Related papers: Volumes and Random Matrices
Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…
In this paper we study the smooth moduli space of closed Riemann surfaces. This smooth moduli is an infinite cover of the usual moduli space $\mathscr{M}_g$ of closed Riemann surfaces, and is identified with the Schottky space of rank $g.$…
The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical…
We propose a new proof, as well as a generalization of Mirzakhani's recursion for volumes of moduli spaces. We interpret those recursion relations in terms of expectation values in Kontsevich's integral, i.e. we relate them to a Ribbon…
We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial…
We exploit the relation among irreducible Riemannian globally symmetric spaces (IRGS) and supergravity theories in 3, 4 and 5 space-time dimensions. IRGS appear as scalar manifolds of the theories, as well as moduli spaces of the various…
We propose a refined expression for the large genus asymptotics of the Weil-Petersson volumes of the moduli space of super-Riemann surfaces with an arbitrary number of boundaries. Our formula leverages the connection between JT supergravity…
We analyze statistical properties of the complex system with conditions which manifests through specific constraints on the column/row sum of the matrix elements. The presence of additional constraints besides symmetry leads to new…
This is an introduction to the geometry of compact Riemann surfaces, largely following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann surfaces with atlases of charts, and as locus of solutions of algebraic equations.…
An introduction to $N=2$ rigid and local supersymmetry is given. The construction of the actions of vector multiplets is reviewed, defining special K\"ahler manifolds. Symplectic transformations lead to either isometries or symplectic…
This is an informal set of lecture notes on moduli spaces of curves based on a set of lectures given at the ICTP last summer. It begins at an elementary level and discusses the genus 1 case in detail. The notes then give an informal…
We review aspects of quantisation of the 11-dimensional supermembrane world volume theory. We explicitly construct vertex operators for the massless states and study interactions of supermembranes. The open supermembrane and its vertex…
We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a…
In these lectures I review the general structure of electric--magnetic duality rotations in every even space--time dimension. In four dimensions, which is my main concern, I discuss the general issue of symplectic covariance and how it…
In this expository review we discuss various aspects of gauge theory. While the focus is on mathematics, wherever possible we make contact with theoretical high energy physics. Particular emphasis is placed on instantons and monopoles,…
The main subject of this expository paper is a connection between Gromov's filling volumes and a boundary rigidity problem of determining a Riemannian metric in a compact domain by its boundary distance function. A fruitful approach is to…
Much recent attention has focused on theories with large extra compactified dimensions. However, while the phenomenological implications of the volume moduli associated with such compactifications are well understood, relatively little…
The materials accompany a lecture short course presented at the 2011 Park City Mathematics Institute, Graduate Summer School on Moduli Spaces of Riemann Surfaces. The lectures were part of/coordinated with an overall program, including…
This in an introduction to random matrix theory, giving an impression of some of the most important aspects of this modern subject. In particular, it covers the basic combinatorial and analytic theory around Wigner's semicircle law,…
In this talk we go over several new developments regarding the techniques for a large class of non-hermitian matrix models with unitary randomness (complex random numbers). In particular, we discuss: (a) - A diagrammatic approach based on a…