• Homotopy theory, manifolds, and field theories
• Introductory School

• Schedule
• Participants
• Poster

There is a relationship between higher algebra/category theory and differential topology. Evidence of this is abundant. For instance, Khovanov homology can be regarded as a knot invariant obtained from a suitable representation of a quantum group, the collection of which forms a braided monoidal category. Also, Turaev-Viro 3-manifold invariants are constructed from a suitably finite monoidal category. In the other direction, deformations of a point in an E_n-scheme over characteristic zero, such as a variety over the complex numbers, are indexed by framed n-manifolds. Hochschild (co)homology is an instance of this for n=1.

This series will dwell on the foundations of this relationship, and less so on examples, for those can be left for invested participants. The talks will be framed by one main result, and a couple formal applications thereof. The main construction is factorization homology with coefficients in higher (enriched) categories. The body of the talks will focus on precise definitions, emphasizing the essential aspects that facilitate the coherent cancellations which support the main result.

Specifically, the content will tour through these topics.

• Stratified spaces, exit-path categories, and a characterization of ∞-categories as sheaves on stratified spaces.
• A recasting of (∞,n)-categories as certain sheaves on a category of framed finely stratified n-manifolds.  A definition of factorization homology.  Likewise for (∞,n)-categories with adjoints.
• Some formal applications of factorization homology concerning embeddings and cobordisms.

The original content of this series is joint with John Francis; various parts of which are also joint with Nick Rozenblyum as well as Hiro Lee Tanaka.

Video recording: Lecture 1, Lecture 2, Lecture 3

Over the last fifteen years, a powerful homotopy-theoretic technique has been developed to study the homology of moduli spaces of smooth manifolds, or in other words, classifying spaces of diffeomorphism groups. It originates in the work of Tillmann, Madsen, and Weiss on the cohomology of diffeomorphism groups of Riemann surfaces, and has been extended by Galatius, and by myself.

It is quite different from the classical approach to these classifying spaces, which passes through (rational) homotopy theory, then surgery theory, then pseudoisotopy theory, then algebraic K-theory of spaces. In many ways it is more elementary. A lot of the key ideas can be seen in quite classical results:

• The case of 0-dimensional manifolds corresponds to the Barratt-Priddy-Quillen-Segal theorem (identifying the homology of the stable symmetric group with that of the free infinite loop space on a point) and Nakaoka's theorem (on the stabilisation of the homology of symmetric groups).
• Describing the zeroth homology of moduli spaces of manifolds of higher (even) dimension corresponds to a theorem of Kreck, on the stable classification of manifolds.

I will first explain the statements of the theorems (of Galatius and myself) which describe the stable homology of moduli spaces of manifolds of even dimension. This will be in enough detail so that participants can now use the theory to study particular manifolds of interest, and may include some worked examples. I will then explain some of the ingredients of the proof.

This talk is not an official part of the Introductory School. There will be no mathematics in it.

Global homotopy theory studies equivariant phenomena that exist for all compact Lie groups in a uniform way. In this series of talks I present a rigorous formalism for this and discuss examples of global homotopy types. The emphasis will be on stable global homotopy theory, and the precise implementation proceeds via a new model structure on the category of orthogonal spectra, with "global equivalences" as weak equivalences.

Looking at orthogonal spectra through the eyes of global equivalences leads to a rich algebraic structure on equivariant homotopy groups, including restriction maps, inflation maps and transfer maps. Many interesting global homotopy types support additional ultra-commutative multiplications, and these gives rise to power operations that interact nicely with the other structure. The localization of orthogonal spectra at the class of global equivalences gives a tensor triangulated category much finer than the traditional stable homotopy category of algebraic topology.

Some examples of global homotopy types that I plan to discuss are:

• global 'Borel type' cohomology theories,
• Eilenberg-MacLane spectra of global Mackey functors,
• global Thom spectra that represent bordism of G-manifolds, respectively a localized 'stable' version thereof,
• global equivariant forms of K-theory.

Book project "Global homotopy theory"

One important role of mathematics is to serve as a language for other sciences. There are a number of success stories - on the theoretical physics side general relativity (Lorentz geometry, Einstein’s equation) and quantum mechanics (functional analysis, Schrödinger’s equation) come to mind. Several mathematical formalisms exist today for quantum field theory, each with their own set of advantages and problems.

In this sequence of lectures at HIM, we’ll first explain classical field theories whose mathematical definition is universally agreed upon and for which many physically relevant examples will be given. We will then discuss how the heuristics of path integrals, or more precisely, integrals over the classical fields, lead to functorial field theories.

These are symmetric monoidal functors from a space-time category to a linear target category and were developed by Atiyah (in the topological case), Segal (in the conformal case) and many others. To give precise mathematical definitions, we're forced to use higher categories by the locality of a QFT, and to understand the geometry of super manifolds by the statistics (bosonic/fermionic) of particles. It turns out that the classifying spaces of functorial field theories lead naturally to commutative ring spectra, which for the lowest possible super-dimensions of space-time include de Rham cohomology and K-theory. In the last lecture, we’ll discuss the various connections between algebraic topological and physical notions that arise. This is a report on joint work with Stephan Stolz and many others.