European Autumn School in Topology 2022

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

Aims and Scope

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.

Program

Lecture series

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

• Paul Goerss: Chromatic Homotopy Theory: Decomposition, Assembly, and Large-Scale Computations

  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

  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.

Preparatory talks

• Chase Vogeli: Classifying spaces and characteristic classes
• Julius Frank: Spectra and the Stable Category
• Coline Emprin: Chern Classes, Pontryagin classes, and Complex Oriented Cohomology Theories
• Leonard Tokic: Cohomology of Groups
• Robin Stoll: The Smale–Hirsch theorem

The topics of the preparatory talks including references are listed here.

Contributed talks

• Connor Malin: Koszul duality and the Lie bracket on the homology of configuration spaces

  Abstract

  Embedding calculus attempts to study framed manifolds \(M\) via the module of configuration spaces \(\mathcal{F}_M\) over the Fulton-MacPherson operad \(\mathcal{F}_n\). The relation between \(\mathcal{F}_M\) and \(\mathcal{F}_n\) endows the homology of the configuration spaces of \(M\) with the structure of a module over the Lie operad; we show this structure depends only on the homotopy type of \(M^+\), the one point compactification.

• Miguel Barrero: Global \(N_\infty\)-operads

  Abstract

  In this talk I will introduce global \(N_\infty\)-operads, that parametrize intermediate levels of commutativity in global homotopy theory. Global homotopy theory is the study of spaces and spectra with compatible actions by all compact Lie groups. I will then classify these global \(N_\infty\)-operads in terms of their associated global transfer systems. The associated global transfer system is the collection of the transfer maps that algebras over a given global \(N_\infty\)-operad posses.

• Christian Carrick: fp-spectra and Real-oriented homotopy theory

  Abstract

  If \(X\) is a spectrum, then the cohomology groups \(H^*(X;\mathbb F_2)\) form a module over the mod \(2\) Steenrod algebra \(A\). Via the Adams spectral sequence, this algebraic structure almost completely determines the homotopy type of \(X\) in many cases. However, \(A\) is a non-Noetherian algebra, so it is often convenient to study \(H^*(X;\mathbb F_2)\) instead as a module over finite subalgebras of \(A\). When \(H^*(X;\mathbb F_2)\) is a finitely presented \(A\)-module, this results in no loss of information, and Mahowald-Rezk demonstrated that such spectra \(X\) - "fp spectra" - admit a host of interesting categorical properties, including a chromatic version of Brown-Comenetz duality. I will discuss some of these properties and go through a number of easy examples of fp spectra, then finish with some new examples that come from Real-oriented homotopy theory. The latter form part of joint work in progress with Mike Hill and Doug Ravenel.

• Andreas Stavrou: Homology of configuration spaces of surfaces as mapping class group representations

  Abstract

  We will introduce configuration spaces and discuss what is known about the action of the mapping class group of a surface on the homology of various flavours of configuration spaces of the surface. Time permitting, we will draw pictures to motivate some of the results.

Registration

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

Travel Information

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.

Organizers

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

Previous schools

Previous editions of this autumn school took place in 2019, in 2018, in 2017, and in 2016, while the 2020 edition had to be canceled because of the corona pandemic and we could not have a 2021 edition for the same reason.