European Autumn School in Topology 2023

This autumn school will take place during the week 18-22 September 2023 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:

• Alexander Berglund: Modern foundations of rational homotopy theory, self-equivalences and arithmeticity

  Abstract Rational homotopy theory owes its existence to the seminal works of Quillen and Sullivan from the 60s-70s. The goal of this lecture series is to bring the participants up to speed with some of the most recent developments, touching on foundational aspects as well as applications.

  In the first part we will discuss a beautiful new approach to the foundations of the subject that extends Quillen's theory beyond the simply connected case. This approach, which can be viewed as a culmination of developments due to Hinich, Getzler, Lazarev-Markl, Buijs-Félix-Murillo-Tanré and others, uses spaces of Maurer-Cartan elements in complete differential graded Lie algebras to model rational homotopy types.

  A classical application of Sullivan's theory was a proof that the group of components of the space of self-homotopy equivalences of a simply connected finite CW-complex is an arithmetic group. In the second part, we will revisit the study of self-equivalences equipped with the new foundations. We will discuss an extension of the arithmeticity result to a space-level statement as well as other new results on spaces of self-equivalences.

• Marc Levine: Motivic homotopy theory and quadratic intersection theory

  Abstract Beginning with Morel's theorem identifying the endomorphism ring of the motivic sphere spectrum over a field k with the Grothendieck-Witt ring of quadratic forms over k, motivic homotopy has provided a powerful framework for the development of a quadratic intersection theory. This gives a refinement of the classical intersection theory based on the Chow ring to one that yields interesting invarients in the Grothendieck-Witt ring. We will discuss both the foundational aspects of this theory, starting with Morel's theorem and his development of the sheaves of Milnor-Witt K-groups, and continuing with motivic theories of cohomology and Borel-Moore homology, as developed by Deglise-Jin-Khan and Panin, Panin-Walter and Ananyevsky, and concluding with several explicit applications to giving "quadratic counts" for several problems in algebraic geometry.

Preparatory talks

• Francesca Pratali: Differential graded Lie algebras and differential graded coalgebras
• Jan McGarry: Rationalization and rational completion
• Federico Ernesto Mocchetti: Quadratic forms
• Anna Savelyeva: Linear algebraic groups
• Edith Hübner: Localization and the unstable motivic homotopy category
• Jan Hennig: Chow rings

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

Contributed talks

• Sebastian Chenery: Loop space Decompositions of Connected Sums and Applications to the Vigué-Poirrier Conjecture

  Abstract Recent work of Beben and Theriault on decomposing based loop spaces of highly connected Poincaré Duality complexes has yielded new methods for analysing the homotopy theory of manifolds. In this talk we will expand upon these methods, which we will then apply to give new examples supporting a long standing question of rational homotopy theory: the Vigué-Poirrier Conjecture.

• Ben Spitz: Mackey and Tambara Functors beyond Equivariant Homotopy

  Abstract In this talk, I’ll introduce a general context for interpreting the notions of Mackey and Tambara functors. This subsumes the classical notion from equivariant algebra, as well as the more recent notions of motivic Mackey and Tambara functors introduced by Bachmann. The aim of this approach is to translate theorems between contexts, enriching the theory and providing “cleaner” proofs of essential facts. To this end, I’ll discuss recent progress (S.) in boosting a foundational result about norms from equivariant algebra to this more general context.

• Coline Emprin: Kaledin obstruction classes and formality criteria

  Abstract A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, an operad, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. Kaledin classes were introduced as an obstruction theory fully characterizing the formality of associative algebras over a characteristic zero field. This construction generalizes to study formality problems in more general contexts, such as colored operads or properads over any coefficient ring. The aim of this talk is to present these classes and new formality criteria based on them.

• Ahina Nandy: An interpolation between the algebraic (co)bordism spectrum \(MGL\), and the special linear algebraic (co)bordism spectrum \(MSL\)

  Abstract Just like the universal complex oriented cohomology theory, complex cobordism (\(MU\) , we have a similar notion of universal oriented cohomology theory in the realm of stable \(A^1\) -homotopy theory. Algebraic cobordism (\(MGL\)) would be our analog of \(MU\) . Similar to \(SU\) -(co)bordism \(MSU\) , there is a notion of “universal” special linear oriented theory in motivic world. \(MSL\) or special linear algebraic (co)bordism plays the role of \(MSU\) here. I would like to talk a bit more about \(MSL\). Conner-Floyd determined the torsions in \(SU\)-bordism already back in the late 60’s. The main ingredient of their work was an interpolation between \(MSU\) and \(MU\). It results into a filtration of \(MU\) in terms of \(MSU\) and facilitates further computation. I will talk about an exactly similar interpolation between \(MSL\) and \(MGL\) we have shown. Now, I am trying to use this filtration and the resulting spectral sequence to get some information about stable homotopy groups of \(MSL\). I would also like to talk a bit about that.

Registration

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

Travel Information

The autumn school starts Monday 18 September in the morning and ends Friday 22 September around noon, so it is recommended that you arrive already on Sunday 17 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 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.