European Autumn School in Topology 2025

This autumn school will take place during the week 15-19 September 2025 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:

• Joana Cirici: Sheaves in Homotopy Theory and Geometric Applications

  Abstract I will explain how to describe rational homotopy types of topological spaces as well as the E-infinity structure on singular cochains, using sheaf-theoretic methods. Then, we will discuss various situations where this approach is useful, mostly for studying homotopy types of algebraic varieties, using mixed Hodge theory and Galois actions in étale cohomology.

• Piotr Pstrągowski: Descent in stable homotopy theory

  Abstract The main calculational tool in the context of stable homotopy theory is given by the Adams spectral sequence. The aim of this lecture series is to give a gentle introduction to this topic, both from the perspective of descent and that of an intermediary between stable and abelian categories. In particular, I plan to cover the following:

  • The Adams spectral sequence of a ring spectrum, some classical calculations
  • Homology theories, epimorphism classes, injective resolutions
  • Encoding spectral sequences using deformations of stable infinity-categories
  • Applications: monoidality of the Adams filtration, coherent multiplication on Moore spectra.

Preparatory talks

• Kai Shaikh: Abelian categories, derived functors, derived categories (60 minutes)
• Georg Jakob: Sheaf theory and sheaf cohomology (60 minutes)
• Elias Klakken Angelsen: Stable \(\infty\)-categories, spectra (60 minutes)
• Qi Zhu: Spectral sequences (60 minutes)
• Maroš Grego: \(E_\infty\)-algebra structure on singular cochains (60 minutes)

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

Contributed talks

• Pedro Meneses Magalhães: Formality of Mixed Hodge Structures and Kähler Manifolds
  Abstract

  The Formality Theorem of Deligne-Griffits-Morgan-Sullivan states that the rational homotopy type of a compact Kähler manifold is entirely determined by its cohomology ring. This provides homotopical obstructions for a compact complex manifold to admit a Kähler metric. On the other hand, by results of Deligne and Morgan, the rational homotopy groups of simply connected Kähler manifolds carry natural mixed Hodge structures. However, there exist diffeomorphic compact Kähler manifolds with the same pure Hodge structures on cohomology but with different mixed Hodge structures on rational homotopy groups. To better understand this phenomenon, in this talk I will introduce a "mixed Hodge formality" theory and provide complex geometric invariants obstructing this stronger notion of formality.

• Bhavna Joshi: Constructing Deformations of Additive \(\infty\)-categories
  Abstract

  We define the notion of homology theories on additive \(\infty\)-categories, and concern ourselves with a class of suitably well behaved ones for constructing a "deformation" category of the additive \(\infty\)-category along them. I will begin with the background story for this construction in the stable setting (developed by Irakli Patchkoria and Piotr Pstragowski), motivating why we care about the deformation category, and addressing the need for generalisation to additive setting. We will then see the proof sketch of the universal property of our construction and comment about other subtleties that arise in additive setting.

• Marius Nielsen: On the geometrization of synthetic spectra
  Abstract

  The category of synthetic spectra is a strong tool for understanding the Adams-Novikov spectral sequence and acts as a 1-parameter deformation between spectra and quasi-coherent sheaves on the moduli of formal groups. In this talk, I will associate a category of synthetic quasi-coherent sheaves to any non-connective spectral stack. Which acts as a 1-parameter deformation of quasi-coherent sheaves on the spectral stack and quasi-coherent sheaves on the associated Dirac stack. If time permits, I will discuss the relationship with the descent spectral sequence and previous definitions of synthetic spectra.

• Francesca Pratali: On \(\infty\)-operads and discrete operads
  Abstract

  In this talk, I will discuss some aspects of the homotopy theory of \(\infty\)-operads, which provide a flexible framework for describing algebraic structures up to homotopy. The main focus will be their relationship with their strict, discrete counterparts - operads without homotopical data-, which I will explore through the lens of localization and straightening theorems. After stating and explaining our main results in this direction, I will discuss their applications and how they open new questions about the interaction between homotopical and strict structures.

Registration

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

We will be able to offer lodging and meals to accepted participants, but we cannot cover any travel expenses.

Travel Information

The autumn school starts Monday 15 September in the morning and ends Friday 19 September around noon, so it is recommended that you arrive already on Sunday 14 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 2024, 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.