Montpellier · 9–10 December 2024

ALPE took place in Montpellier on the 9-10 of December, 2024.

Programme

Titles and abstracts

• Joana Cirici - Universitat de BarcelonaConfiguration spaces of algebraic varietiesAbstract

  I will use the theory of weights in étale cohomology to give a simple and conceptual proof of a theorem of Kriz stating that a rational model for the ordered configuration space of a smooth complex projective variety is given by the second page of the Leray spectral sequence for the rational constant sheaf relative to the obvious inclusion, along with its only non-trivial differential. Our proof builds on Totaro's study of this spectral sequence, combined with a basic observation related to the formality of filtered dg-algebras. An advantage of this proof is that it allows for a generalization to study the p-adic homotopy type of configuration spaces for certain algebraic varieties defined over finite fields. This is joint work in progress with Geoffroy Horel.
• Andrea Di Lorenzo - University of PisaStable maps to quotient stacks with a properly stable pointAbstract

  Given the construction of a moduli stack of curves, it is natural to ask if one can construct a moduli stack of pairs consisting of a smooth curve C together with some extra data. When this extra data is parametrized by an algebraic stack X, this corresponds to constructing a moduli stack of maps from C to X. For enumerative reasons, one wants these stacks of maps to X to be proper. Over the years, the problem of compactifying these stacks of maps to a fixed target X has been solved in several degrees of generality, depending on what kind of object X is. I will present a compactification of the stack of maps to quotient stacks having an integral, projective good moduli space and a properly stable point (plus a mild technical hypothesis, which is automatic for smooth stacks). Examples of new cases which were not covered before include GIT compactifications of stacks of binary forms, of plane cubics, and DM stacks modulo a torus. This is a joint work with Giovanni Inchiostro.
• An-Khuong Doan - KU LeuvenBrill-Noether theory on curves in terms of L_{\infty} pairsAbstract

  We apply the theory of deformations with cohomology constraints in terms of L_{\infty} pairs developed by Budur-Rubió to the case of stable vector bundles on generic smooth projective curves. More precisely, we prove that the associated Brill-Noether loci are locally generic determinantal varieties thereby obtaining important information about their singularities. This is joint work with N. Budur.
• Matthieu Faitg - Université Toulouse III Paul SabatierDerived representations of moduli algebrasAbstract

  Moduli algebras are quantizations of character varieties of surfaces defined by Alekseev-Grosse-Schomerus and Buffenoir-Roche. These algebras admit representations on certain Hom spaces, known as Lyubashenko state spaces. We will show that such representations can be "derived", yielding representations on Ext spaces. We will explain how this generalizes work of Lentner-Mierach-Schweigert-Sommerhauser who constructed derived representations of mapping class groups from Lyubashenko theory. A motivation is that moduli algebras are tightly related to skein theory (we will define them in this way); representation theory of skein algebras is an active topic in quantum topology.
• Bertrand Toën - Université Toulouse III Paul SabatierTannakian duality revisitedAbstract

  The purpose of this talk is to come back to old ideas related to Tannakian duality in the setting of \infty-categories. I'll explain why the naive approach fails in non-zero characteristics and give some ideas of how to improve this. I'll also discuss some examples of Tannakian situations related to local systems and foliations. (Joint work with Nuiten, in progress).
• Lukas Woike - Université de BourgogneThe Cyclic and Modular Microcosm Principle in Quantum TopologyAbstract

  In this talk, I will set up a microcosm principle for cyclic and modular algebras and explain how this can be used to devise a procedure that allows to "integrate" Frobenius algebras in suitable monoidal categories over surfaces. This construction has applications to conformal field theory that will be outlined in the second part of the talk.