# 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 Math Department Youtube channel (click for playlist). Seminars postings and can also be found on the Illinois Mathematics Department Events Calendar.

Archive of past lectures.

**Current Semester**

**12:00 pm, Thursday, December 7, 2017, 243 Altgeld Hall
Least Dilatation of Pure Surface Braids
Marissa Loving (Illinois Math)**

The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface.

For the n=1 case, much is known about this group including upper and lower bounds on the least dilatation of its pseudo-Anosovs due to Dowdall and Aougab—Taylor. I am interested in the least dilatation of pseudo-Anosov pure surface braids for n>1 punctures. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.

**12:00 pm, Tuesday, December 5, 2017, 243 Altgeld Hall
Characteristic random subgroups and their applications
Rostyslav Kravchenko (Northwestern University)**

The invariant random subgroups (IRS) were implicitly used by Stuck and Zimmer in 1994 and defined explicitly by Abert, Glasner and Virag in 2012. They were actively studied since then. We define the notion of characteristic random subgroups (CRS) which are a natural analog of IRSs for the case of the group of all automorphisms. We determine CRS for free abelian groups and for free groups of finite rank. Using our results on CRS of free groups we show that for groups of geometrical nature (like hyperbolic groups, mapping class groups and outer automorphisms groups) there are infinitely many continuous ergodic IRS. This is a joint work with R. Grigorchuk and L. Bowen

Video

**12:00 pm, Thursday, November 30, 2017, 243 Altgeld Hall
Ilya Gekhtman (Yale University)
Green metric, Ancona inequalities and Martin boundary for relatively hyperbolic groups
**Generalizing results of Ancona for hyperbolic groups, we prove that a random path between two points in a relatively hyperbolic group (e.g. a nonuniform lattice in hyperbolic space) has a uniformly high probability of passing any point on a word metric geodesic between them that is not inside a long subsegment close to a translate of a parabolic subgroup. We use this to relate three compactifications of the group: the Martin boundary associated with the random walk, the Bowditch boundary, associated to an action of the group on a proper hyperbolic space, and the Floyd boundary, obtained by a certain rescaling of the word metric. We demonstrate some dynamical consequences of these seemingly combinatorial results. For example, for a nonuniform lattice G in hyperbolic space H^n, we prove that the harmonic (exit) measure on the boundary associated to any finite support random walk on G is singular to the Lebesgue measure. Moreover, we construct a geodesic flow and G invariant measure on the unit tangent bundle of hyperbolic space projecting to a finite measure on T^1H^n/G whose geodesic current is equivalent to the square of the harmonic measure. The axes of random loxodromic elements in G equidistribute with respect to this measure. Analogous results hold for any geometrically finite subgroups of isometry groups of manifolds of pinched negative curvature, or even proper delta-hyperbolic metric spaces.

Video

**12:00 pm, Tuesday, November 28, 2017, 243 Altgeld Hall
Laura Schaposnik (University of Illinois at Chicago)
On Cayley and Langlands type correspondences for Higgs bundles
**The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayleyand Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

Video

**12:00 pm, Tuesday, November 14, 2017, 243 Altgeld Hall
Charles Frohman (University of Iowa)
Skeins and Characters**

Skein theory is a K-theoretic like construction. Think of the underlying three- manifold as a ring, and a link in that manifold as a projective module. Crossings correspond to extensions of one module by another, and the skein relation says that the extension is equivalent to the direct sum of the two links that it extends. The skein module is the K-group from this relation. If the underlying three manifold is a cylinder over a surface, the links act like a category of bimodules, and the skein module is an algebra. In the talk, I will define the Kauffman bracket skein algebra and describe its properties.

Video

**4:00 pm, Monday, November 13, 2017, 343 Altgeld Hall
Jenny Wilson (Stanford University)
Stability in the homology of configuration spaces**

This talk will illustrate some patterns in the homology of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena – relationships between unstable homology classes in different degrees – established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius–Kupers–Randal-Williams.

Video

**12:00 pm, Thursday, October 26, 2017, 243 Altgeld Hall
Jayadev Athreya (University of Washington)
Siegel-Veech transforms are in L2
**Let H denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on R2 is in L2(H,μ), where μ is the Masur-Veech measure on H, and give applications to bounding error terms for counting problems for saddle connections. We will review classical results in the Geometry of Numbers which anticipate this result. This is joint work with Yitwah Cheung and Howard Masur.

