Geometry/Topology Seminar
Winter 2019
Thursdays (and sometimes Tuesdays) 2:303:30pm, in
Ryerson 358

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