12h-14h : buffet 

14h-15h : Dmitry Kubrak

Title : Canonical categorical quantization of symplectic varieties in characteristic p"

Abstract : A (formal) quantization of a Poisson algebraic variety X is a formal (non-commutative) deformation O_h of the sheaf of functions on X that agrees with the Poisson bracket in the "classical limit". In 2003, Bezrukavnikov and Kaledin, building on the work of Fedosov in the real-analytic context, constructed canonical quantizations of affine algebraic symplectic varieties in char 0. In 2005 they extended their construction to char p, producing canonical quantizations of a subclass of so-called restricted (affine) symplectic varieties. However, in both cases it's only the isomorphism class of the quantizations that is canonical, and in fact there are non-trivial obstructions to gluing the algebra O_h globally. It was then suggested by Kontsevich, and proved by Van der Bergh in char 0 in 2006, that the quantization does glue globally if one replaces the algebra O_h by the corresponding category of O_h-modules: this gives a formal deformation of the category QCoh(X) of quasi-coherent sheaves on X, which one calls a categorical quantization. I will talk about joint work with Bogdanova, Travkin and Vologodsky where we construct a canonical categorical quantization in the char p setting. An interesting feature of char p is that this canonical quantization extends from the formal disk (with formal parameter h) to the whole affine line, and can even be compactified to a family of categories over the projective line P^1. This family enjoys certain strong uniqueness properties and is reminiscent of Simpson's twistor space over complex numbers. It also allows to generalize the notion of mod p F-gauges of Bhatt-Lurie to general restricted symplectic varieties X (in such a way that it gives back mod p F-gauges on Y for X=T^*Y).

15h-15h30 : pause café 

15h30-16h30 : Omid Amini 

Title : A decomposition theorem for Lefschetz modules

Abstract : A Lefschetz module is a module M over a graded algebra A that satisfies analogues of Poincaré duality, the Hard Lefschetz property, and the Hodge-Riemann relations with respect to an open convex cone K in the degree one part of A (an abstract analogue of the ample cone). In a joint work with June Huh and Matt Larson, we analyze the decomposition of M into indecomposable modules over subrings of A that are generated by elements in the closure of K (analogue of the nef cone in this abstract setting), establishing a set of structural properties that parallel the decomposition theorem for morphisms of complex projective varieties (in the theory of perverse sheaves). I will present our results and discuss some applications.