| Speaker | Title | Abstract | |
| Yves André | On Galois representations, differential equations and q-difference equations: sketch of a p-adic unification | The title refers to two well-known analogies: between Galois representations of local fields and complex linear differential equations over a punctured disk on one hand, and between differential equations and q-difference equations on the other hand. In the p-adic world, it turns out that these analogies can be substantially strengthened, and we shall indicate how they eventually lead to some equivalences of categories. |
|
| Michel Boileau | Geometry of finite group actions on 3-dimensional manifolds | We will discuss the problem of geometrizing finite group actions on a closed orientable 3-manifold. We will focus on the case of non-free actions where a geometrization theorem is known and give applications to the study of cyclic branched coverings of the 3-sphere along knots. | |
| Antoine Chambert-Loir | Metrized line bundles and measures on Berkovich spaces | The proofs by Ullmo and Zhang of Bogomolov's conjecture concerning the points of small heights in abelian varieties made a crucial use of an equidistribution property of these points in the corresponding complex abelian variety. The question of the p-adic distribution of these points has however only partial results which reveal the necessity of introducing measures on Berkovich space attached to metrized line bundles. Inspired by a formula of Szpiro, Tucker and Pineiro, these measures can also be used in the context of an arithmetical dynamical system over the projective line to derive a dynamical analogue of Mahler's formula for the height of an algebraic number. | |
| Marius Dardalat | Groups, metric spaces and C*-algebras | We plan to survey certain generalizations of amenability such as exactness and Hilbert space uniform embeddability for discrete groups | |
| William Duke | On the mod p reductions of an elliptic curve | In this talk I will first introduce and then describe some recent advances in the study of the mod p reductions of an elliptic curve E defined over the rational numbers. In particular, the distribution of the structures of various groups associated to a reduction as the prime varies will be discussed, motivated by seminal work of Serre and analogies with classical problems from analytic number theory. I will then describe a recent result with A. Cojocaru on the Tate-Shafarevich groups of the reductions, considered as being defined over their function fields. Assuming the Generalized Riemann Hypothesis when the curve has no complex multiplications, we show that the Tate-Shafarevich group is trivial for a positive proportion of primes, provided E has an irrational point of order two. | |
| Matthew J. Emerton | p-adic modular forms, p-adic L-functions, and locally analytic representation theory | In this talk I will first descibe the theory of Jacquet modules in the context of locally analytic representations of p-adic groups. I will then explain how this theory can be applied to construct eigenvarieties that p-adically interpolate systems of eigenvalues attached to automorphic Hecke eigenforms, and (at least in some contexts) to construct several-variable p-adic L-functions along these eigenvarieties. | |
| Gerhard Huisken | Surgery for flows on 3-manifolds and 3-hypersurfaces | Geometric parabolic evolution equations like the Ricciflow and the mean curvature flow can be used to smoothen and deform the Geometry of Riemannian manifolds and of hypersurfaces into more uniform shapes. The lecture discusses recent work of Perelmann on surgeries in Ricciflow and of Huisken and Sinestrari on surgery in mean curvature flow that leads to classification results for 3-manifolds and 3-hypersurfaces. | |
| Boris Khesin | Holomorphic linking numbers and Poisson structures | The Gauss linking number of two curves in the three-space has a complex counterpart. In the talk we define the holomorphic linking number for complex curves in complex three-folds. Moreover, one can define "polar homology" groups of complex projective manifolds by regarding meromorphic forms on their submanifolds as a complex analogue of orientation, and taking the residue as the boundary operator. We also discuss gauge-theoretic aspects of the above correspondence, and, in particular, its relations to the moduli spaces of flat connections and holomorphic Chern-Simons theory. | |
| Gerard Laumon | On the fundamental lemma for unitary groups | Assuming the purity conjecture for the affine Springer fibers which has been formulated by Goresky, Kottwitz and MacPherson, we prove a geometric analog of the fundamental lemma for unitary groups. Our approach is similar to the one of Goresky, Kottwitz and MacPherson. Our main new ingredient is the link between affine Springer fibers and compactified Jacobians. | |
| Joachim Lohkamp | The General Positive Energy Theorem | In this talk we explain the mechanism of how to overcome the previous dimensional and topological restrictions in the proof of the positive energy theorem which is a prototype result for the non-existence of positve scalar curvature metrics on geometrically large manifolds. | |
| Ulrike Tillmann | The space of strings and the stable cohomology of moduli spaces | In this talk we will explain the motivation for studying the string category, and how its properties can be explored in the study of the cohomology of the moduli space of Riemann surfaces. We will include a report on recent results with Ib Madsen and Soren Galatius which determines the divisibility of the stable Miller-Morita-Mumford classes in the torsion free quotient of the integral cohomology ring. The description involves denominators of Bernoulli numbers. (I will try to make this talk complementary to Michael Weiss' talk.) | |
| Boris Tsygan | Algebraic structures on Hochschild complexes and topological quantum field theory | We discuss recent works on the algebraic structures on Hochschild chain and cochain complexes (Kontsevich, Tamarkin and myself, Nest and myself, Caldararu, Markarian) and how they relate to topological quantum field theory. | |
| Michael Weiss | Cohomology of the stable mapping class group | The stable mapping class group is the group of automorphisms of a connected oriented surface of "large" genus. The Mumford conjecture postulates that its rational cohomology is a polynomial ring generated by certain classes of dimension 2i, one for each i greater than 0. Tillmann's insight that the plus construction makes the classifying space of the stable mapping class group into an infinite loop space led Ib Madsen to a stable homotopy theory version of Mumford's conjecture, stronger than the original. This stronger form of the conjecture was recently proved by Madsen and myself. In the second half of my talk, I will outline the strategy of the proof, which is in part a reduction to e result from singularity theory. The first half of the talk will be more historical. | |