| Résume | Après-midi de Topologie - jeudi 23 mai 2024
Organisée par Christian Ausoni (LAGA), Geoffroy Horel (LAGA), Muriel Livernet (IMJ-PRG), Najib Idrissi (IMJ-PRG).
Dan Berwick-Evans (14h) : What is the homotopy type of quantum field theory?
Spaces of quantum theories are the fundamental objects in several modern applications of algebraic topology to theoretical physics. In this talk, I will begin by explaining how twisted equivariant K-theory encodes the homotopy type of the space of (supersymmetric) quantum mechanical systems. Viewing quantum systems as 1-dimensional quantum field theories, generalizing these structures suggests a connection between 2-dimensional (supersymmetric) quantum field theories and twisted equivariant elliptic cohomology, building on ideas of Segal, Stolz and Teichner.
Sacha Ikonicoff (15h15) : Quillen Cohomology of Divided Power Algebras over an Operad
Divided power algebras are algebras equipped with additional monomial operations. They naturally arise in the context of positive characteristic, particularly in the study of simplicial algebras, crystalline cohomology, and deformation theory. An operad is an algebraic object that encodes operations: there is an operad for associative algebras, one for commutative algebras, for Lie algebras, Poisson algebras, and so on. Each operad produces a category of associated algebras, as well as a category of divided power algebras. The purpose of this talk is to demonstrate how Quillen cohomology generalizes to many categories of algebras using the notion of operad. We will introduce the concepts of modules and derivations, as well as an object representing modules - called the universal enveloping algebra - and an object representing derivations - called the module of Kähler differentials - which will allow us to construct an analogue of the cotangent complex. We will show how these concepts allow us to recover known cohomological theories on certain categories of algebras and provide new and somewhat exotic notions when applied to divided power algebras.
Pedro Magalhaes (16h45) : Formality of Kähler manifolds revisited
The interaction of Hodge structures with rational homotopy theory is a powerful tool to provide restrictions on the homotopy types of Kähler manifolds and of complex algebraic varieties. An example is the well-known result of Deligne, Griffiths, Morgan and Sullivan, stating that compact Kähler manifolds are formal. In the simply connected case, it implies, for instance, that the rational homotopy groups of such manifolds are a formal consequence of the cohomology. Despite this fact, the mixed Hodge structure on their rational homotopy groups is not, in general, a formal consequence of the Hodge structures on cohomology. To understand this phenomenon, we will introduce a stronger notion of formality which arises from studying homotopy theory in a category encoding the Hodge structures. We will also introduce obstructions to this strong formality, generalizing the classical ones, and study when are Kähler manifolds formal in this stronger sense. |
