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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.

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.

A pants decomposition of a 2-surface S is a maximal set of mutually non-intersecting closed curves on S whose complement is a union of "pants" (i.e. spheres with 3 holes). A double pants decomposition is a union of two pants decompositions of S. We define a remarkable class DP of double pants decompositions which we call "admissible", as well as a natural class T of transformations acting on double pants decompositions. We show that the transformations from T act transitively on the set of admissible double pants decompositions.