Geometry and Analysis Seminar

I will present a formula that relates a Higgs bundle on an algebraic curve and Gromov-Witten invariants. I will start with the simplest example, which is a rank 2 bundle over the projective line with a meromorphic Higgs field. The corresponding quantum curve is the Airy differential equation, and the Gromov-Witten invariants are the intersection numbers on the moduli space of pointed stable curves. The formula connecting them is exactly the path that Airy took, i.e., from wave mechanics to geometric optics, or what we call the WKB method, after the birth of quantum mechanics. In general, we start with a Higgs bundle. Then we apply a generalization of the topological recursion, originally found by physicists Eynard and Orantin in matrix models, to this context. In this way we construct a quantization of the spectral curve of the Higgs bundle. 

By solving the Homogeneous Monge-Ampere equation on the deformation to the normal cone of a complex submanifold of a Kahler manifold, we get a canonical tubular neighbourhood adapted to both the holomorphic and the symplectic structure. If time permits I will describe an application, namely an optimal regularity result for certain naturally defined plurisubharmonic envelopes.

The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.

In this talk I will discuss results on the geometry of constant mean curvature (H\neq 0) disks embedded in R^3. Among other

things I will prove radius and curvature estimates for such disks. It then follows from the radius estimate that the only complete, simply connected surface embedded in R^3 with constant mean curvature is the round sphere. This is joint work with Bill Meeks.

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will recall how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.