14.00 Charles Vial (Cambridge) Algebraic cycles and fibrations.

The theory of algebraic cycles can be viewed as a homology theory for schemes. Chow groups of algebraic cycles are related to algebraic K-theory via the Riemann-Roch theorem. A natural question is to ask, given a fibration, i.e. a dominant morphism, from a smooth variety X to a smooth variety B, how the Chow groups of X relate to the Chow groups of B and of the fibers of the fibration. I will recall what Chow groups are and give answers to this question when the Chow groups of the fibers are small, in some sense. Time permitting, I will give motivic consequences of such computations.

15.00 Arend Bayer (Edinburgh) Birational geometry of moduli of sheaves on K3s via Bridgeland stability.

I will explain recent results with Emanuele Macri, in which we use wall-crossing for Bridgeland stabiltiy conditions to systematically study the birational geometry of moduli of sheaves on K3 surfaces. In particular, we obtain descriptions of their nef cones via the Mukai lattice of the K3, their moveable cones, their divisorial contractions, and obtain counter-examples to various conjectures in the literature. We also give a proof of a well-known conjecture about Lagrangian fibrations (due to Hassett-Tschinkel/Huybrechts/Sawon). These results are new even for Hilbert schemes on the quartic surface in P^3. Our method is based on a natural map from the space of stability conditions to the movable cone of the moduli space. This gives a systematic connection between wall-crossing and the minimal model program of the moduli space

16.30 Andreas Langer (Exeter) De Rham-Witt complexes and p-adic cohomology.

For an algebraic variety over a finite field of char p one can consider its zeta-function which encodes the number of rational points of the variety over finite field extensions. It is known that this zeta-function is rational , i.e a quotient of polymomials which arise from the Frobenius action on certain finite-dimensional vector spaces over the field of p-adic numbers. These vector spaces occur as suitable p-adic cohomology groups and are defined purely algebraically if the variety is smooth and proper , but involve some p-adic (non-archimedean) analysis otherwise, for example if X is affine. I will explain how these vector-spaces (known as crystalline or rigid cohomology) can be described using (overconvergent) de Rham-Witt complexes. Then I will report on how these cohomology groups can be used to count the number of rational points of the variety and also indicate links to general p-adic Hodge-theory.