## Tondeur Lectures in Mathematics

Spring 2017

### Ulrike Tillmann

Oxford University

#### Topological Quantum Field Theories in Homotopy Theory

**Lecture 1. 4 pm Tuesday, April 25, 314 Altgeld Hall**

A reception will be held following the lecture from 5-6 pm in 239 Altgeld Hall

**Cobordisms: old and new**

Cobordims have played an important part in the classification of manifolds since their invention in the 1950s. In a different way, they are fundamental to the axiomatic approach to Topological Quantum Field Theory. In this colloquium style talk I will explain how recent results have shed new light on both of them.

**Lecture 2. 4 pm Wednesday, April 26, 245 Altgeld Hall**

**Classifying spaces of bordism categories and a filtration of Thom's theory**

We describe a refinement of a theorem with Galatius, Madsen and Weiss which describes the classifying space of bordism categories. In particular this can be interpreted to give evidence for the cobordism hypothesis for invertible TQFTs.

**Lecture 3. 4 pm Thursday, April 27, 245 Altgeld Hall**

**Operads from TQFTs**

Manifolds give rise to interesting operads, and in particular TQFTs define algebras over these operads. In the case of Atiyah's 1+1 dimensional theories these algebras are well-known to correspond to certain algebras. Surprisingly, independent of the dimension of the underlying manifolds, in the topologically enriched setting the manifold operads detect infinite loop spaces. We will report on joint work with Basterra, Bobkova, Ponto, Yeakel.