# 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, 243 Altgeld Hall,Thursday, September 21, 2017
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 Bartholdiâ€”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.