Considering an arbitrary algebraic curve E(x,y)=0, I will build a infinite families of invariants, Fg(E) and Wkg(E), wrt deformations of the complexe and modular structure of the curve. I will show that, when the curve is the spectral curve of a matrix model, i.e. the limit of the loop equations of the model when the size N of the matrix tends to infinity, these objects give the terms of the topological ('t Hooft) expansion of the free energy and the correlations functions of the corresponding matrix model. As an exemple, if E is the spectral curve of the hermitian 2-Matrix Model, one computes the generating functions of 2-colored discretized surfaces closed or open, with boundary operators or not.

The integral over a group of the Harish-Chandra measure has been known for a long time. On the other side, the moments of this measure (or correlation functions) are not known in general and a general formalism to compute them is lacking. I will present a formalism that allows us to compute correlation functions of the Harish-Chandra measure for any of the classical simple groups in terms of integrals over nihilpotent algebras (triangular matrices). This formulas are, in a sense, a generalization of the Duistermaat-Heckman theorem, in other words, a localization formula.

The electric current that flows through a chaotic system like a quantum dot may, as a function of time, be characterized by its moments. The first and second moments are called the conductance and the shot noise. We consider the problem of calculating all higher moments, both within a random matrix theory formulation and by resorting to classical action correlations.

The main result of the talk is a new representation for the Weyl Lagrangian (massless Dirac Lagrangian). As the dynamical variable we use the coframe, i.e. field of orthonormal bases. We write down a simple Lagrangian and show that variation of the resulting action with respect to the coframe produces the Weyl equation. The advantage of our approach is that it does not require the use of spinors, Pauli matrices or covariant differentiation. The only geometric concepts we use are those of a metric, differential form, wedge product and exterior derivative. The construction presented in the talk is similar to that used in the so-called Cosserat theory of elasticity (multipolar elasticity). Reference: D.Vassiliev, Phys. Rev. D75, 025006 (2007).

The destruction of anomalous diffusion of the Harper model at criticality, due to weak non-linearity chi is analyzed. It is shown that the second moment grows sub-diffusively as its expectation value is proportional to t to the power alpha, up to times t star proportional to chi to the power gamma. The exponents alpha and gamma reflect the multifractal properties of the spectra and eigenfunctions of the linear model. For t larger than t star the anomalous diffusion law is recovered, however the evolving profile has different shape with respect to the linear case. Applications to waveguide structures, and arrays of magnetic micro-traps for atomic Bose-Einstein condensates are discussed.

We look at the problem of estimating risk (Operational Risk, Credit Risk and Market Risk) and argue that risk elements, such as processes in an organization, credits in a loan-portfolio or share prices in an investment portfolio cannot be regarded as independent. This naturally leads to formulating risk models as dynamical models of interacting degrees of freedom (particles). The operational risk and credit risk problems can be cast into a language describing heterogeneous lattice gasses, in which interaction parameters and non-uniform chemical potentials have an interpretation in terms of unconditional and conditional failure probabilities. For the market risk problem, a minimal interacting generalization of the classical Geometric Brownian Motion model leads to a formulation of market dynamics that is formally similar to the dynamics of graded response neurons. We describe elements of the statistical mechanical analysis of these models to reveal their macroscopic properties.

Chiral symmetry and its spontaneous breaking (ChSB ) play a major role in the low-energy dynamics of Quantum Chromodynamics (QCD). In the language of Dirac eigenvalues, ChSB imposes strong constraints on Dirac spectra, called Leutwyler-Smilga (LS) spectral sum rules. These sum rules were originally derived for QCD on rather general grounds. I will give an alternative simple combinatorial derivation of the LS sum rules for 1 flavor, based on cluster property and chiral decomposition. Further, I will sketch the exact microscopic (field theory) derivation of them in the closely related to QCD but much simpler 2-dimensional Schwinger model. I will also discuss several related topics including breaking of cluster property in multi-flavor QCD, Random Matrix Theory calculation of the leading mass dependence of the QCD partition function, and the so-called spectral duality.

Using semiclassical periodic orbit theory for a chaotic system, we evaluated the energy level correlation depending on the magnetic field as an external parameter. The result is in agreement with the prediction of parameter-dependent random matrix theory.

I will discuss the behavior of eigenvalues of Hermitian random matrices in certain critical regimes, related to possible changes in the number of intervals in the limiting spectrum as the size of the matrices tends to infinity. In the critical regimes local eigenvalue correlation functions are described by Painleve transcendents.

The algorithm by M.Bauer and O.Golinelli to evaluate the moments of the spectral density of the incident matrix of random graphs is very useful also for other matrix ensembles: real symmetric, real anti-symmetric, real, laplacian, Wishart,... In most cases it may efficiently be performed by computer. The moments thus evaluated presumably will be useful to complement numerical simulations. In the large n limit, they are a transparent way to examine possible n-dependent rescaling then the emergence of universality and the validity of the addition theorem for random matrices.

We develop a functional integral formalism to find an exact representation for the electron Green function of the Luttinger liquid (LL) in the presence of a single backscattering impurity. This allows us to reproduce results (well known from the bosonization techniques) for the suppression of the electron local density of states (LDoS) at the position of the impurity and for the Friedel oscillations at finite temperature. In addition, we have extracted from the exact representation an analytic dependence of LDoS on the distance from the impurity and shown how it crosses over to that for the pure LL.

The random planar curves that occur, for example, as the boundaries of spin clusters in the Ising model or of percolation clusters in two dimensions have the special property of being conformally invariant in the scaling limit. I describe a new approach, called tochastic Loewner evolution (SLE), to describing the measure on such curves and to computing many of their properties.

We review some correspondences between YM2 and the theory of random walks. In this spirit, we then consider YM2 in a non conventional large N limit, in which the coupling constant A is not fixed ('t Hooft scaling) but scales with a factor logN. In this regime the effective number of d.o.f. of the model is proportional to N to the power k, with k(A) less than 2, rather than to N squared. Moreover, a transition corresponding to the cutoff transition in random walks occurs when k equal to zero. This transition may be thought of as a step further in the spirit of the Douglas-Kazakov transition.

We construct a new family of invariant measures for products of random matrices with gamma-distributed entries. The invariant measures generalize the well know family of GIG laws studied in the context of random continued fractions by Letac and Seshadri. The associated Lyapunov exponents are computed explicitly and the result is applied to the Anderson model with gamma-distributed potential yielding an explicit estimate on the localization rate.

We define some elementary concepts concerning the description of random matrix ensembles in terms of symmetric spaces. We also discuss examples of how the mapping to symmetric spaces can be used in some physical contexts in which a random matrix description is applied.

We analyze the eigenvalue spectrum of the staggered Dirac matrix in two-color QCD at nonzero chemical potential when the eigenvalues become complex. The quasi-zero modes and their role for chiral symmetry breaking and the deconfinement transition are examined. The bulk of the spectrum and its relation to quantum chaos is considered. A comparison with predictions from random matrix theory is presented. We further provide first evidence that matrix models describe the low lying complex Dirac eigenvalues in a theory with dynamical fermions at nonzero chemical potential. Lattice data for two-color QCD with staggered fermions are compared to detailed analytical results from matrix models in the corresponding symmetry class, the complex chiral symplectic ensemble. They confirm the predicted dependence on chemical potential, quark mass and volume.