Archive | math summer workshop

WEEK 3:  INTEGRABLE SYSTEMS METHODS IN ENUMERATIVE GEOMETRY  In the early 90’s Witten, motivated by a matrix model description of topological gravity, conjectured that the intersection numbers of tautological classes on moduli spaces of curves are governed by the KdV integrable hierarchy. Witten’s conjecture was first proven by Kontsevich, and his proofs as well as […]

WEEK 2: RELATIVE AND LOGARITHMIC GROMOV-WITTEN THEORY Gromov-Witten theory, developed in the early 1990s, provides a system of curve-counting invariants of symplectic manifolds or non-singular algebraic varieties. In the context of symplectic geometry, these invariants are defined via moduli spaces of J-holomorphic curves, where J is an almost complex structure on the target symplectic manifold. […]

WEEK 1: THE SYZ CONJECTURE AND COLLAPSING CALABI-YAU MANIFOLDS The Strominger-Yau-Zaslow (SYZ) conjecture dates from 1996. It was proposed as a geometric mechanism underlying mirror symmetry for Calabi-Yau manifolds. The proposal is that, at least near the “large complex structure” limit in moduli space, a Calabi-Yau manifold has a fibration whose generic fibers are Special […]