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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

Cluster algebras

All informations about the course can be found on the webpage https://people.sissa.it/~mbersht/Cluster2025.html

Random matrices

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).

From quadratic forms to modular forms to quantum modular forms

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

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. This experiment opened the avenue to the discover of slow thermalization processes and it is the first historical example of prethermalization.

 

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.

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