Geometry/Topology Seminar
Spring 2018
Thursdays (and sometimes Tuesdays) 2:303:30pm, in
Ryerson 358

 Thursday October 11 at 34pm in Ry 358
 Rita Gitik, University of Michigan

On Tame Subgroups of Finitely Presented Groups

Abstract: We describe several examples of tame
subgroups of finitely presented groups and prove that the
fundamental groups of certain finite graphs of groups are
locally tame.

 Thursday October 18 at 34pm in Ry 358
 Daniel Ramras, IUPUI
 Homological stability for spaces of representations

Abstract: I will discuss recent work with Mentor
Stafa on homological stability for various spaces built from
commuting elements in Lie groups. These results depend on
rational models for these spaces due to T. Baird, and
utilize J. Wilson's theory of FI_{W}modules. A key
aspect of this work is the relationship between stability in
ordinary and in equivariant (co)homology. Time permitting,
I'll also discuss what is known regarding stability for
character varieties of free groups and surface groups.

 Thursday October 25 at 34pm in Ry 358
 Daniel Studenmund, Notre Dame
 Commensurability growth of nilpotent groups

Abstract: A classical area of study in geometric
group theory is subgroup growth, which counts the number of
subgroups of a given group Gamma as a function their index.
We will study a richer function, the commensurability
growth, associated to a subgroup Gamma in an ambient group
G. The main result of this talk concerns the case that Gamma
is an arithmetic subgroup of a unipotent group G, starting
with the simplest example of the integers in the real line.
This is joint work with Khalid BouRabee.

 Thursday November 01 at 34pm in Ry 358
 Yulan Qing, University of Toronto
 Loops with Large Twist Get Short Along Quasigeodesics in Out(Fn)

Abstract: We study the behaviour of quasigeodesics
in Out(Fn) equipped with word metric. Given an element
\phi of Out(Fn), there are several natural paths
connecting the origin to \phi in Out(Fn). We show
that these paths are, in general, not quasigeodesics in
Out(Fn). In fact, we clear up the current misunderstanding
about distance estimating in Out(Fn) by showing that there
exists points in Out(Fn) where all quasigeodesics between
them backtracks in all of the current Out(Fn) complexes .

 Thursday November 08 at 34pm in Ry 358
 Kevin Schreve, University of Chicago
 Action Dimension of Simple Complexes of Groups

Abstract: The action dimension of a discrete group G
is the minimal dimension of contractible manifold that
admits a properly discontinuous G action. I will compute the
action dimension of several examples,including Artin groups,
graph products, and hyperplane arrangements.

 Thursday November 15 at 34pm in Ry 358
 Trevor Hyde, University of Michigan
 Moduli of multivariate irreducible polynomials and liminal reciprocity

Abstract: I will share some recent results on the
moduli space of multivariate polynomials which are
irreducible over a field K, including a surprising
connection between univariate polynomials and the limiting
space of irreducibles in infinitely many variables.

 Thursday November 29 at 34pm in Ry 358
 Fedor Manin, Ohio State University
 Geometric shadows of rational homotopy theory

Abstract: Given compact simplicial complexes or
Riemannian manifolds X and Y, what is
the least Lipschitz constant of a nullhomotopy of an
LLipschitz map X \to Y, as a function
of L? If X is the circle, then this is
certainly bounded below by the square root of the Dehn
function of \pi_{1}(Y), which for certain
Y grows faster than any computable function. On
the other hand, if Y is simply connected, then
for any X whatsoever the answer is
O(L^{2}) (this is almost sharp in some
cases.) This, among other results, follows from a geometric
upgrade to a fundamental correspondence in algebraic
topology due to Sullivan, which I will attempt to describe.

 Thursday January 10 at 3:304:30pm in Ry 358
 Izzet Coskun, UIC
 The stable cohomology of moduli spaces of sheaves on surfaces

Abstract: Moduli spaces of Gieseker semistable
sheaves on surfaces play a central role in mathematics and
have many applications to cycles and linear systems on
surfaces, Donaldson's 4manifold invariants and mathematical
physics. In this talk, I will describe a conjecture with
Matthew Woolf on the cohomology of these moduli spaces. We
conjecture that the Betti numbers of these moduli spaces
stabilize as the discriminant tends to infinity and that the
stable numbers are independent of the rank and the first
Chern class. In particular, calculations of Gottsche
determine the stable numbers. I will give some evidence for
the conjecture. This is joint work with Matthew Woolf.

