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).

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).

We consider the system of nonlinear differential equations
\begin{equation}
(1) \qquad
\begin{cases}
\dot u_n(t) + \lambda^{2n} u_n(t) 
- \lambda^{\beta n} u_{n-1}(t)^2 + \lambda^{\beta(n+1)} u_n(t) u_{n+1}(t) = 0,\\
u_n(0) = a_n, n \in \mathbb{N}, \quad \lambda > 1, \beta > 0.
\end{cases}
\end{equation}
In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A. Cheskidov proved that the system (1) has a unique "strong" solution if $\beta \le 2$, whereas the "strong" solution does not exist if $\beta > 3$. (Note, that the 3D-Navier-Stokes equations correspond to the value $\beta = 5/2$.) We show that for sufficiently "good" initial data the system (1) has a unique Leray-Hopf solution for all $\beta > 0$.