The recent surge of interest in positive characteristic birational geometry provided us with a fairly good understanding of the geometry of threefolds in characteristic p>5 and surfaces in any characteristic, but, so far, little was known beyond these cases. In my talk I will explain what are the main issues with extending latest advances to lower characteristics or higher dimensions, and what are the new phenomena that needs to be understood. The talk will be partially based on a joint project with Christopher Hacon.

Following last week's discussion of p-divisible groups over commutative ring spectra, we'll define formal groups over commutative ring spectra. One key example is the Quillen formal group associated to a complex oriented ring spectrum. We'll introduce the idea of an orientation of a formal group, which amounts to an isomorphism with the associated Quillen formal group. This will allow us to state the new construction of Morava E-theory -- it is the object representing oriented deformations of a height n formal group over a perfect field.

In this talk I will revisit the computation, originally due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras for perfect fields of positive characteristic. The original proof relied on an understanding of cyclic polytopes in order to determine the genuine equivariant homotopy type of the cyclic bar construction for a suitable monoid. Using the Nikolaus-Scholze framework for topological cyclic homology the  same result is achieved using only the homology of said cyclic bar construction, as well as the action of Connes’ operator.

Time permitting, I will sketch how to use this method to make new computations of K-theory, in particular for the coordinate axes in affine d-space over perfect fields of positive characteristic. This extends work by Hesselholt in the case d = 2. The analogous results for fields of characteristic zero were found by Geller, Reid and Weibel in 1989. I also extend their computations to base rings which are smooth Q-algebras.

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 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.

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