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.  

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 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 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.