∙ 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

## Introduction to sub-Riemannian geometry

- Isoperimetric problem and Heisenberg group.
- Sub-Riemannian length and metric.
- Rashevskii-Chow theorem.
- Existence of length-minimizers.
- Normal and abnormal geodesics.
- Hamiltonian setting; Hamiltonian characterization of geodesics.
- The endpoint map and the exponential map; conjugate and cut points.
- Nonholonomic tangent space.
- Popp volume and Hausdorff measure.
- Sub-Laplacian and sub-Riemannian heat equation.
- Lie groups and left-invariant sub-Riemannian structures.

## Mathematical Colloquia 2018-2019

Date | Speaker | Seminar |

October 3, 2018 | Jean-Pierre Demailly | Analytic and arithmetic solutions of algebraic equations |

January 29, 2019 | Patrick Gerard | Complex Cauchy matrices, inverse spectral problems and colourful invariant tori |

February 13, 2019 | Massimo Marinacci | Functional Analysis and Theoretical Economics |

June 7, 2019 | Eitan Tadmor | Emergent behavior in collective dynamics |