Purdue Topology Seminar, Fall 2018

The Purdue Topology Seminar is held Tuesdays 1:30pm-2:30pm at MA 431 unless otherwise noted.

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

August 21: no seminar
August 28: Cihan Bahran (U Minnesota)
Title: Flows in configuration spaces and representation stability
Abstract: We study two settings with quite general conditions in which the ever-present “FI-structure” on ordered configuration spaces can be extended to larger categories. This results in improvements in the stable ranges of the representation stability phenomena.
September 4: Jeremy Miller (Purdue)
Title: Integral generation of Steinberg modules
Abstract: Assuming the generalized Riemann hypothesis, we show that the
Steinberg module of SL_n of a number ring is generated by integral apartments if
and only if the ring is Euclidean. Our methods give new examples of rings where
the cohomological dimension of SL_n agrees with the virtual cohomological
dimension. This is joint with Rohit Nagpal, Peter Patzt, Jennifer Wilson, and
Dan Yasaki.
September 11: Paul VanKoughnett (Purdue)
Title: Formal groups, elliptic curves, and homotopy theory
Abstract: This expository talk will introduce the chromatic approach to
stable homotopy theory. We’ll focus on the use of formal groups to create
designer cohomology theories, and what these theories are able to tell us
about geometry and homotopy theory. In particular, we’ll discuss one of the
field’s greatest success stories, the topological modular forms spectrum —
a cohomology theory built out of elliptic curves.
September 18: no seminar
September 25: Ralph Kaufmann (Purdue)
Title: Surfaces, strings, arcs and configurations
Room Change: UNIV 301 Abstract: Together with M. Livernet and Bob Penner, we defined a cyclic and
modular operad structures based on surfaces with arcs.
This has yielded several results, including proofs of various versions and
generalizations of Deligne’s conjectures on operations on the Hochschild
complex, such as string topology. In another direction, we applied a stabilization
technique (essentially with respect to genus) to provide a surface
construction of an E_\infty operad and with it an Omega-spectrum.

In newer developments, we have (a) provided the explicit link to
configuration spaces in the genus 0 case together with Yongheng Zhang and
(b) elucidated some of the actions such as a co-product we had previously
defined that has re-appeared in the symplectic setting as the
Goreszky–Hingston
co-product.

We will give the general background and an executive summary of the results
before entering into the newer territory and speculations on future
applications.
October 2: no seminar
October 9: October break
October 16: Sarah Percival (Purdue) and Nathanael Cox (Purdue)
Title: Reeb Spaces of Definable Maps
Abstract: We prove that the Reeb space of a proper definable map in an
arbitrary o-minimal expansion of the reals is realizable as a proper definable
quotient. We demonstrate that Betti numbers of a Reeb space can be arbitrarily
large compared to those of X, unlike in the more particular case of Reeb graphs
of manifolds. Nevertheless, we present a singly exponential upper bound on the
Betti numbers of a Reeb space of f, f : X -> Y , assuming that X is a closed
and bounded semi-algebraic set, in terms of the numbers and degrees of the
polynomials defining X; Y and f.
October 23: Talk cancelled
October 30: David Recio-Mitter (Lehigh)
Title: Topological robotics and braid groups
Abstract: One of the main problems in robotics is that of motion planning. It consists of finding an algorithm which takes pairs of positions as an input and outputs a path between them. It is not always possible to find such an algorithm which depends continuously on the inputs. Studying this problem from a topological perspective, in 2003 Michael Farber introduced the topological complexity of a space, which measures the minimal (unavoidable) discontinuity of all motion planners on a given topological space. The topological complexity TC(X) turns out to be a homotopy invariant of the space X.

In this talk we will determine (or narrow down to a few values in some cases) the topological complexity of the unordered configuration spaces of aspherical surfaces (including surfaces with boundary and non-orientable surfaces). We will also see how this can be understood as the topological complexity of the surface braid groups.

This is joint work with Andrea Bianchi.
November 6: Talk cancelled
November 13: no seminar
November 20: Thanksgiving
November 27: Andrew Salch (Wayne State)
Title: What we know so far about “topological Langlands correspondences.”
Abstract: I’ll give a survey of some relationships between Galois representations and stable homotopy groups of finite CW-complexes which suggest the possibility of “topological Langlands correspondences.” I’ll explain what such correspondences ought to be, what their practical consequences are for number theory and for algebraic topology, and I’ll explain the cases of such correspondences that are known to exist so far. As an application of one family of known cases, I’ll give a topological proof of the Leopoldt conjecture for one particular family of number fields. Some of the results in this talk are joint work with M. Strauch.
December 4: Jennifer Wilson (U Michigan)
Title: Quillen’s method in representation stability
Abstract: In this talk, I will describe a classical strategy due to Quillen for establishing that a family of groups is homologically stable. In work joint with Jeremy Miller and in part with Peter Patzt, we applied a representation-theoretic adaptation of this strategy to prove stability results for families such as configuration spaces of points in a manifold, and congruence subgroups of linear groups and of mapping class groups.