Past Geometry and Analysis Seminar

The lecture will introduce the notion of a folded 4-dimensional hyperkähler manifold, give examples and prove a local existence theorem from boundary data using twistor methods, following an idea of Biquard.  

The moduli space of Higgs bundles on a hyperbolic Riemann surface is a complex analytic variety which has a hyperkahler metric on its smooth locus. As such it has several associated symmetry groups including the group of complex analytic automorphisms and the group of isometries. I will discuss the classification of these and some other related groups.

Kodaira dimension provides a very successful classification scheme for complex manifolds. The notion was extended to symplectic 4-manifolds. In this talk, we will define the Kodaira dimension for 3-manifolds through Thurston’s eight geometries. This is compatible with other Kodaira dimensions in the sense of “additivity”. This idea could be extended to dimension 4. Finally, we will see how it is sitting in a potential classification of 4-manifolds by exploring its relations with various Kodaira dimensions and other invariants like Gromov norm.

Hitchin's existence theorem asserts that a stable Higgs bundle of rank two carries a unitary connection satisfying Hitchin's self-duality equation. In this talk we discuss a new proof, via gluing methods, for
elements in the ends of the Higgs bundle moduli space and identify a dense open subset of the boundary of the compactification of this moduli space.
 

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic K3 fibrations whose mirror images are also elliptic K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.

 I will report on recent work on a tropical/symplectic approach to the Horn inequalities. These describe the possible spectra of Hermitian matrices which may be obtained as the sum of two Hermitian matrices with fixed spectra. This is joint work with Anton Alekseev and Maria Podkopaeva.

I will describe a generalization of toric symplectic geometry to a new class of Poisson manifolds which are
symplectic away from a collection of hypersurfaces forming a normal crossing configuration.  Using a "tropical
moment map",  I will describe the classification of such manifolds in terms of decorated log affine polytopes,
in analogy with the Delzant classification of toric symplectic manifolds. 

 I will discuss joint work with Daniel Greb and Matei Toma in which we introduce a notion of Gieseker-stability that depends on several polarisations.  We use this to study the change in the moduli space of Giesker semistable sheaves on manifolds giving new results in dimensions at least three, and to study the notion of Gieseker-semistability for sheaves taken with respect to an irrational Kahler class.

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.

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.