# 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, 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.

**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.

**12:00 pm, Tuesday, October 31, 2017, 243 Altgeld Hall
CANCELLED
**

**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.

**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.