Wednesday

The first part of the talk will review briefly a number of current research areas in which “multivalued” solutions of elliptic equations appear. These are solutions taking values in a flat bundle over the complement of a codimension-2 submanifold. In particular, we will describe their relevance to adiabatic models for special holonomy manifolds fibred by hyperkahler 4-manifolds. In the second part of the talk we will discuss semi-explicit solutions in Euclidean space which can be found using classical integral representations and twistor theory—making connections with complex geometry--- and many open questions in this direction.

I will report on joint work with Penka Geogieva on a structure theorem for real Gromov-Witten invariants of Calabi-Yau 3-folds in the presence of an anti-symplectic involution. Our results were motivated by the Gopakumar-Vafa and Walcher conjectures, and generalize earlier joint work with Thomas Parker and with Aleksander Doan and Thomas Walpuski proving the integrality and respectively finiteness part of the Gopakumar-Vafa conjecture for the Gromov-Witten invariants of 3-folds.

I will describe a general approach to defining virtual fundamental classes of moduli spaces, regarded as families over a space of parameters. While this was designed for Gromov-Witten moduli spaces, it should be broadly applicable in gauge theory. This is joint work with Eleny Ionel.

Coassociatives are 4-dimensional calibrated submanifolds of G2 manifolds, interesting both as examples of minimal submanifolds and for the role they play in gauge theory on G2 manifolds. I will discuss certain non-compactness phenomena related to a generalized connected sum gluing construction for compact coassociatives.

This talk is motivated by an old problem of constructing Donaldson-Thomas invariants for symplectic six-manifolds. The basic idea is to count connections satisfying a 6d instanton equation. Alas, such a count is not an invariant and has to be corrected by contributions from pseudo-holomorphic curves. Making this proposal rigorous is a fascinating yet challenging programme plagued with analytic difficulties. I will discuss a way of defining correction terms, following Haydys-Walpuski, and then focus on the case of particularly simple six-manifolds: fibrations over Riemann surfaces with fibers diffeomorphic to the resolution of C^2 /Z_2. Based on an ongoing project with Greg Parker and Thomas Walpuski.