, we provide a number of simple criteria that ensure that a mapping class is a normal generator, with normal closure equal to the whole group. We then apply these criteria to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator whenever genus is at least 3. We also show that every pseudo-Anosov mapping class with stretch factor less than √2 is a normal generator. Showing that pseudo-Anosov normal generators exist at all answers a question of Darren Long from 1986. In addition to discussing these results on normal generators, we will describe several ways in which they can be leveraged to answer other questions about mapping class groups. This is joint work with Dan Margalit.

**12:00 pm, 243 Altgeld Hall,Thursday, September 20, 2018
Moira Chas (Stony Brook)
The generalization of the Goldman bracket to three manifold and its relation to Geometrization**

Abstract: In the eighties, Bill Goldman discovered a Lie algebra structure on the free abelian group with basis the free homotopy classes of closed oriented curves on an oriented surface S. In the nineties, jointly with Dennis Sullivan, we generalized this Lie algebra structure to families of loops (defining the equivariant homology of the free loop space of a manifold). This Lie algebra, together with other operations in spaces of loops is now known as String Topology. The talk will start with a discussion of the Goldman Lie bracket in surfaces, and how it "captures" the geometric intersection number between curves. It will continue with the description of the string bracket, which generalizes of the Goldman bracket to oriented manifolds of dimension larger than two, and the space of families of loops where the string bracket is defined. The second part of the lecture describes how this structure in degrees zero and one plus the power operations in degree zero recognizes key features of the Geometrization, the above mentioned joint work. The lions share of effort concerns the torus decomposition of three manifolds which carry mixed geometry. This is joint work with Siddhartha Gadgil and Dennis Sullivan.

**12:00 pm, 243 Altgeld, Tuesday, September 11, 2018
Eric Samperton (UCSB)
From the dynamics of surface automorphisms to the computational complexity of 3-manifolds**

Abstract: Every 3-manifold admits a Heegaard splitting, and many 3-manifold invariants admit formulas using Heegaard splittings. These facts are one starting point for a common theme in the study of 3-manifolds: one can relate various topological or geometric properties of 3-manifolds to dynamical systems in 1 or 2 dimensions. We’ll explore this theme in the context of computational complexity. I’ll start with two examples (coloring invariants and the Jones polynomial) that translate dynamical properties of mapping class group actions into complexity-theoretic hardness properties of 3-manifold invariants. I’ll conclude with some brainstorming about future directions. I will introduce all of the necessary complexity theory as we go.