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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

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.

Excursion in Integrability

The goal of the course is an excursion on algebraic aspects of integrable systems. The topics are

  1. KP equation and Sato-Segal-Wilson Grassmannian.
    Tau function and Hirota bilinear relation.
    Plücker coordinates.
    Schur function expansion.
  2. Hermitian random matrix models.
    Orthogonal polynomials, partition function and Schur expansion.
    Combinatorial interpretation of correlator expansion of classical matrix.
    Integrals: ribbon graphs and Hurwitz numbers.

References:

Derived categories in algebraic geometry

Abstract A quick introduction to the techniques of derived categories and functors among them, with a focus on examples rather than proofs.

Hamiltonian methods for integrable systems

The course is centered on the Hamiltonian aspects of integrable systems of Ordinary and, especially, Partial Differential Equations, with a focus on the geometrical side. Integrability will mean existence of a “sufficient number” of conservation laws.

Introduction to Topological Field Theories

The course provides a brief introduction to Topological Field Theories as infinite dimensional generalisation of classical localisation formulae in equivariant cohomology.

It starts with an introduction to these latter subjects (Duistermaat-Heckman theorem, equivariant cohomology and Atiyah-Bott formula) and their extension on supermanifolds. It then continues supersymmetric quantum mechanics and its relation with Morse theory, gradient flow lines and Morse-Smale-Witten complex.

Topics in complex geometry

This year we will focus on some aspects of two different topics:

 (a) Hodge-theoretic mirror symmetry for projective toric manifolds;

(b) K-stability for polarised manifolds and its generalisations.

In the last part of the course we will explain some nontrivial connections between (a) and (b).

 

Plan of the course:

1) Background

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