Matrizes Aleatórias em Matemática e Física

Objectives

Introduce the study of random matrices alongside an overview of some of its applications within mathematics and physics, with emphases on applications within modern research trends.

Syllabus

Basics: Random matrices; Gaussian, circular and β-ensembles; Correlation functions, statistical properties, Hermitian matrices; 't Hooft limit; Saddle- point analysis and orthogonal polynomials. Loop Equations and Large N Expansion: Loop equantions, spectral curve, topological expansion, multi- resolvent correlators; Topological recursion; Applications; Large N limits. DSL Limit and CFT: Criticality and fluctuating geometries, 2d quantum gravity, Painlevé equations, KdV hierarchy; Basis of (B)CFT, Liouville theory, ZZ/FZZT- branes; More on large N limits. Nonperturbative Approaches: WKB wave functions; Integrable systems, Riemann-Hilbert problems, tau- function, Baker- Akhiezer functions, determinant formulae; Resurgence and transseries. Generic Models: Chain of matrices; Two-matrix models; Multi-matrix models, large N duality and Calabi-Yau spaces; Supersymmetric models; Chern-Simons models; Yang-Mills models.

Prerequisites

Basic knowledge of quantum mechanics and of Riemannian.

Cross Competence Component

The course unit fosters the development of transversal skills in critical thinking, creativity, and problem-solving strategies, both in class, through independent work, and across the various assessment components.

Ethical Principles

All member of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.