| Résume | Après-midi de Topologie - 6 novembre 2024
Organisée par Christian Ausoni (LAGA), Geoffroy Horel (LAGA), Muriel Livernet (IMJ-PRG), Najib Idrissi (IMJ-PRG).
Victor Carmona : AQFTs vs. Factorization Algebras: toward a higher comparison
Quantum Field Theory (QFT) is an exciting yet elusive domain within mathematics and physics. Despite the lack of rigorous foundations to support many advancements made by physicists, mathematicians have engaged in a fruitful endeavor to formalize QFTs. In current times, we find ourselves at a crossroads: while we still lack the comprehensive techniques and language to fully grasp QFT, numerous distinct axiomatic frameworks are attempting to capture its essence. A natural question
arises: how are these approaches connected? This talk will focus on two such frameworks: Algebraic Quantum Field Theories (AQFTs) and Factorization Algebras (FAs), both of which encapsulate the algebraic structure carried by observables in a QFT. The significance of these frameworks is motivated, among other things, by rigorous programs led by Fredenhagen-Rejzner and Costello-Gwilliam to construct perturbative QFTs using AQFTs and FAs, respectively. Recent contributions from Gwilliam-Rejzner and Benini-Musante-Schenkel establish a relationship between these two programs, vaguely speaking. At a more structural level, Benini-Perin-Schenkel have established an equivalence of 1-categories between specific subcategories of AQFTs and (time-orderable pre-)FAs. The goal of this talk is to present our strategy for establishing an even broader equivalence between ∞-categories of AQFTs and tpFAs. This additional layer of generality is crucial for accommodating gauge theories, which are QFTs with non-trivial homotopical content. This talk is based on ongoing joint work with M.Benini, A.Grant-Stuart and A.Schenkel.
Vadim Lebovici : Théorèmes de décomposabilité des modules de persistance multiparamétriques
La décomposabilité des modules de persistance uniparamétriques en somme directe de modules intervalles --- l'existence des fameux "codes-barres"
--- est la clé de voûte de la théorie de l'homologie persistante et de ses applications en analyse topologique de données et en géométrie symplectique. L'impossibilité de trouver de telles décompositions dans le cas multiparamétrique a suscité le développement de diverses approches. Dans cet exposé, je présenterai l'une d'entre elles, consistant à exhiber des sous-classes de modules de persistance admettant effectivement une décomposition en somme de modules intervalles. En outre, l'appartenance à ses sous-classes peut-être testée localement, i.e., sur des sous-ensembles très simples de l'espace des paramètres. Cet exposé est basé sur des travaux en collaboration avec Magnus B. Botnan, Jan-Paul Lerch et Steve Oudot.
Tasos Moulinos : Lifting the Hilbert additive group to the sphere
The Hilbert additive group scheme arises in algebraic geometry as the "unipotent completion" of the integers over Spec(Z). At the level of functions, it carries a canonical filtration compatible with the group structure. This is intimately related to the HKR filtration on Hochschild homology. I will review the above story and then discuss ongoing work (joint with Alice Hedenlund) on lifting the Hilbert additive group, together with its filtration on functions, to the setting of derived algebraic geometry over the sphere spectrum. This crucially uses the yoga of even filtrations, introduced by Hahn-Raksit-Wilson, which I will briefly review as well. |