 Thursday January 24 at 3:304:30pm in Ry 358
 Nate Harman, Chicago
 Effective and infinite rank superrigidity for special linear groups

Abstract: Superrigidity for
SL_{n}(Z) tells us that any finite
dimensional complex representation virtually extends to
(i.e. agrees along a finite index subgroup with) an
algebraic representation of SL_{n}(C). We
look at effective improvements of this statement, and
explore the interaction of superrigidity with representation
stability. This leads to a formulation of superrigidity for
the infinite rank special linear group  despite the fact
that it has no nontrivial finite dimensional representations
or finite index subgroups.

 Thursday February 7 at 3:304:30pm in Ry 358
 Kasia Jankiewicz, Chicago
 Groups acting and not acting on CAT(0) cube complexes

Abstract: CAT(0) cube complexes are built by gluing
Euclidean cubes of various dimensions together, such that a
certain combinatorial condition is satisfied, which
guarantees that they are as least as nonpositively curved as
the Euclidean space. The combinatorial nature of CAT(0) cube
complexes gives them a feel of simplicial trees, yet the
family of groups admitting actions on CAT(0) cube complexes
is rich and various. There are strong relations between
actions on CAT(0) cube complexes and algebraic properties of
groups. Constructing group actions on CAT(0) complexes is
usually done by a standard construction, and obstructing
them is in general harder. In my talk I will survey
properties and examples of groups acting nicely on CAT(0)
cube complexes. I will also discuss some examples of groups
that do not admit such actions, including the 4strand braid
group (joint work with J. Huang and P. Przytycki).

 Thursday February 14 at 3:304:30pm in Ry 358
 Jingyin Huang, Ohio State University
 Commensurability and virtually specialness of some uniform cubical lattices

Abstract: A classical result by Bieberbach says that
uniform lattices acting on Euclidean spaces are virtually
free abelian. On the other hand, uniform lattices acting on
trees are virtually free. This motivates the study of
commensurability classification of uniform lattices acting
on RAAG complexes, which are cube complexes that
"interpolate" between Euclidean spaces and trees. We show
that the tree times tree obstruction is the only obstruction
for commmensurability of labelpreserving lattices acting on
RAAG complexes. Some connections of this problem with
Haglund and Wise's work on special cube complexes will also
be discussed.

 Tuesday March 05 at 3:304:30pm in Ry 358
 Christin Bibby, University of Michigan
 Supersolvable posets and arrangements

Abstract: The structure of a supersolvable geometric
lattice has proven to be fruitful in the theory of
hyperplane arrangements, where it arises as the intersection
poset of a fibertype arrangement. A nice partition of the
atoms in the poset determines the roots of the
characteristic polynomial, thus giving a factorization of
the Poincare polynomial of the arrangement complement. The
cohomology ring (the OrlikSolomon algebra) is a Koszul
algebra, which allows one to extract information about the
rational homotopy theory of the complement. We explore these
ideas for toric and elliptic arrangements, where the
analogue of the intersection poset is not even a semilattice
but a notion of supersolvability can still be applied. The
main motivating example is an analogue of reflection
arrangements, where the complement is an orbit configuration
space and the poset is a generalization of partition
lattices. Based on joint work with Emanuele Delucchi.

 Thursday March 07 at 3:304:30pm in Ry 358
 Sebastian Hensel, LMU Munchen
 Virtual homology representations of mapping class groups and topology

Abstract: Given a mapping class f of a surface S (or
an automorphism of a free group), one can extract basic
information about f from the action on the first homology of
S. While this representation is very well understood, it is
also very coarse  most interesting topological or group
theoretic properties of f are not determined by the homology
action. Somewhat surprisingly, this picture changes
drastically if one is willing to consider the homology of
finite covers as well. We will discuss theorems that give
examples of this behaviour, in particular concering the
question when mapping classes extend over handlebodies.

