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 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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Stanford University
Tue 12 Dec 2017, 3:00pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Parametrized morse theory, cobordism categories, and positive scalar curvature
ESB 4133 (PIMS Lounge)
Tue 12 Dec 2017, 3:00pm-4:00pm

Abstract

In this talk I will show how to use parametrized Morse theory to construct a map from the infinite loopspace of certain Thom spectrum, MTSpin(d), into the space of positive scalar curvature metrics on a closed spin manifold of dimension d > 4. My main novel construction is a cobordism category consisting of cobordisms equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. My first result identifies the homotopy type of the classifying space of this topological category with the infinite loopspace of another Thom spectrum related that is related to MTSpin(d), and built out of the space of Morse jets on Euclidean space. The result can viewed as an analogue of the well known theorem of Galatius, Madsen, Tillmann, and Weiss, for manifolds equipped with the extra geometric structure of a choice of admissible Morse function.

In the second part of the talk I will show how to use this cobordism category to probe the homotopy type of the space of positive scalar curvature metrics, R^{+}(M), on a closed, spin manifold M when dim(M) > 4. This uses a parametrized version of the Gromov-Lawson construction developed by Walsh and Chernysh. Our main result detects many non-trivial homotopy groups in the space of positive scalar curvature metrics R^{+}(M). It in particular gives an alternative proof and extension of a recent breakthrough theorem of Botvinnik, Ebert, and Randal-Williams.
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PhD Candidate: Xinyu Liu
Mathematics, UBC
Wed 13 Dec 2017, 12:30pm SPECIAL
Room 203, Graduate Student Centre, UBC
PhD Oral Exam: Schwartz Analysis and Intertwining Distributions
Room 203, Graduate Student Centre, UBC
Wed 13 Dec 2017, 12:30pm-2:30pm

Details

Abstract: In this dissertation, we combine the work of A. Aizenbud and D. Gourevitch on Schwartz functions on Nash manifolds, and the work of F. du Cloux on Schwartz inductions, to develop a toolbox of Schwartz analysis on algebraic groups. We then use these tools to study the intertwining operators between parabolic inductions, and the behaviour of intertwining distributions on certain open subsets. Finally we use our results to give new proof of results of F. Bruhat, on irreducibilities of degenerate principal series and minimal principal series.

Note for Attendees

Latecomers will not be admitted.
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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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Université de Sherbrooke
Thu 14 Dec 2017, 3:00pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Heegaard Floer homology as immersed curves.
ESB 4133 (PIMS Lounge)
Thu 14 Dec 2017, 3:00pm-4:00pm

Abstract

The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. I'll set up this machinery and describe some of the applications that follow. This is joint work with Jonathan Hanselman and Jake Rasmussen.
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