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

Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations

Dubrovin B, Pavlov MV, Zykov SA. Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations. Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290 [Internet]. 2011 . Available from: http://hdl.handle.net/1963/6430

Integrable functional equations and algebraic geometry

Dubrovin B, Fokas AS, Santini PM. Integrable functional equations and algebraic geometry. Duke Mathematical Journal. Volume: 76, Issue: 2, Pages: 645-668 [Internet]. 1994 . Available from: http://hdl.handle.net/1963/6482

On an isomonodromy deformation equation without the Painlevé property

Dubrovin B, Kapaev A. On an isomonodromy deformation equation without the Painlevé property. [Internet]. 2014 . Available from: http://hdl.handle.net/1963/6466

Recurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory

Dubrovin B, Maltsev AYa A. Recurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory. SISSA; 1999. Available from: http://hdl.handle.net/1963/6489

Quantum Geometry and Matter 2013 (QGM13)

Read more about Quantum Geometry and Matter 2013 (QGM13)

Hilbert Schemes, Sheaves and Representations (VBAC 2013)

Read more about Hilbert Schemes, Sheaves and Representations (VBAC 2013)

Virtual Classes

Topics:

1. A review of Fulton-MacPherson intersection theory.
2. Cones, normal cones, lci morphisms and reduced cotangent complex.
3. Groupoids, 2-categories and stacks.
4. Deligne-Mumford and Artin algebraic stacks.
5. Abelian cone stacks and derived category; normal cone stack.
6. Perfect obstruction theories and virtual classes.
7. Functorial properties of virtual classes and pullbacks.
8. Gromov-Witten invariants: definition and proof of the
Kontsevich-Manin axioms.