Past Event

ALPE took place in Montpellier on 19-20 of March, 2024.

Programme

List of talks and abstracts.

Titles and abstracts

• Pieter Belmans - University of Luxembourg Hochschild cohomology of Hilbert schemes of points ▼
  I will present a formula describing the Hochschild cohomology of symmetric quotient stacks, computing the Hochschild-Kostant-Rosenberg decomposition of this orbifold. Through the Bridgeland-King-Reid-Haiman equivalence this allows the computation of Hochschild cohomology of Hilbert schemes of points on surfaces. These computations explain how this invariant behaves differently from say Betti or Hodge numbers, which have been studied intensively in the past 30 years, and it allows for new deformation-theoretic results. This is joint work with Lie Fu and Andreas Krug.
• Emma Brakkee - Leiden University Singular symplectic varieties via Prym fibrations ▼
  TWe construct new examples of singular symplectic varieties, as relative Prym varieties associated to linear systems on surfaces with a double cover of a K3 surface. This construction has been studied before for the anti-canonical linear system on low degree del Pezzo surfaces, and for Enriques surfaces. We expand on this by considering arbitrary surfaces with a K3 double cover. I will explain the construction and discuss criteria for when the resulting variety is primitive symplectic or irreducible symplectic. This is joint work in progress with C. Camere, A. Grossi, L. Pertusi, G. Saccà and A. Viktorova.
• Sophie d’Espalungue d’Arros - Université de Lille Generalized operads for multi dimensional algebra ▼
  While non-symmetric operads are shaped on the monoidal structure of the set $\mathbb N$ of natural numbers, symmetric operads are shaped on the symmetric monoidal structure of the symmetric groupoid $\mathfrak S$.
  It is well known that $E_n$-operads can not be described as non-symmetric operads for $n\neq 1$, resulting in a complex combinatorial structure that still lacks a purely algebraic characterization for intermediate values of $n$.
  This expository talk is intended to promote an operadic-like algebraic framework which fits the structural constraints arising from interchange. For this purpose, I propose to revisit the theory of operads and their algebras from a conceptual point of view, with the aim of providing a formal notion of operad that is contingent upon a specified structure.
  I will explain how operads build on an iterated monoidal framework are likely to provide an efficient characterization of $n$-fold loop spaces and discuss the prospects regarding the identification of polytopes analogous to Stasheff's associahedra.
• Mario Fuentes - IMT Derivations and moduli spaces of some rational homotopy types ▼
  Thanks to Daniel Quillen's approach to rational homotopy theory, we can study topological spaces through their respective `models' of Lie algebras. For example, if we fix a homology, we can ask which are the different homotopy types of topological spaces that share that homology (and similarly with homotopy groups or cohomology algebras). From the algebraic point of view, this question corresponds to finding the different differentials that we can place in a certain fixed Lie algebra $L$. This question induces a deformation problem and, following Deligne's principle, these deformations will be given by the Maurer-Cartan elements in the derivations of L. In this talk, we will introduce these concepts and study some simple properties of the moduli space associated with the Maurer-Cartan elements of complete Lie algebras of derivations.
• Jeroen Hekking - University of Regensburg Reduction of stabilizers for derived Artin stacks ▼
  For an Artin stack X, stabilizer reduction is a procedure to resolve the stackiness of X to produce a Deligne-Mumford stack X˜ by a sequence of (modified) blow-ups. This was first carried out by Kirwan for smooth quotient stacks using GIT [Kir85], and later generalized by Edidin–Rydh [ER21] and independently by Kiem–Li–Savvas [KLS17, Sav20] to Artin stacks admitting a good moduli space. Applications include a partial desingularization of the good moduli space, and the construction of virtual fundamental cycles in the (-1)-shifted symplectic case. In this talk we will look at a derived enhancement of this algorithm. In order to carry this out, we first introduce a derived analogue of Alper's good moduli spaces, and define derived (saturated) blow-ups. If time permits, we will see how the picture can be described étale-locally using derived quotient stacks and derived GIT quotients. This is joint work with Eric Ahlqvist, Michele Pernice, David Rydh and Michail Savvas.
• Joan Millès - IMT André—Quillen cohomology of curved algebras ▼
  Curved algebras appear in several fields such as symplectic and complex geometry. In this talk, we present cofibrant resolutions in a model category structure tailored for curved algebras. We use these resolutions to describe a cohomology theory for curved algebras. As for dg algebras, we construct these resolutions by means of bar and cobar constructions alongside Koszul duality theory, leading to homotopy versions of curved algebras. Key examples include curved unital associative algebras and curved complex Lie algebras. Additionally, we compute the André—Quillen cohomology for specific (homotopy) curved complex Lie algebras, revealing an interesting connection with derived complex geometry.