08:30 to 09:55 Registration 09:55 to 10:00 Welcome - David Wallace 10:00 to 11:00 Partitions, matrix models, and geometry Sums of partitions with the Plancherel measure appear in various problems of statistical physics and mathematics (crystal growth, point processes, Gromov-Witten invariants). We rewrite such sums as matrix integrals, which allow to compute their large size expansion to all orders. The coefficients in the expansion are geometric objects called symplectic invariants of spectral curves. This makes a link between combinatorics of partitions, matrix models, algebraic geometry, integrability, quantum field theory and topological string theory. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 The continuous limit of random planar maps Planar maps are graphs embedded in the two-dimensional sphere $S^2$, considered up to continuous deformation. They have been studied extensively in combinatorics, but they also have significant geometrical applications. Random planar maps have been used as models of random geometry in theoretical physics. Our goal is to discuss the convergence of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map $M(n)$, which is uniformly distributed over the set of all planar maps with $n$ vertices in a certain class. We equip the set of vertices of $M(n)$ with the graph distance rescaled by the factor $n^{-1/4}$. We then discuss the convergence in distribution of the resulting random metric spaces as $n\to\infty$, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit can be written as a quotient space of the so-called Continuum Random Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. This limiting random metric space had been introduced by Marckert and Mokkadem and called the Brownian map. It can be viewed as a Brownian surface'' in the same sense as Brownian motion is the limit of rescaled discrete paths. We show that the Brownian map is almost surely homeomorphic to the sphere $S^2$, although it has Hausdorff dimension $4$. Furthermore, we are able to give a complete description of the geodesics from a distinguished point (the root) of the Brownian map, and in particular of those points which are connected by more than one geodesic to the root. As a key tool, we use bijections between planar maps and various classes of labeled trees. Related Links http://www.dma.ens.fr/users/legall/ - contains links to papers related to the talk INI 1 12:30 to 13:30 Lunch at Churchill College 14:00 to 15:00 O Bernardi ([CNRS])A bijection for covered maps on orientable surfaces A map of genus g is a graph together with an embedding in the orientable surface of genus g. For instance, plane trees can be considered as maps of genus 0 and unicellular maps (maps with a single face) are a natural generalisation of plane trees to higher genus surfaces. In this talk, we consider covered maps, which are maps together with a distinguished unicellular spanning submap. We will present a bijection between covered maps of genus g with n edges and pairs made of a plane tree with n edges and a bipartite unicellular map of genus g with n+1 edges. This bijection allows to recover bijectively some very elegant formulas by Mullin and by Lehman and Walsh. We will also show that our bijection generalises a bijection of Bouttier, Di Francesco and Guitter (which, in turns, generalises a previous bijection of Schaeffer) between bipartite maps and some classes of labelled trees. INI 1 15:00 to 15:30 Tea 15:30 to 16:30 Vacancy localisation in the square dimer model, statistics of geodesic in large quadrangulations I shall present two independent projects carried out last year, illustrating the interests of a typical combinatorial statistical mechanist. - I first consider the classical dimer model on a square lattice with a single vacancy, and address the question of the possible motion of the vacancy induced by dimer slidings. Adapting a variety of techniques (Temperley bijection, matrix tree theorem, finite-size analysis, numerical simulations), we find that the vacancy remains localized albeit in a very weak fashion, leading to non-trivial diffusion exponents. [joint work with M. Bowick, E. Guitter and M. Jeng] - I next consider the statistical properties of geodesics in large random planar quadrangulations. Introducing "spine trees" extending Schaeffer's well-labeled tree construction, we obtain in particular the generating function for quadrangulations with a marked geodesic. We deduce exact statistics for large quadrangulations both in the "local" and "scaling" limits. [joint work with E. Guitter] INI 1 16:30 to 17:30 Large deviation of the top eigenvalue of a random matrix The statistical properties of the largest eigenvalue of a random matrix are of interest in diverse fields ranging from disordered systems to string theory. In this talk I'll discuss some recent developements on the theory of extremely rare fluctuations of the largest eigenvalue and its various applications. Related Links http://fr.arxiv.org/abs/cond-mat/0609651 - paper published in Phys. Rev. Lett., 97, 160201 (2006)http://fr.arxiv.org/abs/0801.1730 - to appear in Phys. Rev. E (2008)http://fr.arxiv.org/abs/cond-mat/0701371 - published in J. Phys. A: Math. Theor. 40(16) (2007) 4317-4337 INI 1 17:30 to 18:30 Welcome Wine Reception 18:30 to 19:00 Dinner at Churchill College (Residents only)