Talk

Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let G be a p-subgroup of the k-automorphism group of C such that |G| > 2pg/(p-1). Then, C -->C/G is an étale cover of the affine line Spec k[X] totally ramified at infinity. To study such actions, we focus on the second ramification group G_2 of G at infinity, knowing that G_2 actually coincides with the derived group of G. We first display realizations of such actions with G_2 abelian of arbitrary large exponent . Our examples come from the construction of curves with many rational points using ray class field theory for global function fields. Then, considering additive polynomials of k[X], we obtain a structure theorem for the functions parametrizing the Artin-Schreier cover C --> C/G_2, in the case of a p-elementary abelian G_2. We finally emphasize the link between the curves obtained in this last case and supersingular curves (i.e. curves whose Jacobian is isogeneous to a product of supersingular elliptic curves).

The talk will be devoted to the problem of classification of all natural operations acting on the Hochschild cohomology of an associative algebra, by which one usually means the understanding of the homotopy type of the operad B formed by these operations. We will explain what a "natural operation" is and present two equivalent definitions of completely different natures. We then describe the homotopy type of B and show how is this descrition related to the Deligne Hochschild cohomology conjecture.

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface X endowed with an antiholomorphic involution which determines topologically the original surface S. In this talk, we relate dianalytic vector bundles over S and holomorphic vector bundles over X, devoting special attention to the implications this has for moduli spaces of semistable bundles over X. We construct, starting from S, Lagrangian submanifolds of moduli spaces of semistable bundles of fixed rank and degree over X. This relates the present work to constructions of Ho and Liu over non-orientable compact surfaces with empty boundary.

It's an active trend to exhibit classical invariants like lattices or representations as traces of categorical invariants. In this talk, lifts of the Milnor lattice of certain surface singularities to categories will be discussed. As it turns out, there are different possibilities, but we will also point out a way to compare them. The original content mentioned here is from joint work with Wolfgang Ebeling and Chris Brav.

We introduce p-adic Kummer spaces of continuous functions on $Z_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and some decompositions. This theory can be transferred to values of Dirichlet L-functions at negative integer arguments in residue classes. That leads to a conjecture about their structure supported by several computations using a link to p-adic functions that are related to Fermat quotients. Finally, we present a conjectural formula of the structure of the classical Bernoulli and Euler numbers.

This is an purely introductory talk on the subject of motives of smooth, projective varieties. It assumes only rudimentary acquaintance with algebraic cycles, Chow groups, and the basic operations intersection product, flat pullback, proper push forward as discussed in the previous lecture in this seminar (December 21,2009). The purpose of the talk is to introduce the category of motives and to discuss some basic examples.