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.
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 on 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.
There will be additional talks providing background, and participants will have the chance to give short presentations about their own work.
Registration
If you like to participate in the autumn school, please
fill out the the registration form. The
deadline for registrations is 10 June 2023. Due to
the limited capacity, we may not be able to admit all
registered participants. The decisions will be made soon
after the deadline.
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 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.