Purdue Topology Seminar, Spring 2019

The Purdue Topology Seminar is held Wednesdays 1:30pm-2:30pm in MA431 unless otherwise noted.

Primary contact is Peter Patzt (ppatzt at purdue dot edu)

January 9: No Seminar
January 16: No Seminar
January 22: Oscar Randal-Williams (Cambridge, UK) Talk cancelled
January 30: Phil Tosteson (Michigan)
Title: Operads and combinatorial categories
Abstract: Recent applications of the representation theory of category of
finite sets and injections to topology have spurred research into the
representation theory of other “combinatorial categories.” For every
operad $P$, there is an associated combinatorial category— its PROP or
“wiring category”, C. We will apply this connection between operads and
combinatorial categories to the representation theory of C, and to the
description of actions on the homology of moduli spaces and configuration
February 6: Christin Bibby (Michigan)
Title: Combinatorics of orbit configuration spaces
Abstract: From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this “orbit configuration space” is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its topology. We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement) which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations akin to those of (Totaro ’96). Joint work with Nir Gadish.
February 13: Javier Zuniga (Universidad del Pacifico, Peru)
Title: Modular Operations on Semi-stable Ribbon Graphs
Abstract: Modular operations appear in moduli spaces of Riemann surfaces as an “opposite” operation to geodesic degeneration. This was studied initially as Modular operads by Getzler and Kapranov. Graph homology was developed by Kontsevich from stable ribbon graphs which arise from a cellular decomposition of moduli space. Using and extension of this model via semi-stable graphs it is possible to define modular operations that can be used to give a combinatorial solution to the Master Equation.
February 20: Daniel Studenmund (Notre Dame)
Title: Semiduality in group cohomology
Abstract: A duality group has a pairing exhibiting isomorphisms between its homology and cohomology groups. Examples include solvable Baumslag-Solitar groups and arithmetic groups over number fields, by work of Borel and Serre. Many naturally occurring groups fail to be duality groups, but are morally very close. In this talk we make this precise with the notion of a semiduality group. We then make a general conjecture, building on the result of Borel-Serre, that certain arithmetic groups in positive characteristic are semiduality groups, and discuss aspects of the proof in the rank 1 case. This talk covers work joint with Kevin Wortman.
February 27: Maxim Bergeron (Chicago)
Title: The topology of representation varieties
Abstract: Let G be a complex reductive algebraic group (such as a the general linear group) and let K be a maximal compact subgroup of G (such as the unitary group). I will discuss exceptional classes of finitely generated groups \Gamma for which the space of representations Hom(\Gamma,G) admits a deformation retraction onto the subspace Hom(\Gamma,K).
March 6: Rohit Nagpal (Michigan)
Title: Finiteness properties of the Steinberg representation
Abstract: We will show that the Steinberg modules for the general linear groups form a Koszul monoid in an appropriate symmetric monoidal category. Using this we will find bounds on the codimension-one cohomology of level-3 congruence subgroups. This Koszulness result can also be used to show Ash–Putman–Sam homological vanishing theorem for the Steinberg representations. This is a joint work with Jeremy Miller and Peter Patzt.
March 13: Spring Break
March 20: no seminar
March 27: Zachary Himes (Topic Exam)
Title: Homological stability of groups
Abstract: In this talk, I will describe an argument due to Quillen for detecting when a family of groups (for example, the symmetric groups) have stable homology groups along with several generalizations of this phenomenon. For an application, I will also discuss how homological stability can be applied to study the homology of the moduli space of hyperelliptic curves.
April 3: no seminar
April 10: Dan Yasaki (UNC Greensboro)
Title: On the growth of torsion in the cohomology of arithmetic groups
Abstract: Bergeron and Venkatesh give a precise conjecture about the growth of the order of the torsion subgroup of homology groups over a tower of cocompact congruence subgroups. In this talk, we describe our computational investigation of this phenomena. We consider the cohomology of several (non-cocompact) arithmetic groups, including GL_n(Z) for n = 3, 4, 5 and GL_2(O) for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron-Venkatesh. This is joint work with Avner Ash, Paul Gunnells, and Mark McConnell.
April 17: Steven Sam (San Diego)
Title: Representations of surjections and compactifications of moduli spaces
Abstract: Recently, the (co)homology of families of moduli spaces (e.g., configuration spaces, moduli space of curves with n marked points) have been fruitfully studied using the technology of FI-modules (functors on the category of finite sets and injective functions) to uncover patterns and stabilization results. However, many of these moduli spaces admit natural compactifications for which these techniques fail to be useful. I will explain some recent work which repairs this deficiency by using the category of finite sets and surjective functions.
April 24: Avner Ash (Boston College)
Title: Resolutions of the Steinberg module for GL(n)
Abstract: The Steinberg module St(n,K) for the general linear group G of a
number field K is the dualizing module for arithmetic subgroups of G. So
St(n,K) can be used in studying the cohomology of arithmetic groups, and is
especially suitable for computing Hecke operators. I will present several
resolutions of St(n,Q) that are helpful in this regard. I will discuss
applications to computations for congruence subgroups of SL(4,Z) (with Paul
Gunnells and Mark McConnell) and some work related to Serre-type
conjectures for GL(n,Z) (with Darrin Doud.)