Geometry/Topology Seminar
Spring 2016
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday March 31 at 34pm in Eck 308
 Kevin Kordek, Texas A&M
 Mapping class groups and the topology of the hyperelliptic locus

Abstract: The hyperelliptic mapping class group is
the subgroup of the mapping class group of a closed
orientable surface whose elements commute with a fixed
hyperelliptic involution. This group and its principal
congruence subgroups are important not only in geometric
topology and group theory, but also in algebraic geometry,
where they appear as fundamental groups of the components of
the hyperelliptic loci in various moduli spaces of Riemann
surfaces. In this talk, I will summarize what is known about
the grouptheoretic and topological structure of these
objects, describe a few open problems, and report on some
recent partial progress.

 Wednesday April 13 at 1:302:30pm in RY358
 Zinovy Reichstein, UBC
 Minicourse  Introduction to essential dimension

Abstract: Informally speaking, the essential
dimension of an algebraic object is the minimal number of
independent parameters one needs to define it. In
particular, one can associate a nonnegative integer to
every algebraic group G by considering the maximal value of
essential dimension for Gtorsors over fields. This
numerical invariant, called the essential dimension of G,
has been extensively studied since the mid1990s; yet, its
exact value remains unknown for many groups. The purpose of
my lecture will be to survey some of the work in this area.
After motivating and defining the notion of essential
dimension, I will focus on two specific techniques for
proving lower bounds on the essential dimension of a linear
algebraic group: the fixed point method and the Brauer
obstruction. I will also discuss a number of examples and
open problems. (The minicourse will also meet on Friday at
the same time and place)

 Thursday April 14 at 34pm in Eck 308
 Haomin Wen, Notre Dame
 Lens rigidity and scattering rigidity in two dimensions.

Abstract: Scattering rigidity of a Riemannian
manifold allows one to tell the metric of a manifold with
boundary by looking at the directions of geodesics at the
boundary. Lens rigidity allows one to tell the metric of a
manifold with boundary from the same information plus the
length of geodesics. There are a variety of results about
lens rigidity but very little is known for scattering
rigidity. We will discuss the subtle difference between
these two types of rigidities and prove that they are
equivalent for a large class of twodimensional Riemannian.
In particular, twodimensional simple Riemannian manifolds
(such as the flat disk) are scattering rigid since they are
lens/boundary rigid (PestovUhlmann, 2005).

 Thursday April 28 at 34pm in Eck 308
 Matthew Day, University of Arkansas
 Link splitting complexes for rightangled Artin groups

Abstract: I will review several topics in geometric
group theory: automorphism groups, row reduction of
matrices, Stallings's folding of graphs, actions on cube
complexes, and curve complex analogues. I will also connect
all these things: we use actions on cube complexes to build
a curve complex analogue for rightangled Artin groups, and
we use it define algorithms relating to their automorphism
groups. I will give many examples. This is work in progress,
joint with Henry Wilton.

 Thursday May 5 at 34pm in Eck 308
 Nate Harman, MIT
 The completed space of symmetric group representation categories

Abstract: Deligne defined a family of categories
Rep(S_{t}) depending on a continuous
parameter t, which he claimed "interpolate" the
categories of representations of symmetric groups over a
field of characteristic zero. I will provide a different
interpretation for what these categories are doing and
explain the role they play in what I call the completed
space of symmetric group representation categories. Special
emphasis will be put on the relationship between this theory
and the theory of FImodules and representation stability.

 Thursday June 2 at 34pm in Eck 308
 Spencer Dowdall, Vanderbilt
 Splittings, suspension flows, and polynomials for freebycyclic groups

Abstract: A freebycyclic group (i.e., semidirect
product of a finiterank free group with the integers) can
often be expressed as such a product in infinitely many
ways. This talk will explore this phenomenon and work
towards 1) describing the structure of the family of such
splittings of a given group, and 2) looking for for
relationships between the splittings themselves. This
discussion will involve BieriNeumannStrebel invariants,
the dynamics of semiflows and free group automorphisms, and
a polynomial invariant for certain freebycyclic groups
that ties all of this together. Joint work with Ilya
Kapovich and Christopher Leininger.
For questions, contact