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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

Introduction to Smooth Ergodic Theory

Smooth Ergodic Theory is the study of dynamical systems on smooth manifolds from a probabilistic and statistical perspective.

Integrable systems and wave motion

Course content

  1. From the incompressible Euler equation to the Korteweg - de Vries (KdV) equation
  2. KdV and Schrödinger: The Inverse Scattering Method.
  3. The Hamiltonian and bi-Hamiltonian settings for equations of KdV type.
  4. Group actions and Hamiltonian reductions. Systems of Calogero-Moser and Toda type.
  5. Stratified flows. The Green-Naghdi and Miyata-Camassa-Choi equations.
  6. Hamiltonian aspects of sharply stratified flows and boundary effects."

Log Calabi-Yau geometry

Calabi-Yau manifolds play a central role in algebraic/differential

Algebraic Geometry

This is a first course in algebraic geometry, roughly covering sections 2.1 to 2.8 of Hartshorne's GTM 52 book. 

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