Imperial College

Calabi-Yau manifolds without flux are perhaps the best-known
supergravity backgrounds that leave some supersymmetry unbroken. The
supersymmetry conditions on such spaces can be rephrased as the
existence and integrability of a particular geometric structure. When
fluxes are allowed, the conditions are more complicated and the
analogue of the geometric structure is not well understood.

In this talk, I will define the analogue of Calabi-Yau geometry for
generic D=4, N=2 backgrounds with flux in both type II and
eleven-dimensional supergravity. The geometry is characterised by a
pair of G-structures in 'exceptional generalised geometry' that
interpolate between complex, symplectic and hyper-Kahler geometry.
Supersymmetry is then equivalent to integrability of the structures,
which appears as moment maps for diffeomorphisms and gauge
transformations. Similar structures also appear in D=5 and D=6
backgrounds with eight supercharges.

As a simple application, I will discuss the case of AdS5 backgrounds
in type IIB, where deformations of these geometric structures give
exactly marginal deformations of the dual field theories.

I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of special orthogonal groups and certain inner forms of symplectic groups $G$ over a number field $F$. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set $S$ of real places of $F$ such that $G$ has discrete series at places in $S$ and is quasi-split at places outside $S$, and restricting to automorphic representations of $G(A_{F})$ which have algebraic regular infinitesimal character at the places in $S$. In particular, this proves the general multiplicity formula for groups $G$ such that $F$ is totally real, $G$ is compact at all real places of $F$ and quasi-split at all finite places of $F$. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois
gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

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.

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 use Carleman type techniques to address uniqueness of self-shrinkers of mean curvature flow with given asymptotic behaviors.

Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).             

I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.

Counting curves with given topological properties in a variety is a very old question. Example questions are: How many conics pass through five points in a plane, how many lines are there on a Calabi-Yau 3-fold? There are by now several ways to count curves and the numbers coming from different curve counting theories may be different. We would then like to have methods to compare these numbers. I will present such a general method and show how it works in the case of stable maps and stable quasi-maps.