Mathematics

Description:
In the first part of the course we will cover the basic theory of compact complex manifolds, in particular those admitting a strongly compatible Riemannian metric (i.e. a Kähler metric), especially in the case of vanishing Ricci curvature (i.e. Calabi-Yau manifolds). In the second part we will study a remarkable class of submanifolds of Calabi-Yau manifolds, known as special Lagrangian submanifolds, following ideas of Thomas, Yau, Joyce and Li.

The course will discuss rigorous methods for the study of quantum statistical mechanics models of importance in condensed matter physics. The focus will be on analytic techniques that allow to describe in a quantitative way the large scale behavior of many-body systems in the thermodynamic limit.

Location: room 139. Exceptions: April 20 in room 137, April 23 in room 134.

Program:

This course explores the topology of random algebraic hypersurfaces. The starting point is Hilbert sixteenth Problem, which asks for the possible topologies of real algebraic hypersurfaces of degree $d$. In the case of plane curves, Harnack’s inequality provides a bound on the number of connected components, but as d grows the number of isotopy types grows super-exponentially, making a deterministic classification hopeless. 

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 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.

The course introduces linear systems of differential equations in the complex domain, their monodromy and essential monodromy data. Then, the theory of isomonodromy deformations will be studied, with applications to some examples. These notions play an important role in modern mathematical physics, for example in integrable systems.

Prerequisites: some knowledge of classical mechanics will be useful.

Contents:
Integrable Systems: 2 cycles 40 hours

Timetable

• Tuesday 14-16 (Sissa)
• Wednesday 9-11 (Sissa)
• Thursday 10-11 (Aula D Edificio A Università s di Trieste)

The course will start October 7, 2025.

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

All informations on the course can be found on the webpage https://people.sissa.it/~mbersht/RT2025.html