A quantum curve is a magical object. It conjecturally captures information of quantum topological invariants in a concise manner. In this talk we will discuss a method of constructing quantum curves by quantizing the spectral curves of Hitchin systems in terms of the WKB method. The quantization is performed by applying the Eynard-Orantin theory, which is a generalization of the topological recursion formalism developed by Eynard and Orantin in 2007. The talk is based on my joint work with Olivia Dumitrescu.

Let B be a quaternionic algebra over a totally real field F, and p be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B* of level prime to p. A generalization of Deligne-Carayol's "modèle étrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic p fiber of X, and show that each closed Goren-Oort stratum is an iterated P1-fibration over another quaternionic Shimura variety in characteristic p. Now suppose that [F:Q] is even and that p is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic p fibre of Hilbert modular varieties of prime-to-p level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

Page web du séminaire

In the context of Loop Quantum Gravity, Black Holes are closely related to Chern-Simons theory on a punctured 2-sphere with SU(2) gauge group. Using this link, one can describe precisely the space of microstates for the Black Holes and compute the corresponding statistical entropy. However, it turns out that the entropy depends on the unphysical Immirzi parameter γ. But, using a suitable analytic continuation of γ to complex values, we show that the entropy reproduces the expected Bekenstein-Hawking expression when γ = ± i at the semi-classical limit. This remarkable result has a nice and clear geometric interpretation and many very interesting physical consequences. In particular, we show that, at the semi-classical limit, the Black Hole microstates (at the vicinity of the horizon) are particles in equilibrium at the Unruh temperature.

In this talk I will discuss some of the ideas behind the difficult problem of computing the mean values of L-functions on the critical line. After a basic and brief discussion I will consider the same problem for function fields over a finite field, and in this case something `deeper’ can be said.

I will introduce the notion of fundamental groups and etale sheaves at a very elementary level. These are generalizations of the classical Galois theory and the usual fundamental groups. Also I will discuss certain problems on ramification of sheaves and degeneration of varieties.

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.