Course overview -- Wintersemester 2015/2016 (Archive)

The moduli space of certain one-parameter families of Calabi-Yau manifolds is governed by the (generalized) hypergeometric differential equation. We discuss the analytic continuation of its solutions.

Hamiltonian PDE, arising e.g. in classical field theories and quantum mechanics, can be viewed as infinite-dimensional Hamiltonian systems. In this talk I show that analogues of the classical rigidity results from symplectic topology, such as Gromov's nonsqueezing theorem and the Arnold conjecture, also hold for these Hamiltonian PDE. In order to establish the existence of the relevant holomorphic curves, I use the surprising fact from logic that each separable symplectic Hilbert space is contained in a symplectic vector space which behaves as if it were finite-dimensional. As a concrete result I show (without experiment !) that every Bose-Einstein condensate, which is constrained to a circle and annoyed by a time-periodic exterior potential, has infinitely many time-periodic quantum states.

One construction of Frobenius manifolds, originating in Saito's work on singularities, uses a Landau-Ginzburg superpotential. This is a family of curves equipped with meromorphic functions whose critical values are canonical coordinates on the Frobenius manifold. Conversely, Dubrovin showed how to produce any semi-simple conformal Frobenius manifold this way. In joint work with Dunin-Barkowski, Orantin, Popolitov and Shadrin we apply topological recursion to the superpotential and prove that it retrieves the Frobenius manifold. This is useful in both directions - it tells us more about the Frobenius manifold and more about topological recursion.

There are particularly nice holomorphic Lagrangian subvarieties in the moduli space of Higgs pairs on a compact Riemann surface, called Hitchin sections. Physicist Gaiotto conjectured that there should be a canonical procedure to quantize the Hitchin spectral curves of the Higgs pairs on this Lagrangian into a family of holomorphic connections, called "opers," on the same Riemann surface. Recently this conjecture has been solved in a joint work by Dumitrescu-Fredrickson-Kydonakis-Mazzeo-Mulase-Neitzke. In this talk a holomorphic construction of the conjectured opers will be given. We will see that the quantum deformation parameter, the Planck constant, is identified as the coordinate of a sheaf cohomology group naturally associated with the Hitchin section.