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 […]
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 […]
We consider valued fields of equicharacteristic zero equipped with a continuous derivation. This class of structures is rather diverse, including both monotone differential fields and asymptotic differential fields. (These terms will be defined.) Nevertheless, some results can be established uniformly for the entire class: algebraic extensions, construction of residue field extensions, the Equalizer Theorem, construction of immediate extensions, differential-henselianity. Next I will revisit Scanlon's thesis on the model theory of differential-henselian monotone differential fields with enough constants. Time permitting I will add some remarks on the case of asymptotic differential […]
We describe a recent program for analyzing definable sets and groups in certain model theoretic settings. Those settings include:(a) o-minimal structures (M, P), where M is an ordered group and P is a real closed field defined on a bounded interval (joint work with Peterzil),(b) tame expansions (M, P) of a real closed field M by a predicate P, such as expansions with o-minimal open core (work in progress with Gunaydin and Hieronymi).The analysis of definable groups first goes through a local level, where a pertinent notion of a pregeometry […]
The Tate-Voloch conjecture is a statement about p-adic distance from torsion points to subvarieties in a semi-abelian variety defined over C_p. The use of Galois equations on torsion points by Pink and Rossler to prove the Manin-Mumford conjecture can be adapted to prove that conjecture in the case where both the semi-abelian variety and its subvariety are defined over a finite extension of Q_p.In this talk, we will present such a proof, and try to give an insight on how this proof differs from the model-theoretic one given by Scanlon.
In the spirit of work by Pila-Wilkie (2006) and by Pila (2009), we will present bounds on the number of points of bounded height in the non-archimedean context. An important tool to make the determinant method work is provided by a non-archimedean version of the Yomdin - Gromov parameterizing lemma. We wil explain these results, obtained in joint work with G. Comte and F. Loeser.
