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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

Random polynomial systems, Kahler geometry and the momentum map

Lecture 1: On counting solutions of polynomial systems

  •   Bézout's theorem
  •   Smale's 17-th problem
  •   Shortcomings of Bézout's theorem
  •   Sparse polynomial systems, and the mixed volume

Lecture 2: Differential forms

  •   Multilinear algebra over R
  •   Complex differential forms
  •   Kähler geometry
  •   The coarea formula, using bundles.
  •   Projective space

Lecture 3: Reproducing kernel spaces

Introduction to sub-Riemannian geometry

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

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