Past

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

Abstract: Flagella and cilia are ubiquitous in biology as a means of motility and critical for male gametes migration in reproduction, to mucociliary clearance in the lung, to the virulence of devastating parasitic pathogens such as the Trypanosomatids, to the filter feeding of the choanoflagellates, which are constitute a critical link in the global food chain. Despite this ubiquity and importance, the details of how the ciliary or flagellar waveform emerges from the underlying mechanics and how the cell, or the environs, may control the beating pattern by regulating the axoneme is far from fully understood. We demonstrate in this talk that mechanics and modelling can be utilised to interpret observations of axonemal dynamics, swimming trajectories and beat patterns for flagellated motility impacts on the science underlying numerous areas of reproductive health, disease and marine ecology. It also highlights that this is a fertile and challenging area of inter-disciplinary research for applied mathematicians and demonstrates the importance of future observational and theoretical studies in understanding the underlying mechanics of these motile cell appendages.

The purpose of this talk is to provide a large class of examples of large initial data which gives rise to a global smooth solution. We shall explain what we mean by large initial data. Then we shall explain the concept of slowly varying function and give some flavor of the proofs of global existence.

The signature of the path is an essential object in rough path theory which takes value in tensor algebra and it is anticipated that the expected signature of Brownian motion might characterize the rough path measure of Brownian path itself. In this presentation we study the expected signature of a Brownian path in a Bananch space E stopped at the first exit time of an arbitrary regular domain, although we will focus on the case E=R^{2}. We prove that such expected signature of Brownian motion should satisfy one particular PDE and using the PDE for the expected signature and the boundary condition we can derive each term of expected signature recursively. We expect our method to be generalized to higher dimensional case in R^{d}, where d is an integer and d >= 2.

All this is related to my ongoing research projects with Chris Douglas and Arthur Bartels, in which we investigate conformal nets from a category
theoretical
perspective.

Abstract: We construct solutions to Burgers type equations perturbed by a multiplicative

space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation. We show that our solutions are stable under smooth approximations of the driving noise. A more general class of approximations will also be discussed. This is joint work with Martin Hairer and Jan Maas.

We present theory and numerics of affine processes and several of their applications in finance. The theory is appealing due to methods from probability theory, analysis and geometry. Applications are diverse since affine processes combine analytical tractability with a high flexibility to model stylized facts like heavy tails or stochastic volatility.

We are interested in measure theory and integration theory that ¯ts into the
o-minimal context. Therefore we introduce the following de¯nition:
Given an o-minimal structure M on the ¯eld of reals and a measure ¹ de¯ned on the
Borel sets of some Rn, we call ¹ M-tame if there is an o-minimal expansion of M such
that for every parameter family of functions on Rn that is de¯nable in M the family of
integrals with respect to ¹ is de¯nable in this o-minimal expansion.
In the ¯rst part of the talk we give the de¯nitions and motivate them by existing and
many new examples. In the second one we discuss the Lebesgue measure in this context.
In the ¯nal part we obtain de¯nable versions of important theorems like the theorem of
Radon-Nikodym and the Riesz representation theorem. These results allow us to describe
tame measures explicitly.
1

This talk will focus on a family of Bayesian inference algorithms built around Gaussian processes. We firstly introduce an iterative Gaussian process for multi-sensor inference problems. Extensions to our algorithm allow us to tackle some of the decision problems faced in sensor networks, including observation scheduling. Along these lines, we also propose a general method of global optimisation, Gaussian process global optimisation (GPGO). This paradigm is extended to the Bayesian decision problem of sequential multi-scale observation selection. We show how the hyperparameters of our system can be marginalised by use of Bayesian quadrature and frame the selection of the positions of the hyperparameter samples required by Bayesian quadrature as a sequential decision problem, with the aim of minimising the uncertainty we possess about the values of the integrals we are approximating.

In various fields of application, one must solve very large scale boundary value problems using parallel solvers and supercomputers. The domain decomposition approach partitions the large computational domain into smaller computational subdomains. In order to speed up the convergence, we have several ``optimized'' algorithm that use Robin transmission conditions across the artificial interfaces (FETI-2LM). It is known that this approach alone is not sufficient: as the number of subdomains increases, the number of iterations required for convergence also increases and hence the parallel speedup is lost. A known solution for classical Schwarz methods as well as FETI algorithms is to incorporate a ``coarse grid correction'', which is able to transmit low-frequency information more quickly across the whole domain. Such algorithms are known to ``scale weakly'' to large supercomputers. A coarse grid correction is also necessary for FETI-2LM methods. In this talk, we will introduce and analyze coarse grid correction algorithms for FETI-2LM domain decomposition methods.

This talk will be an introduction to the space of Bridgeland stability conditions on a triangulated category, focussing on the case of the derived category of coherent sheaves on a curve. These spaces of stability conditions have their roots in physics, and have a mirror theoretic interpretation as moduli of complex structures on the mirror variety.

For curves of genus g > 0, we will see that any stability condition comes from the classical notion of slope stability for torsion-free sheaves. On the projective line we can study the more complicated behaviour via a derived equivalence to the derived category of modules over the Kronecker quiver.

Considering kinetic equations (Boltzmann, BGK, say...) in the small mean free path regime lead to conservation laws (the Euler system, typically) When the problem is set in a domain, boundary layers might occur due to the fact that incoming fluxes could be far from equilibrium states. We consider the problem from a numerical perspective and we propose a definition of numerical fluxes for the Euler system which is intended to account for the formation of these boundary layers.

A brief survey of the above.

As many of you are no doubt aware, a Study Group of particular significance will take place in Thuwal, Saudi Arabia, from 23rd January - 26th January 2011.

The problem statements, in their preliminary form can be found at:

http://people.maths.ox.ac.uk/bentahar/KSG/