Geometry/Topology Seminar
Spring 2016
Thursdays (and sometimes Tuesdays) 3-4pm, in
Eckhart 308
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- Thursday March 31 at 3-4pm in Eck 308
- Kevin Kordek, Texas A&M
- Mapping class groups and the topology of the hyperelliptic locus
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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 group-theoretic and topological structure of these
objects, describe a few open problems, and report on some
recent partial progress.
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- Wednesday April 13 at 1:30-2:30pm in RY358
- Zinovy Reichstein, UBC
- Mini-course -- Introduction to essential dimension
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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 non-negative integer to
every algebraic group G by considering the maximal value of
essential dimension for G-torsors over fields. This
numerical invariant, called the essential dimension of G,
has been extensively studied since the mid-1990s; 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 mini-course will also meet on Friday at
the same time and place)
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- Thursday April 14 at 3-4pm in Eck 308
- Haomin Wen, Notre Dame
- Lens rigidity and scattering rigidity in two dimensions.
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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 two-dimensional Riemannian.
In particular, two-dimensional simple Riemannian manifolds
(such as the flat disk) are scattering rigid since they are
lens/boundary rigid (Pestov--Uhlmann, 2005).
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- Thursday April 28 at 3-4pm in Eck 308
- Matthew Day, University of Arkansas
- Link splitting complexes for right-angled Artin groups
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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 right-angled 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.
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- Thursday May 5 at 3-4pm in Eck 308
- Nate Harman, MIT
- The completed space of symmetric group representation categories
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Abstract: Deligne defined a family of categories
Rep(St) 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 FI-modules and representation stability.
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- Thursday June 2 at 3-4pm in Eck 308
- Spencer Dowdall, Vanderbilt
- Splittings, suspension flows, and polynomials for free-by-cyclic groups
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Abstract: A free-by-cyclic group (i.e., semi-direct
product of a finite-rank 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 Bieri-Neumann-Strebel invariants,
the dynamics of semi-flows and free group automorphisms, and
a polynomial invariant for certain free-by-cyclic groups
that ties all of this together. Joint work with Ilya
Kapovich and Christopher Leininger.
For questions, contact