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Geometry and Mathematical Physics

∙ Integrable systems in relation with differential, algebraic and symplectic geometry, as well as with the theory of random matrices, special functions and nonlinear waves, Frobenius manifolds • Deformation theory, moduli spaces of sheaves and of curves, in relation with supersymmetric gauge theories, strings, Gromov-Witten invariants, orbifolds and automorphisms
• Quantum groups, noncommutative Riemannian and spin geometry, applications to models in mathematical physics
• Mathematical methods of quantum mechanics
• Mathematical aspects of quantum Field Theory and String 
Theory
• Symplectic geometry, sub-riemannian geometry

• Geometry of quantum fields and strings

An introduction to long time prethermalization

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.

Homotopical algebra (with a look toward algebraic geometry)

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.

Complex algebraic surfaces

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.
Formative objectives
In the first part of the course (8 lectures) the students will learn the basics of
the theory of algebraic surfaces. The main topics are
• Riemann-Roch theorem.

Topics on Poisson geometry

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

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.

Representations of classical infinite-dimensional Lie algebras and ind-geometry

The course will consist of two parts, roughly ten hours each. The first part will start with a brief recollection of basics from the representation theory of classical finte-dimensional Lie algebras, followed by an introduction to Lie algebras of infinite matrices. The main topic of the fist part will be a discussion of the categories of tensor modules over Mackey Lie algebras.
 

Random matrices, orthogonal polynomials, and asymptotic analysis

Topics:
  1. definition and basic properties of determinantal point processes;
  2. short introduction to the theory of orthogonal polynomials;
  3. unitary ensembles of random matrices.
  4. Gaussian Unitary Ensemble. Semicircle law and local properties of the eigenvalues (sine-kernel and Airy-kernel processes). Asymptotic analysis of integrals. 

 Frobenius manifolds: analytic theory

The aim of the course is to introduce the audience to the analytic theory of Dubrovin-Frobenius manifolds.

The theory of Frobenius manifolds was constructed by B. Dubrovin to formulate in geometrical terms the WDVV equations of associativity of 2D topological field theories.

It has links to many branches of mathematics, like singularity theory and reflection groups, algebraic and enumerative geometry, quantum cohomology, theory of isomonodromic deformations, boundary value problems and Painlevé equations, integrable systems and non-linear waves.

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