A generalised power series (in several variables) is a series with real nonnegative exponents whose support is contained in a cartesian product of well-ordered subsets of the real line. Let A be the collection of all convergent generalised power series. I will show that, if f(x_1,...,x_n,y) is in A, then the solutions y=g(x_1,...,x_n) of the equation f=0 can be expressed as terms of the language which has a symbol for every function in A and a symbol for division. The construction of the terms is rather explicit. If instead of […]

Let K be a non-archimedean complete normed field, K_alg the algebraic closure of K and let L be the language of normed fields augmented with symbols for the strictly convergent powerseries over K. Strictly convergent rigid subanalytic sets over K are the subsets of (K_alg)^n definable in L. I will survey what is known about these sets, including recent joint results with Raf Cluckers.

The dimension of the étale cohomology groups, with coefficients in Z/lZ, of a scheme of finite type over an algebraically closed field of characteristic different from l, is computable in the sense of Church-Turing. To prove this, we construct a hypercovering of X by schemes (analogous to Artin's ?Roegood neighborhoods?R) having algorithmically testable geometric properties which allow to reduce the computation of the cohomology of X to that of their completed fundamental group.

(Joint work with Pierre Simon) I will present some new results on definably amenable groups in NIP theories (typical examples of which are definably amenable groups in o-minimal theories, algebraically closed valued fields and p-adics). In particular I will demonstrate that in this context various notions of genericity coincide (answering some questions of Newelski and Petrykowski) and a characterization of ergodic measures will be given. Arguments rely on the theory of forking for types and measures in NIP theories and the so-called (p,q)-theorem from combinatorics.If time permits, I will describe […]