Mathematics

We will discuss the main facts of random matrix theory from the viewpoint of mathematical analysis. We will introduce the Gaussian Unitary Ensemble as the main example and study its properties. Its properties are shared by many other ensembles of random matrices due to universality. We will emphasise the connection to orthogonal polynomials and asymptotic analysis (steepest descent and Riemann-Hilbert problem methods).

The theory of quadratic forms and the theory of modular forms are two of the pillars of classical number theory.

Thermalization is the process through which a physical system evolves from an out-of-equilibrium state to a thermal state, where it can be described by statistical mechanics. In general, the thermalization process is rather complicate and it is not yet fully understood. This problem is known since the seminal work of Fermi, Pasta, Ulam, who studied numerically a simple one-dimensional model of a nonlinear crystal. This experiment opened the avenue to the discover of slow thermalization processes and it is the first historical example of prethermalization.

Description
Derived algebraic geometry is the natural setting in which mathematicians can study
deformation theory of schemes, moduli spaces, and highly singular objects employing the
ideas and the technical machinery of homological and categorical algebra. In a broader
sense, it also provides a unifying theoretical and conceptual framework encompassing
simultaneously the worlds of algebraic geometry, algebraic topology, homotopy theory and
higher category theory.

Topic of the course
The course will focus on algebraic surfaces. The first part will concern smooth
ones, mostly following ”Complex Algebraic Surfaces” by Arnaud Beauville, while
the second part will focus on singular surfaces and their resolutions.

The course is an introduction to Poisson geometry. In the first part, we shall introduce the basic notions related to Poisson geometry and we shall prove some basic results as the Weinstein s plitting theorem and the existence of symplectic realization.

In the last part of the course, we shall introduce the basic notions about Lie algebroids and Lie groupoids and explain theire relationship with Poisson structures.

Syllabus

1st lecture: Poisson brackets. Linear Poisson brackets on the dual of a Lie algebra.

Mathematics of many-body quantum systems

The course will discuss rigorous methods for the study of many-body systems of importance in quantum statistical mechanics and in condensed matter physics. The focus will be on analytic techniques that allow to describe in a quantitative way the large scale behavior of physically relevant systems. Specifically, we will discuss the following topics.

Part 1: continuous symmetry breaking in quantum spin systems.