Comme chaque année, une demi-journée d'exposés est organisée pour permettre aux doctorant.e.s de l'équipe de présenter leurs travaux. Cette année, nous aurons le plaisir d'écouter des exposés de :

• Jordan Levin : Manifold Structures and Cohomology
  In differential geometry, there is the fundamental problem of classifying smooth manifolds up to diffeomorphism. Similarly, in algebraic topology, there is the question of classifying spaces up to homotopy equivalence. Both of these classification problems are notoriously difficult in general. Remarkably, if we fix a homotopy type together with suitable auxiliary data, the problem of classifying compatible manifolds admits an algebraic description through the machinery of Surgery Theory. Roughly speaking, the relevant algebraic invariants arise as the homotopy groups of generalized cohomology theories called Algebraic L-theory. Though much of this theory was developed in the last century, my work concerns modern refinements arising from new perspectives in higher algebra and stable homotopy theory.
• Michele Tamagnone : The Decomposition Theorem for the Beauville-Mukai system
  Ngô’s refinement of the BBDG decomposition theorem is a strong tool for the study of the cohomology of Lagrangian fibrations: it allows to describe the cohomology of the total space in terms of certain strings identified by local systems on the base.
  We will see this at work for the Beauville-Mukai system, a deformation of the Hitchin system parametrizing sheaves on a K3 surface. Ngô’s theorem will reduce the proof of the full support property for this system to the study of the irreducible components of the compactified Jacobian of a non-reduced curve with non-smooth reduction, an argument not deeply explored in the literature.
• Tianyang Wang : Obstruction de Brauer-Manin transcendante pour les espaces homogènes
  L'obstruction de Brauer-Manin est un outil arithmétique fondamental introduit pour expliquer les défauts au principe de Hasse et à l'approximation faible sur les variétés algébriques. Son étude sur les espaces homogènes s'est historiquement concentrée sur la partie algébrique du groupe de Brauer, aboutissant notamment aux résultats de Borovoi en 1996 pour les stabilisateurs abéliens ou connexes.
  Durant cette présentation, je rappellerai d'abord un exemple de défaut au principe de Hasse et l'émergence de l’obstruction de Brauer-Manin avant de présenter la divergence de nature entre les obstructions algébriques et transcendantes. J’illustrerai ensuite la nécessité de cette composante transcendante en détaillant un contre-exemple explicite et récent à l'approximation faible sur un espace homogène.
• Ronghan Yuan : Skein Algebras at Roots of Unity
  Skein algebras can be thought of as noncommutative versions of the coordinate rings of SL2C-character varieties. Bonahon–Wong and Frohman–Kania-Bartoszynska–Lê identified the center of these algebras and showed that it is isomorphic to the coordinate ring of the character variety. They also proved that, after localizing the center appropriately, the skein algebra is Azumaya.
  Later, Ganev–Jordan–Safronov and also Detcherry–Santharoubane showed that the skein algebra is Azumaya over the smooth locus of the character variety. In this short talk, I will discuss how to prove that this Azumaya algebra is nontrivial, using its Brauer class as a cohomological invariant.

Les exposés commenceront à 14h30 et se termineront à 17h30. Ils seront suivis d'un buffet convivial.