Imperial College London

A Landau-Ginzburg B-model is a smooth scheme $X$, equipped with a global function $W$. From $(X,W)$ we can construct a category $D(X,W)$, which is called by various names, including ‘the category of B-branes’. In the case $W=0$ it is exactly the derived category $D(X)$, and in the case that $X$ is affine it is the category of matrix factorizations of $W$. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction.

I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $W$ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.

Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!

A well known theorem of Coleman states that an overconvergent modular eigenform of weight k>1 and slope less than k-1 is classical. This theorem was later reproved and generalized using a geometric method very different from Coleman's cohomological approach. In this talk I will explain how one might go about generalizing the cohomological method to some higher-dimensional Shimura varieties.

In this talk I will explain a notion of p-adic functoriality for inner forms of definite unitary groups. Roughly speaking, this is a morphism between so-called eigenvarieties,  which are certain rigid analytic spaces parameterizing p-adic families  of automorphic forms. We will then study certain properties of classical Langlands functoriality that allow us to prove p-adic functoriality in some "stable" cases.

The derivatives of PDE models are key ingredients in many

important algorithms of computational science. They find applications in

diverse areas such as sensitivity analysis, PDE-constrained

optimisation, continuation and bifurcation analysis, error estimation,

and generalised stability theory.

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These derivatives, computed using the so-called tangent linear and

adjoint models, have made an enormous impact in certain scientific fields

(such as aeronautics, meteorology, and oceanography). However, their use

in other areas has been hampered by the great practical

difficulty of the derivation and implementation of tangent linear and

adjoint models. In his recent book, Naumann (2011) describes the problem

of the robust automated derivation of parallel tangent linear and

adjoint models as "one of the great open problems in the field of

high-performance scientific computing''.

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In this talk, we present an elegant solution to this problem for the

common case where the original discrete forward model may be written in

variational form, and discuss some of its applications.

String theory on a torus requires the introduction of dual coordinates

conjugate to string winding number. This leads to physics and novel geometry in a doubled space. This will be

compared to generalized geometry, which doubles the tangent space but not the manifold.

For a d-torus,   string theory can be formulated in terms of an infinite

tower of fields depending on both the d torus coordinates and the d dual

coordinates. This talk focuses on a finite subsector  consisting of a metric

and B-field (both d x d matrices) and a dilaton all depending on the 2d

doubled torus coordinates.

The double field theory is constructed and found to have a novel symmetry

that reduces to diffeomorphisms and anti-symmetric tensor gauge

transformations in certain circumstances. It also has manifest T-duality

symmetry which provides a generalisation of the usual Buscher rules to

backgrounds without isometries. The theory has a real dependence on the full

doubled geometry:  the dual dimensions are not auxiliary. It is concluded

that the doubled geometry is physical and dynamical.

Given a diffusion in R^n, we prove a small-noise expansion for its density. Our proof relies on the Laplace method on Wiener space and stochastic Taylor expansions in the spirit of Benarous-Bismut. Our result applies (i) to small-time asymptotics and (ii) to the tails of the distribution and (iii) to small volatility of volatility.

We shall study applications of this result to stochastic volatility models, recovering the Berestycki- Busca-Florent formula (using (i)), the Gulisashvili-Stein expansion (from (ii)) and Lewis' expansions (using (iii)).

This is a joint work with J.D. Deuschel (TU Berlin), P. Friz (TU Berlin) and S. Violante (Imperial College London).

Ultracold atomic gases have recently proven to be enormously rich

systems from the perspective of a condensed matter physicist. With

the advent of optical lattices, such systems can now realise idealised

model Hamiltonians used to investigate strongly correlated materials.

Conversely, ultracold atomic gases can exhibit quantum phases and

dynamics with no counterpart in the solid state due to their extra

degrees of freedom and unique environments virtually free of

dissipation. In this talk, I will discuss examples of such behaviour

arising from spinor degrees of freedom on which my recent research has

focused. Examples will include bosons with artificially induced

spin-orbit coupling and the non-equilibrium dynamics of spinor

condensates.

Some striking, and potentially useful, effects in electrokinetics occur for

bipolar membranes: applications are in medical diagnostics amongst other areas.

The purpose of this talk is to describe the experiments, the dominant features observed

and then model the phenomena: This uncovers the physics that control this process.

Time-periodic reverse voltage bias

across a bipolar membrane is shown to exhibit transient hysteresis.

This is due to the incomplete depletion of mobile ions, at the junction

between the membranes, within two adjoining polarized layers; the layer thickness depends on

the applied voltage and the surface charge densities. Experiments

show that the hysteresis consists of an Ohmic linear rise in the

total current with respect to the voltage, followed by a

decay of the current. A limiting current is established for a long

period when all the mobile ions are depleted from the polarized layer.

If the resulting high field within the two polarized layers is

sufficiently large, water dissociation occurs to produce proton and

hydroxyl travelling wave fronts which contribute to another large jump

in the current. We use numerical simulation and asymptotic analysis

to interpret the experimental results and

to estimate the amplitude of the transient hysteresis and the

water-dissociation current.