For algebraic varieties defined over the complex numbers, one can study geometry using both algebraic and analytic methods. Over a non-Archimedean field, one can try to do the same thing using Berkovich spaces. I will discuss positivity notions for metrics on line bundles on varieties defined over discretely or trivially valued fields.

I will discuss problems around definably amenable groups in NIP theories, informed by some invariants coming from topological dynamics.

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 […]