# Past Geometry and Analysis Seminar

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

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

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.

From a geometric viewpoint the irregular Riemann-Hilbert correspondence can be viewed as a machine that takes as input a simple

`additive' symplectic/Poisson manifold and it outputs a more complicated `multiplicative' symplectic/Poisson manifold. In the

simplest nontrivial example it converts the linear Poisson manifold Lie(G)^* into the dual Poisson Lie group G^* (which is the Poisson

manifold underlying the Drinfeld-Jimbo quantum group). This talk will firstly describe some more recent (and more complicated) examples of

such `nonperturbative symplectic/Poisson manifolds', i.e. symplectic spaces of Stokes/monodromy data or `wild character varieties'. Then

the natural generalisations (`fission algebras') of the deformed multiplicative preprojective algebras that occur will be discussed, some

of which are known to be related to Cherednik algebras.

A meromorphic connection on a complex curve can be interpreted as a representation of a simple Lie algebroid. By integrating this Lie algebroid to a Lie groupoid, one obtains a complex surface on which the parallel transport of the connection is globally well-defined and holomorphic, despite the apparent singularities of the corresponding differential equations. I will describe these groupoids and explain how they can be used to illuminate various aspects of the classical theory of singular ODEs, such as the resummation of divergent series solutions. (This talk is based on joint work with Marco Gualtieri and Songhao Li.)