European Autumn School in Topology 2024

This autumn school will take place during the week 16-20 September 2024 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:

• Bjørn Ian Dundas: Trace methods in algebraic K-theory and equivariant stable homotopy theory (notes)
  Abstract

  Quillen's algebraic K-theory was a revolutionary invention opening a coherent and structural approach to a wide array of mathematical phenomena spanning from number theory to manifolds and beyond. Naturally, the wider the scope and the deeper the grasp, the harder it is to obtain information about an invariant. Algebraic K-theory turned out to be very hard indeed. It started out very promising with Quillen himself providing the first calculations relevant for the Adams conjecture. Bright brains like Suslin and Thomason's made tremendous contributions, but for twenty years calculations of even the simplest cases seemed far out of reach.

  However, two major breakthroughs came towards the end of the millennium. First it was realized that given the right target, trace methods could expand the already existing calculations enormously and secondly the motivic program finally succeeded.

  Since then our understanding of algebraic K-theory has expanded beyond anyone’s wildest expectation. Not only have we successfully calculated K-theory in key situations, the methods have shed light on central aspects of fields not immediately associated with algebraic K-theory, as for instance in the disproof of the telescope conjecture.

  These talks aim at exploring trace methods spanning from the very first calculations to the most recent applications. This will have to include introductions to

  • algebraic K-theory
  • topological cyclic homology

  but we also will have to familiarize ourselves with the calculational tools available and look at some basics about the topics investigated through these invariants.

• Markus Hausmann: Equivariant homotopy theory (Lecture notes)
  Abstract

  Equivariant topology and the study of symmetries of spaces has a long history. The aim of these lectures will be to give an overview of classical and modern aspects of the theory. In particular, I aim to cover the following:

  • Bredon homology, Smith theory and the localization theorem
  • Modern approaches to the homotopy theory of \(G\)-spectra and their relationship: Inverting representation spheres, spectral Mackey functors, Tate gluing
  • Complex bordism with reality \(\textrm{MR}\) vs. equivariant complex bordism \(\textrm{MUG}\): The role of the former in Hill-Hopkins-Ravenel's solution to the Kervaire invariant one problem, and the role of the latter for an equivariant version of chromatic homotopy theory

Preparatory talks

• Max Blans: Simplicial methods and ∞-categories (60 minutes)
• Maxime Wybouw: Homotopy (co)limits (60 minutes)
• Robert Szafarczyk: Spectra (60 minutes)
• Qi Zhu: Equivariant homotopy theory (30 minutes)
• Jackson Morris: Hochschild homology (30 minutes)
• Alba Sendón Blanco: Algebraic K-theory (60 minutes)

More details about the preparatory talks including references are listed here.

Contributed talks

• Emma Brink: Equivariant Bordism and Thom Spectra
  Abstract

  For a compact Lie group \(G\), we describe equivariant versions of geometric bordism with stable tangential structures. These define \(\mathbb Z\)-graded equivariant homology theories which admit a Thom-Pontryagin comparison map to the homology theory represented by associated Thom spectra.

  If \(G\) has non-abelian connected identity component (e.g. \(SU(2)\)), the geometric bordism theory is not represented by a genuine G-spectrum as it fails to admit certain Wirthmüller isomorphisms. In particular, the Thom-Pontryagin map can not be an isomorphism. We show that if \(G\) has abelian identity component, the Thom-Pontryagin map is an isomorphism. This follows from the non-equivariant statement and a comparison of geometric fixed points; analogous to the statement for unoriented bordism proven in [tom Dieck, 1972] and [Schwede, 2018].

  If time permits, we will explain that for suitable multiplicative tangential structures, the Thom-Pontryagin map becomes an isomorphism after localization at a family of inverse Thom classes.

• Branko Juran: The algebraic \(K\)-theory of algebraic tori via equivariant homotopy theory
  Abstract

  There is an equivalence of categories between algebraic tori over a field \(F\) and topological tori with an action of the absolute Galois group of \(F\). We prove that the algebraic \(K\)-theory of an algebraic torus coincides with the equivariant homology of the associated topological torus with respect to the homology theory represented by the \(K\)-theory of the base field \(F\) and its field extensions. This is joint work in progress with Qingyuan Bai, Shachar Carmeli and Florian Riedel.

• Leonard Tokic: A \(K(2)\)-local splitting of a finite Galois extension of \(\mathrm{MString}\)
  Abstract

  Using a Milnor-Moore type argument one can see that, \(K(2)\)-locally at the prime 2, MString tensored with a finite (degree 96) Galois extension of the sphere splits as a direct sum of Morava E-theories, and that the Ando-Hopkins-Rezk orientation admits a unital section after tensoring with this Galois extension. Using equivariant complex orientations one can then show that a degree 4 extension of MString splits as a direct sum of suspensions of \(\mathrm{TMF}_0(3)\).

  In this talk I want to discuss where the key input for the Minor-Moore argument comes from, compare this result to the Anderson-Brown-Peterson splitting of \(\mathrm{MSpin}\), and explain how one could try to use the theory of cubical structures to determine the splitting maps.

• Noah Wisdom: Fields in Equivariant Algebra
  Abstract

  Tambara functors generalize finite Galois extensions and are of independent interest to equivariant homotopy theorists, appearing as the homotopy groups of ring spectra. Nakaoka defined the notion of a field-like Tambara functor (for which Galois extensions provide an example). In this talk, we will introduce Tambara functors and classify field-like Tambara functors for the group \(C_{p^n}\). Additionally, in collaboration with Ben Spitz and Jason Schuchardt, we will study what it means for such a Tambara functor to be algebraically closed, or "Nullstellensatzian" (in the sense of Burklund-Schlank-Yuan), with an eye towards an equivariant chromatic Nullstellensatz.

Registration

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

Travel Information

The autumn school starts Monday 16 September in the morning and ends Friday 20 September around noon, so it is recommended that you arrive already on Sunday 15 September. 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 2023, in 2022, 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.