# Topology Seminar

This seminar is held on Tuesdays at 3pm in 210 Deady.

### Fall Quarter, 2017

- September 26,
**Nathan Perlmutter** (Stanford)

Parametrized Morse Theory, Cobordism Categories, and Positive Scalar Curvature
**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 manifolds 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 category with the infinite loopspace of another Thom spectrum that is related to MTSpin(d) and the space of Morse jets on Euclidean space. This 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, with critical points confined to a range of prescribed degrees.

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 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. In particular, it gives an alternative proof and extension of a recent breakthrough theorem of Botvinnik, Ebert, and Randal-Williams.

- October 3,
**John Lind** (Reed)

T-duality in physics and topology
**Abstract**: T-duality arose as an agreement between the predictions of different versions of string theory. There is an underlying topological agreement as well, which can be expressed as an isomorphism between the twisted K-theory of certain circle bundles. I will carefully explain this through examples, and then I will discuss my work with Westerland and Sati on T-duality for more general sorts of fiber bundles. In particular, I will describe the universal T-duality theory for sphere bundles of a fixed rank and its relationship with algebraic K-theory. Time permitting, I will discuss current work on a T-duality isomorphism in chromatic homotopy theory.

- October 10,
**Nikolai Saveliev** (U. Miami)
- October 24,
**Eric Hogle** (UO)
- October 31,
**Clover May** (UO)
- November 7,
**Keegan Boyle** (UO)
- November 14,
**Beibei Liu** (UC Davis)
- November 21,
**Mauricio Gomez Lopez** (UO)
- December 5,
**Ben Knudson** (Harvard)

Topology Seminar

Previous years: 2017 2016 2015 2013 2012 F11 S10 W10 F09 F08 S08 W08 F07 S07 W07 F06 F05-S06 F04-S05 F03-S04 F02-S03 F00-S01 F99-S00 F98-S99 F97-S98