∙ 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

## Algebra Superiore/Representation Theory

- Finite groups, character theory and irreducible representations.
- Complex Lie Groups. Solvable and semisimple groups, irreducible finite dimensional representations. Compact forms.
- Algebraic homogeneous spaces and Borel Weil Bott theorem.
- Affine group schemes. Hopf algebras. Introduction to quantum groups.

## Intersection Theory

- Algebraic cycles, flat pullback, proper pushforward.
- Rational and algebraic equivalence. Chow groups.
- Degree of a zero cycle. Numerical equivalence.
- Vector bundles, cell decompositions.
- Pseudodivisors, first Chern class of a line bundle.
- Chern and Segre classes.
- Cones, abelian cones, normal cones.
- Degeneration to the normal cone.
- Gysin pullback.
- Properties of the Gysin pullback.
- Statement of Grothendieck-Riemann-Roch and applications. If time allows,sketch of proof.

## Algebraic Geometry

Il corso coprirà il materiale descritto nelle sezioni 1-8 del secondo capitolo del testo di R. Hartshorne, Algebraic Geometry, GTM 52. In dettaglio: fasci, schemi, sottoschemi, proprietà degli schemi e dei loro morfismi, criteri valutativi, fasci coerenti e quasicoerenti, fibrati, fascio cotangente relativo e assoluto, fibrati lineari e divisori, morfismi proiettivi e loro proprietà.

## Integrable Systems

- Definition of an integrable system in finite dimension.
- Lax pair.
- Infinite dimensional integrable systems.
- An example: the Toda lattice equation. Soliton solution and periodic solutions.
- Toda lattice and Hermitian random matrices.
- The beta-ensamble (non integrable).

## Quantum Mechanics

- scratch course of spectral theory of operators on Hilbert spaces
- a reminder of classical probability
- fundamental postulates of quantum physics
- Heisenberg uncertainty principle
- the von Neumann-Lüders postulate, and the issue of individual states
- symmetries and the Wigner-Kadison theorem
- symmetry groups
- projective representations
- eliminability of multipliers for one parameter groups
- Weyl relations
- von Neumann uniqueness
- CCR and uncertainties

## Characteristic Classes

We will cover the most important characteristic classes such as Stifel-Whitney classes, Chern classes and Pontrjagin classes as well as their relations to obstruction theory. Also, various application will be discussed. Hopefully we will keep an eye on the subject of elliptic genus, which is a quite active field since 1990s'.

Reference:

Characteristic Classes by Milnor and Stasheff.

Differential Forms in Algebraic Topology by Bott and Tu.

Manifolds and Modular Forms by Hirzebruch.

## Mathematical Physics

- Electromagnetic field. Matter and gauge fields. Geometrical background.
- Instantons.
- Global analysis of gauge theories.
- Space of connections, group of gauge transformations and orbit space.
- Outline of functional integral quantization.