 Tuesday March 12 at 3:304:30pm in Ry 358
 John WiltshireGordon, Wisconsin
 Configuration space in a product

Abstract: Write Conf(n,X) for the space of
injections {1,...,n} > X. For example, the space
Conf(n, R) is homotopy equivalent to a discrete space with
cardinality n!. In contrast, the space Conf(n, R x R) seems
much more interesting and complicated. In this talk, we
explain how to compute the homotopy type of Conf(n, X x Y)
using only information about configurations in each factor.
As an application, we show that configuration space
distinguishes the two real line bundles on a circle, and
find a homological stability result for trivial complex
vector bundles of high rank.

 Thursday March 28 at 3:304:30pm in Ry 358
 Wolfgang Lueck, Bonn
 Universal L^{2}torsion, L^{2}Euler characteristic, Thurston norm and polytopes (joint with S. Friedl)

Abstract: Given an L^{2}acyclic
connected finite CWcomplex, we define its
universal L^{2}torsion in terms of the
chain complex of its universal covering. It takes values in
the weak Whitehead group \Wh^{w}(G). We
study its main properties such as homotopy invariance, sum
formula, product formula and Poincaré duality. Under
certain assumptions, we can specify certain homomorphisms
from the weak Whitehead group \Wh^{w}(G)
to abelian groups such as the real numbers or the
Grothendieck group of integral polytopes, and the image of
the universal L^{2}torsion can be
identified with many invariants such as the
L^{2}torsion, the
L^{2}torsion function, twisted
L^{2}Euler characteristics and, in the
case of a 3manifold, the Thurston norm and the
(dual) Thurston polytope.

 Thursday April 4 at 2:303:30pm in Ry 358
 Weiyan Chen, Minnesota
 Cohomology of the space of complex irreducible polynomials in
several variables

Abstract: We will show that the space of complex
irreducible polynomials of degree d in n variables satisfies
two forms of homological stability: first, its cohomology
stabilizes as d increases, and second, its compactly
supported cohomology stabilizes as n increases. Our
topological results are inspired by counting results over
finite fields due to Carlitz and Hyde.

 Friday April 12 at 45pm in Ry 251
 Vladimir Markovic, California Institute of Technology
 Namboodiri Lecture 1: The Surface Subgroup Problem

Abstract: The surface subgroup problem asks whether
a given group contains a subgroup that is isomorphic to the
fundamental group of a closed surface. In this talk I will
survey the role the surface subgroup problem plays in some
important solved and unsolved problems in 3manifold
topology, the theory of arithmetic manifolds and geometric
group theory.

 Monday April 15 at 45pm in Ry 251
 Vladimir Markovic, California Institute of Technology
 Namboodiri Lecture 2: Rigidity and geometry of Harmonic Maps

Abstract: Harmonic maps play a prominent role in
geometry. I will explain some of these applications
including Siu's rigidity of negatively curved Kahler
manifolds and the CorletteGromovSchoen rigidity of
representations theorem, as well as the very recent results
of Markovic and BenoistHulin about the existence and
uniqueness of harmonic maps between rank1 symmetric spaces.

 Tuesday April 16 at 45pm in E 206
 Vladimir Markovic, California Institute of Technology
 Namboodiri Lecture 3: Teichmueller flow and complex geometry of Moduli Spaces

Abstract: I will explain why in general the
Caratheodory and Teichmueller metrics do not agree on
Teichmueller spaces and why this yields a proof of the
convexity conjecture of Siu. Moreover, I will illustrate how
deep theorems in Teichmueller dynamics play an important
role in classifying Teichmueller discs where the two metrics
agree.

 Thursday April 18 at 2:303:30pm in Ry 358
 Sam Nariman, Northwestern
 Dynamical and cohomological obstruction to extending group actions

Abstract: For any 3manifold M with torus
boundary, we find finitely generated subgroups of
\Diff_{0}(\partial M) whose actions do
not extend to actions on M; in many cases, there
is even no action by homeomorphisms. The obstructions are
both dynamical and cohomological in nature. We also show
that, if \partial M = S^{2}, there is no
section of the map \Diff_{0}(M) \to
\Diff_{0}(\partial M). This is joint work with
Kathryn Mann.

 Thursday May 9 at 2:303:30pm in Ry 358
 Alex Duncan, South Carolina
 TBA

Abstract: TBA

 Thursday May 16 at 2:303:30pm in Ry 358
 Kathryn Mann, Brown
 TBA

Abstract: TBA

 Monday May 27 at 2:303:30pm in Ry 358
 Corey Bregman, Brandeis
 TBA

Abstract: TBA
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact