Topology and Geometric Group Theory Seminars at the Ohio State University

Jan 13 -

Jan 15 - 

Jan 20 - Jan 22 - Special events: Zassenhaus Lecture series by June Huh 

Jan 27 - 

Jan 29 - 

Feb 3 - 

Feb 5 - Doron Puder (IAS, Tel Aviv)    

Joint speaker with the Combinatorics Seminar

 Invariants of words from random matrices

Let w be a word in a free group and let G be a finite (or more generally compact) group. A w-random element of G is obtained by substituting the letters of w with uniform random elements from G. For example, if w=xyxy^{-2}, the random element is ghgh^{-2}, with g and h independent uniformly random elements of G. Composing with linear representations of G, we get w-random matrices. A series of works over the last decade has revealed many intriguing phenomena around w-random elements in nice families of groups, such as the symmetric groups or the unitary groups. In particular, many invariants of words, some new and some well-known, play significant roles in this theory.

This story involves probability, combinatorics, topology, algebra and representation theory. In the talk, where all notions will be explained, I will try to give a flavor of this interesting theory.

Feb 10 - 

Feb 12 - Zhenghao Rao (Rutgers)

Immersed incompressible surfaces in hyperbolic manifolds

        The study of surface subgroups in 3-manifolds has drawn sustained attention for decades, motivated both by their intrinsic geometric richness and by their broad consequences in geometric topology, geometric group theory, and dynamics. A landmark result is the Surface Subgroup Theorem of Kahn–Markovic, which asserts that every cocompact Kleinian group contains a closed surface subgroup. In this talk, we will survey key developments in the subject and highlight our recent progress, including joint work with Jeremy Kahn and with Xiaolong Han and Jia Wan.

Feb 17 - 

Feb 19 - Katherine Goldman (McGill)

Injective metrics and affine hyperplane arrangements

A complex affine hyperplane arrangement is a locally finite collection of affine hyperplanes (complex codimension-1 subspaces) in a finite dimensional complex affine space. Since these subspaces have complex codimension 1, the complement of their union is a connected manifold. It is a broad, longstanding problem with many connections to different areas of mathematics to determine the arrangements for which this manifold is aspherical (has contractible universal cover). A subset of this problem dating back to the 1970s, commonly attributed to Arnol’d, Brieskorn, Thom, and Pham, concerns arrangements arising from reflection groups in real affine space.

One approach that has seen success is to construct a cell complex which is homotopy equivalent to this complement and endow it with some kind of ("singular") non-positive curvature. Along these lines, by showing that a specific cell complex (based on a construction of Falk) carries a so-called injective metric, we show that a broad class of affine arrangements (including the infinite families of affine reflection arrangements, modulo a conjecture about D_n-type) have aspherical complement. In particular, this provides some of the first examples of infinite affine arrangements which have aspherical complement, but do not arise from reflection groups. This is joint work with Jingyin Huang.

Feb 24 -  Inhyeok Choi  (KIAS)

Random subgraphs of the mapping class group

In percolation theory, one samples a random subgraph of a given graph by independently deleting each edge with probability 1-p. Benjamini and Schramm asked if a Cayley graph is nonamenable if and only if there exists a number p such that the random p-subgraph has infinitely many unbounded components. In this talk, I will explain an answer to this conjecture for the mapping class group and irreducible cubical groups. This is joint work with Donggyun Seo.

Feb 26 - Pratit Goswami (University of Oklahoma)

Dehn Functions of Coabelian Subgroups

The study of Dehn functions has developed into a major area of research in geometric group theory mainly because the growth types of these functions are quasi-isometry invariants of finitely presented groups. The Dehn function of a finitely presented group G is also connected to the complexity of solving the word problem in G namely, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation is recursive. In this talk, we will discuss new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. This is joint work with Noel Brady and Rob Merrell.

Mar 3 - Leslie Mavrakis (Utah)

Mar 5 - Xiangdong Xie (BGSU)

       Mar 10 - John Etnyre (Georgia Tech)

Mar 12 - Wenyu Pan (U Toronto)

Mar 24 - 

Mar 26 - Gil Goffer (UCSD)

Mar 31 - Thang Nguyen (FSU)

April 2 - 

April 7 -  Corey Bregman (Tufts)

April 9-  Hyunki Min (UGa)

April 14 - Lizzy Teryoshin (UCSD)

April 16 - Yunhui Wu (Tsinghua University)

April 21 - Ishan Banerjee (OSU)

April 23 - Ishan Banerjee (OSU)