(Oxford University)

One of many overlaps between logic and topology is duality: Stone duality links Boolean algebras with zero-dimensional compact Hausdorff spaces, and gives a useful topological way of describing certain phenomena in first order logic; and there are generalisations that allow one to study infinitary logics also. We will look at a couple of ways in which this duality theory is useful.'

This is the 4th talk of the study group on Beilinson's approach to p-adic Hodge theory, following the notes of Szamuley and Zabradi.

I shall finish the computation of the module of differentials of the ring of integers of the algebraic closure of Q_p and describe a universal thickening of C_p.

I shall also quickly introduce the derived de Rham algebra. Kevin McGerty will give a talk on the derived de Rham algebra in W5 or W6.

In this lecture we describe the effective Chebotarev Theorem for global function fields and show how this can be used to describe the statistics of a polynomial map f in terms of its monodromy groups. With this tool in hand, we will provide a strategy to remove the remaining heuristic in the quasi-polynomial time algorithm for discrete
logarithm problems over finite fields of small characteristic.

I will describe a sketch of the proof of Grothendieck conjecture on fundamental groups.
 

This is the first talk of the workshop organised by F. Brown, M. Kim and D. Rössler on Beilinson's approach to p-adic Hodge theory. 

In this talk, we shall give the definition and recall various properties of the cotangent complex, which was originally defined by L. Illusie in his monograph "Complexe cotangent et déformations" (Springer LNM 239, 1971).

I will explain how to find an explicit embedding of the Kummer variety of a higher genus curve into projective space and discuss applications of such an embedding to the study of rational points on Jacobians of curves, as well as the original curves.