This autumn school will take place during the week September 26-30, 2022 in a conference center close to Utrecht, The Netherlands.

The idea of this autumn school is to bring together a group of about 25 participants in a remote place in order to learn about advanced topics in Algebraic Topology. The talks are meant to be accessible to first or second year PhD students.

The following lecture series constitute the core program of the autumn school:

**Paul Goerss**: Chromatic Homotopy Theory: Decomposition, Assembly, and Large-Scale Computations#### Tentative Abstract:

Chromatic is a word with two meanings. It can refer to color and how we assemble light into what we see, and it is also a musical term, describing how the tonal scale builds what we hear. In the 1970s algebraic topologists came to understand that stable homotopy theory can similarly be assembled from specific layers. Following the work of Quillen, homotopy theorists noted that both the quantitative and qualitative aspects of the field could be organized using the algebraic geometry of smooth 1-parameter formal groups; the link was through cohomology theories with Chern classes. The layers were then determined by an invariant of formal groups known as height. This paradigm is now so classical that we have forgotten the initial shock of these ideas: before this, no one had any organized plan to find or understand large-scale phenomena in stable homotopy theory. One of points of these lectures is to recapture some of this revolutionary spirit. To do this I will take a quasi-historical approach, beginning with what we can learn from K-theory. From there I will lay out the more general theory, the groundbreaking results of the '70s and '80s, focusing on what can be said at large primes. This will leave us well-placed to discuss more recent work, such as the interplay between chromatic layer 2 and the theory of elliptic curve. I will end with a discussion of higher order invariants, such as Picard groups.

**Alexander Kupers**: Homotopical approaches to spaces of diffeomorphisms and embeddings#### Tentative Abstract

A classical problem in topology is the classification of manifolds, and a modern reinterpretation and generalisation of this asks for the determination of the homotopy type of moduli spaces of manifolds. Some of the major open questions in this direction concern the "symmetry groups" of manifolds, that is, their diffeomorphisms. To understand the space of diffeomorphisms of a manifold it is helpful to understand how it differs from that for other manifolds; the relative terms are spaces of embeddings. The goal of this series of lectures is to explain recent progress in understanding spaces of diffeomorphisms and embeddings. This story is not yet finished, and has tantalising connections to many subjects in modern homotopy theory. We intend to cover the following topics:

- - The homotopy type of cobordism categories.
- - The calculus of embeddings.
- - The Weiss fiber sequence.
- - The rational homotopy type of diffeomorphism groups of discs.

The registration deadline has passed, and registration is no longer possible.

The autumn school starts Monday September 26 in the morning and ends Friday 30 September around noon, so it is recommended that you arrive already on Sunday September 25. Travel details will be given in due time before the autumn school.

The autumn school is organized by Gijs Heuts, Lennart Meier, Ieke Moerdijk, and Steffen Sagave.