Mathematics Dept.
  Events
Stanford University
Mon 11 Dec 2017, 3:00pm
Department Colloquium
MATX 1100
Cobordism categories and moduli spaces of manifolds
MATX 1100
Mon 11 Dec 2017, 3:00pm-4:00pm

Abstract

Let M be a smooth manifold, let Diff(M) denote the topological group of sel-diffeomorphisms, and let BDiff(M) denote the "classifying space”. For any paracompact space X, there is a one-one correspondence between the set of homotopy classes [X, BDiff(M)] and the set of isomorphism classes of fibre bundles over X with fibre M. The classifying space BDiff(M) is referred to as the "moduli space of manifolds of type M". The study of the homotopy type of these spaces occupies a central place in smooth topology. 
 
In this talk I will discuss some contemporary approaches to studying the homotopy/homology of BDiff(M), for varying M. In particular I will discuss the results of Madsen and Weiss on the stable moduli spaces of Riemann surfaces and the results of Galatius and Randal-Williams on the stable moduli spaces of manifolds of dimension 2n. I will then present recent work of mine pertaining to the moduli spaces of odd dimensional manifolds, and manifolds with boundary, and discuss connections to cobordism categories and surgery theory.

Note for Attendees

Refreshments will be served in MATX 1100 at 2:30p.m. before this Mathematics Colloquium.
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Université de Sherbrooke
Wed 13 Dec 2017, 3:00pm
Department Colloquium
MATX 1100
A snapshot of Heegaard Floer theory
MATX 1100
Wed 13 Dec 2017, 3:00pm-4:00pm

Abstract

Heegaard Floer homology provides a suite of invariants for studying three-manifolds, introduced by Ozsváth and Szabó. This theory has, more recently, been expanded to treat manifolds with boundary through bordered Floer homology, providing the tools required to answer natural questions that arise when decomposing a three-manifold along a surface. This talk aims to provide a brief overview of Heegaard Floer theory, give a sense for some of the questions driving its study, and point to some recent progress on answering these.

Note for Attendees

Refreshments will be served in MATX 1100 at 2:30 p.m. before this Mathematics Colloquium.
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