Donaldson-Thomas theory for Calabi-Yau 3-folds is a complexification of Chern-Simons theory. In this talk, I will discuss joint work with Naichung Conan Leung on the complexification of Donaldson theory.

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.

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.

In this talk we discuss a utility-risk portfolio selection problem. By considering the first order condition for the objective function, we derive a primitive static problem, called Nonlinear Moment Problem, subject to a set of constraints involving nonlinear functions of “mean-field terms”, to completely characterize the optimal terminal wealth. Under a mild assumption on utility, we establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems, their positive answers have long been missing in the literature. In particular, the existence result in utility-downside-risk problem is in contrast with that of mean-downside-risk problem considered in Jin-Yan-Zhou (2005) in which they prove the non-existence of optimal solution instead and we can show the same non-existence result via the corresponding Nonlinear Moment Problem. This is joint work with K.C. Wong (University of Hong Kong) and S.C.P. Yam (Chinese University of Hong Kong).