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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

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

Topics in advanced algebras

An introduction to the theory of derived functors in homological algebra, and its applications to sheaves and other geometric and algebraic objects.
  • Basic notions: categories, functors, abelian categories, complexes
  • Derived functors: injective objects, right derived functors, long exact sequence of a derived functor, acyclic resolutions, delta-functors.
  • Introduction to sheaves: presheaves, sheaves, étalé space, direct and inverse images

Algebraic Geometry

The course aims to introduce the student to the language of schemes, the central object of study in modern algebraic geometry.

C*-Algebras that one can see

C*-algebras are operator algebras that form the conceptual foundation of noncommutative geometry.Since commutative C*-algebras yield categories anti-equivalent to categories of locally compactHausdorff spaces by the celebrated Gelfand-Naimark equivalence theorem, noncommutative C*-algebrasare viewed as function algebras on quantum spaces. Their study from this point of view is referred toas noncommutative topology. Here KK-theory and index theory are among prime tools leading tosignificant applications.

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