The Ehrhart polynomial counts the number of lattice points in integer dilates of a lattice polytope. The h*-polynomial encodes the Ehrhart polynomial in a particular basis. In this talk we give an introduction to the method of interlacing polynomials which is a powerful tool to prove that a polynomial has only real roots and present applications to h*-polynomials of zonotopes and dilated lattice polytopes.

A new feature of varieties in positive characteristic is the occurence of morphisms between smooth varieties whose fibres are all singular.  This provides an obstacle to using induction by restriction to a general fibre.  One must overcome this by working with the generic fibre instead, which is a regular but not smooth variety over an imperfect field. In this talk I will discuss progress in developing the LMMP for such varieties, and in controlling their bad behaviour in situations of interest to the LMMP such as Mori fibre spaces. This talk will be based on joint work with Zsolt Patakfalvi and works in progress with Omprokash Das and Lena Ji.

We will discuss the smash product.

We prove the \'etale comparison theorem for prismatic cohomology.

I will discuss what we know about the relation between p-adic etale cohomology and de Rham cohomology of rigid analytic varieties, where, in general, both cohomology groups are infinite dimensional. This is based on a joint work with Pierre Colmez and Gabriel Dospinescu.

We consider deformations of objects over non-commutative rings, which are more natural than commutative ones in some cases. As examples, we treat deformations of perverse coherent sheaves, and semi-orthogonal decompositions of derived categories corresponding to singularities.

The cycle map from Chow groups to Hodge cohomology has a long and illustrious history. This talk will concern a variant adapted to log-smooth pairs, mapping out of the “log Chow groups” and taking values in log Hodge cohomology (using differential forms with log poles).

One way to investigate whether two links are equivalent is to use link invariants. One of the most famous ones is Khovanov-Rozansky triply graded homology. It is defined algebraically and is notoriously hard to compute. Gorsky, Negut and Rasmussen conjectured that it can be calculated as cohomology of certain coherent sheaves on the flag Hilbert scheme. A link invariant of a similar nature has been constructed by Oblomkov and Rozansky in terms of matrix factorizations. The aim of this joint work in progress with Roman Bezrukavnikov, Alexei Oblomkov and Andrei Negut is to prove the conjecture by reformulating Oblomkov-Rozansky using work of Ben-Zvi and Nadler et al. together with work of Arkhipov and Kanstrup