Algèbre et géométrie

Following the spirit of Grothendieck's Esquisse d'un Programme, the Ihara/Oda-Matsumoto conjecture predicted a combinatorial description of the absolute Galois group of Q based on its action on geometric fundamental groups of varieties. This conjecture was resolved in the 90's by Pop using anabelian techniques. In this talk, I will discuss the proof of stronger variant of this conjecture, using mod-ell two-step nilpotent quotients, while highlighting some connections with model theory.

Soulignée par Nash dans les années 60, l'interaction entre la géométrie des espaces d'arcs et la théorie des singularités s'est fortement amplifiée sous l'influence de la théorie de l'intégration motivique notamment. Dans cet exposé, nous introduirons le schéma des arcs associé à une variété algébrique et donnerons quelques illustrations de cette interaction. Parmi elles, nous parlerons de l'interprétation (possible) du point de vue des singularités d'un théorème de Drinfeld et Grinberg-Kazhdan démontré au début des années 2000. (Cette dernière partie de l'exposé s'appuie sur une collaboration avec David Bourqui.)

Given a system of polynomial equations in m complex variables with solution set of dimension d, if we take finite subsets X_i of C each of size at most N, then the number of solutions to the system whose ith co-ordinate is in X_i is easily seen to be bounded as O(N^d). We ask: when can we improve on the exponent d in this bound?Hrushovski developed a formalism in which such questions become amenable to the tools of model theory, and in particular observed that incidence bounds of Szemeredi-Trotter type […]

Originating in the theory of unitary group representations, the notion of spectral gap has played a huge role in many of the deep results in the theory of von Neumann algebras in the last couple of decades. Recently, with my collaborators, we are slowly understanding the model-theoretic significance of spectral gap, in particular its connection with definability. In this talk, I will discuss a few of our recent observations in this direction and speculate on some further possible developments. I will assume no knowledge of von Neumann algebras nor continuous […]

The Chabauty method often allows one to find the rational points on curves of genus at least 2 over the rationals, but has a lot of limitations. On a theoretical level, the Mordell-Weil rank of the Jacobian of the curve has to be strictly smaller than its genus. In practice, even when this condition is satisfied, the relevant Coleman integrals can usually only be computed for hyperelliptic curves. We will report on recent work of ours (with different combinations of collaborators) on extending the method to more general curves. In […]

Zilber conjectured that the complex field equipped with the exponential function is quasiminimal: every definable subset of the complex numbers is countable or co-countable. If true, it would mean that the geometry of solution sets of complex exponential-polynomial equations and their projections is somewhat like algebraic geometry. If false, it is likely that the real field is definable and there may be no reasonable geometric theory of these definable sets.I will report on some progress towards the conjecture, including a proof when the exponential function is replaced by the approximate […]