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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

Noncommutative Geometry 2

In the second half of the course, we would like to illustrate some basic ideas and tools in NCG through one example: noncommutative two tori, the simplest but most revealing one. We shall explain in great detail how to adapt questions in differential geometry (metric structure and index problem) into an operator theoretical framework. Full description can be found in the attached PDF.

Morse Theory and Related Topics

The course includes:

  • Classical Morse theory for smooth functions and regular variational problems.
  • Morse theory for nonsmooth and, in particular, piecewise smooth functions.
  • Symplectic interpretation of the Lagrange multipliers rule. Maslov index and its relation to the Morse index.
  • Floer homology.

The only prerequisite beyond basic mathematical background is the elementary homology theory. No knowledge of symplectic or Riemannian geometry is expected.

I do not plan to follow any book. Standard sources are:

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