University of Chicago

Wed 25 Jan 2017, 4:15pm
SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)

Quantitative Nullcobordism and the (in)effectiveness of algebraic topology.

ESB 4133 (PIMS Lounge)
Wed 25 Jan 2017, 4:15pm5:15pm
Abstract
Topology is full of ineffective arguments constructing objects and equivalences by algebra.
One of the great early achievements of algebraic topology was the work of Thom, followed by Milnor and Wall, on cobordism theory, which describes when a compact smooth (oriented) manifold is the boundary of some compact manifold with boundary. This method is typical of the problems that arise in the use of algebraic methods and is an early example of one of the dominant philosophies of geometric topology. The question we study is to what extent the complexity of a manifold can be used to bound, when it exists, the minimum necessary complexity of something that it bounds.
The goal of this talk is to explain generally some of the issues of making topology less ineffective.
We shall show that there are polynomial size nullcobordisms in a suitable sense. This is joint work with Greg Chambers, Dominic Dotterer and Fedor Manin.
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University of Oregon

Wed 1 Feb 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)

TBA

ESB 4133 (PIMS Lounge)
Wed 1 Feb 2017, 3:15pm4:15pm
Abstract
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North Carolina State University

Wed 15 Feb 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)

Representation stability in configuration spaces via Whitney homology of the partition lattice

ESB 4133 (PIMS Lounge)
Wed 15 Feb 2017, 3:15pm4:15pm
Abstract
In recent years, important families of symmetric group representations have come to be better understood through the perspective of representation stability, a viewpoint introduced and developed by Thomas Church, Jordan Ellenberg, and Benson Farb, among others. A fundamental example of representation stability is the $S_n$module structure for the $i$th cohomology of the configuration space of $n$ distinct, labeled points in the plane, or more generally in a connected, orientable manifold, as $i$ is held fixed and $n$ grows. For the plane, this translates to Whitney homology of the partition lattice via an $S_n$equivariant version of the GoreskyMacPherson formula. This talk will survey the combinatorial literature regarding the partition lattice and discuss what new things this can tell us about representation stability for configuration spaces. In particular, we deduce new, sharp stability bounds and representation theoretic structure through a combination of symmetric function technology and poset topology. This is a joint work with Vic Reiner.
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