Video

**12:00 pm, Thursday, October 19, 2017, 243 Altgeld Hall
Maxime Bergeron (University of Chicago)
The Topology of Representation Varieties
**Let H be a finitely generated group, let G be a complex reductive algebraic group (e.g. a special linear group) and let K be a maximal compact subgroup of G (e.g. a special unitary group). I will discuss exceptional classes of groups H for which there is a deformation retraction of Hom(H,G) onto Hom(H,K), thereby allowing us to obtain otherwise inaccessible topological invariants of these representation spaces.

Video

**
12:00 pm, Tuesday, October 10, 2017, 243 Altgeld Hall
Alex Wright (Stanford)
Dynamics, geometry, and the moduli space of Riemann surfaces
**The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

Video

**12:00 pm, Thursday, September 28, 2017, 243 Altgeld Hal
Mark Pengitore (Purdue University)
Effective Twisted Conjugacy Separability of Nilpotent Groups
**There has been a recent interest in providing effective proofs of separability properties such as residual finiteness and conjugacy separability. Unlike residual finiteness, conjugacy separability does not respect finite extensions. Thus, we introduce twisted conjugacy separability, originally defined by Fel'shtyn, in order to study effective conjugacy separability of finite extensions of conjugacy separable groups. In joint work with Jonas Dere, we provide an effective proof of twisted conjugacy separability of finitely generated nilpotent groups. That, in turn, provides an effective bound for conjugacy separability of all finite extensions of a fixed nilpotent group in terms of the asymptotic behavior of conjugacy separability of the base nilpotent group.

**12:00 pm, Tuesday, September 26, 2017, 243 Altgeld Hal
David Dumas (University of Illinois at Chicago)
Limits of cubic differentials and projective structures
**A construction due independently to Labourie and Loftin identifies the moduli space of convex RP^2 structures on a compact surface S with the bundle of holomorphic cubic differentials over the Teichmueller space of S. We study pointed geometric limits of sequences that go to infinity in this moduli space while remaining over a compact set in Teichmueller space. For such a sequence, we construct a local limit polynomial (in one complex variable) which describes the rate and direction of accumulation of zeros of the cubic differentials about the sequence of base points. We then show that this polynomial determines the convex polygon in RP^2 that is the geometric limit of the images of the developing maps of the projective structures. This is joint work with Michael Wolf.

Video

**12:00 pm, Thursday, September 21, 2017, 243 Altgeld Hal
Xin Zhang (University of Illinois at Urbana-Champaign)
Pair correlation in Apollonian circle packings
**Abstract: Consider four mutually tangent circles, one containing the other three. An Apollonian circle packing is formed when the remaining curvilinear triangular regions are recursively filled with tangent circles. The extensive study of this object in the last fifteen years has led to many beautiful theorems in number theory, graph theory, and homogeneous dynamics. In this talk I will discuss a new type of problems, which concern the fine scale structure of Apollonian circle packings. In particular, I will show that the limiting pair correlation of circles exists. A critical tool we use is an extended version of a theorem of Mohammadi-Oh on the equidistribution of expanding horospheres in infinite volume hyperbolic spaces. This work is motivated by an IGL project that I mentored in Spring 2017.

**12:00 pm, Thursday, September 14, 2017, 243 Altgeld Hall
**

**Kelly Yancey (Institute for Defense Analyses)**

**Self-Similar Interval Exchange Transformations**

Abstract: During this talk we will discuss the class of self-similar 3-IETs and show that they satisfy Sarnak's conjecture. We will do this by appealing to the theory of joinings. Specifically we will show how to prove the property of minimal self-joinings for substitution systems (self-similar IETs can be thought of in this context).

**12:00 pm, Tuesday, September 5, 2017, 243 Altgeld Hall
Moon Duchin (Tufts University)
Curvature of graphs
**Abstract: I'll discuss some ideas for measuring curvature of graphs that carry over to the setting of large finite graphs, including discrete Ricci curvature and cotangent-weighting. There's an interesting interplay of ideas from pure math (geometric group theory) and theoretical computer science (mesh clustering and smoothing), with potential practical applications to the study of electoral redistricting.

**12:00 pm, 243 Altgeld Hall,Tuesday, August 29, 2017
Rebecca Winarski (University of Wisconsin, Milwaukee)
The twisted rabbit problem via the arc complex
**Abstract: The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one? After remaining open for 25 years, this problem was solved by Bartholdi—Nekyrashevych using iterated monodromy groups. In joint work with Lanier and Margalit, we formulate the problem topologically and solve the problem using the arc complex.