Le théorème de Nekhoroshev garantit la stabilité, pendant un intervalle de temps exponentiellement long, des systèmes hamiltoniens qui sont proches des systèmes intégrables. C’est un résultat théorique, qui est hélas difficilement applicable à des systèmes hamiltoniens concrets. Le but de l’exposé est d’expliquer une variante de ce théorème de Nekhoroshev, plus flexible, que l’on espère pouvoir appliquer à des systèmes concrets tels que le problème à trois corps Soleil-Jupiter-Saturne.

Compactifications of M5-branes on non-trivial 3-manifolds lead to a N=2 supersymmetric theories in 3 dimensions. If the 3-manifold in question is a knot complement, various Chern-Simons amplitudes – or corresponding knot invariants – for this knot determine the properties of the resulting 3d, N=2 theory. This relation between N=2 theories and Chern-Simons theory is referred to as the 3d-3d correspondence. M-theory realization of this correspondence implies that N=2 theories obtained in this way possess one special flavor symmetry, which is related to certain deformation of Chern-Simons theory and homological knot invariants. In this talk I will discuss properties of these refined/homological invariants, and their role in the 3d-3d correspondence.

Dans cet exposé, j’expliquerai le comportement de la conjecture de Birch et Swinnerton-Dyer dans la Zp-extension cyclotomique : si p est un nombre premier ordinaire, il existe une fonction L p-adique classique, avec des invariants canoniques qui fournissent une description adéquate de ce comportement. Si p est un nombre premier supersingulier, on peut le décrire par une paire de fonctions L p-adiques (avec une paire d’invariants), auquel cas il y a des phénomènes mystérieux.

We define the characteristic cycle of a locally constant étale sheaf on a smooth variety in positive characteristic, ramified along the boundary, as a cycle in the cotangent bundle of the variety, at least on a neighborhood of the generic point of the divisor on the boundary.

We discuss a compatibility with pull-back and local acyclicity in non-characteristic situations. We also give a relation with the characteristic cohomology class and the Euler-Poincaré characteristic.

All necessary biological knowledge for understanding the hot biological problem(s) underlying the modeling will be provided.

We study two classes of n-generated quadratic algebras over a field K. The first is the class Cn of all n-generated PBW algebras with polynomial growth and finite global dimension. We show that a PBW algebra A is in Cn iff its Hilbert series is HA(z) = 1/(1-z)n. Furthermore each class Cn contains a unique (up to isomorphism) monomial algebra A = K ‹ x1, … , xn › / (xj xi | 1 ≤£ i < j £ n). The second is the class of n-generated quantum binomial algebras A, where the defining relations are nondegenerate square-free binomials xy – cxy zt, with nonzero coefficients cxy. Our main result shows that the following conditions are equivalent: (i) A is a Yang-Baxter algebra, that is the set of quadratic relations R defines canonically a solution of the Yang-Baxter equation. (ii) A is an Artin-Schelter regular PBW algebra. (iii) A is a PBW algebra with polynomial growth. (iv) A is a binomial skew polynomial ring. (v) The Koszul dual A! is a quantum Grassmann algebra.

Given an F-crystal C over a scheme S of positive characteristic p, one can associate many reduced locally closed subschemes T of S defined by the property that a suitable invariant of F-crystals is constant on the fibres of C over points of S. A basic problem is to study different properties of such locally closed subschemes T. In this talk, we present a survey of purity results for locally closed subschemes T of S, including some of them which were obtained by us and by the graduate student Jinghao Li.

A technical ingredient in Faltings’ original approach to p-adic comparison theorems involves the construction of K(π, 1)-neighborhoods for a smooth scheme X over a mixed characteristic dvr with a perfect residue field: every point x ∈ X has an open neighborhood U whose general fiber is a K(π, 1) scheme (a notion analogous to having a contractible universal cover). I will show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p-adic Hodge theory.

We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation.

We prove that despite the chiral nature of the model anomalies in the isometry currents do not appear for CP(N-1) at any N. This is in contradistinction with the minimal heterotic model (with no right-moving fermions) which is anomaly-free only for N=2, i.e. in CP(1). We also consider the N=(0,2) supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the « Konishi anomaly. » This gives us another method for finding exact β functions.

A clear–cut parallel between N=1 4D Yang-Mills and N=(0,2) 2D sigma models is revealed.

In my talk I will discuss the possible hypothetical generalization of the Langlands correspondence for two-dimensional local fields and two-dimensional arithmetic schemes. A two-dimensional local field appear naturally from a point and a formal stalk of irreducible curve on a surface such that this curve contains the point. An example of two-dimensional local field on an algebraic surface is the field of iterated Laurent series.

Neutron star and black hole magnetospheres, converting rotational energy to Poynting flux, are often modeled using force-free electrodynamics, since the energy of the field dominates that of the charged matter. In this peculiar scheme, the requirement that the 4-force on the current vanish plays the role of a non-linear field equation, with the current defined by the field via Maxwell’s equations. This talk will show how the theory becomes remarkably simple and elegant when treated relativistically with the help of differential forms, and some applications and new exact solutions will be discussed.

Journée autour des formes modulaires et amplitudes en théorie des cordes.