Geometry and Analysis Seminar

From a complex analytic perspective, both Teichmüller spaces and
symmetric spaces can be realised as contractible bounded domains, that
have several features in common but also exhibit many differences. In
this talk we will study isometric maps between these two important
classes of bounded domains equipped with their intrinsic Kobayashi metric.

We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result

I will discuss some properties of Hermitian metrics on compact complex manifolds, having constant Chern scalar curvature, focusing on the existence problem in fixed Hermitian conformal classes (the "Chern-Yamabe problem"). This is joint work with Daniele Angella and Simone Calamai.

Let X be a smooth projective curve (of genus greater than or equal to 2) over C and M the moduli space of vector bundles over X, of rank 2 and with fixed determinant of degree 1.Then the Fourier-Mukai functor from the bounded derived category of coherent sheaves on X to that of M, given by the normalised Poincare bundle, is fully faithful, except (possibly) for hyperelliptic curves of genus 3,4,and 5

 This result is proved by establishing precise vanishing theorems for a family of vector bundles on the moduli space M.

 Results on the deformation  and inversion of Picard bundles (already known) follow from the full faithfulness of the F-M functor

A long-standing problem in almost complex geometry has been the question of existence of (complete) inhomogeneous nearly Kahler 6-manifolds. One of the main motivations for this question comes from $G_2$ geometry: the Riemannian cone over a nearly Kahler 6-manifold is a singular space with holonomy $G_2$.

Viewing Euclidean 7-space as the cone over the round 6-sphere, the induced nearly Kahler structure is the standard $G_2$-invariant almost complex structure on the 6-sphere induced by octonionic multiplication. We resolve this problem by proving the existence of exotic (inhomogeneous) nearly Kahler metrics on the 6-sphere and also on the product of two 3-spheres. This is joint work with Lorenzo Foscolo, Stony Brook.