Past

We highlight a technique for studying edge colourings of multigraphs, due to Tashkinov. This method is a sophisticated generalisation of the method of alternating paths, and builds upon earlier work by Kierstead and Goldberg. In particular we show how to apply it to a number of edge colouring problems, including the question of whether the class of multigraphs that attain equality in Vizing's classical bound can be characterised.

This talk represents joint work with Jessica McDonald.

In field theory simple forms of certain scattering amplitudes in four dimensional theories with massless particles are known. This has been shown to be closely related to underlying (super)symmetries and has been a source of inspiration for much development in the last years. Away from four dimensions much less is known with some concrete development only in six dimensions. I will show how to construct promising on-shell superspaces in eight and ten dimensions which permit suggestively simple forms of supersymmetric four point scattering amplitudes with massless particles. Supersymmetric on-shell recursion relations which allow one to compute in principle any amplitude are constructed, as well as the three point `seed' amplitudes to make these work. In the three point case I will also present some classes of supersymmetric amplitudes with a massive particle for the type IIB superstring in a flat background.

We give a characterization of divergence-free vector fields on the plane such that the Cauchy problem for the associated continuity (or transport) equation has a unique bounded solution (in the sense of distribution).

Unlike previous results in this directions (Di Perna-Lions, Ambrosio), the proof relies on a dimension-reduction argument, which can be regarded as a variant of the method of characteristics. Note that our characterization is not stated in terms of function spaces, but is based on a suitable weak formulation of the Sard property for the potential associated to the vector-field.

This is a joint work with S. Bianchini (SISSA, Trieste) and Gianluca Crippa (Parma).

I will describe why e^{\pi\sqrt{163}} is almost an integer and how this is related to Q(\sqrt{-163}) having class number one and why n^2-n+41 is prime for n=0,...,39. Bits and pieces about Gauss's class number problem, Heegner numbers, the j-invariant and complex multiplication on elliptic curves will be discussed along the way.

Numerical Approximations of Non-linear Stochastic Systems. Abstract:  The explicit solution of stochastic differential equations (SDEs can be found only in a few cases. Therefore, there is a need fo accurate numerical approximations that could, for example, enabl  Monte Carlo Simulations. Convergence and stability of these methods are well understood for SDEs with Lipschit  continuous coefficients. Our research focuses on those situations wher  the coefficients of the underlying SDEs are non-Lipschitzian  It was demonstrated in the literature,  that in this case using the classical methods we may fail t  obtain numerically computed paths that are accurate for small step-sizes, or to obtain qualitative information about the behaviour of numerical methods over long time intervals. Our work addresses both of these issues, giving a customized analysis of the most widely used numerical methods.

We consider parabolic scalar conservation laws perturbed by a (conservative) noise. Large deviations are investigated in the singular limit of jointly vanishing viscosity and noise. The model is supposed to feature the same behavior of "asymmetric" particles systems (e.g. TASEP) under Euler scaling.

A first large deviations principle is obtained in a space of Young measures. A "second order" large deviations principle is then discussed, including connections with the Jensen and Varadhan functional. As time allows, more recent "long correlation" models will be treated.

We derive a general methodology to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approach is suitable to compute expectations of functions of arithmetic or geometric means. In the context of small SDE coefficients, we establish an expansion, which terms are explicit and easy to compute. We also provide non asymptotic error bounds. Applications to the pricing of basket options, Asian options or commodities options are then presented. This talk is based on a joint work with M. Miri.

Eukaryotic systems migrate in response to gradients in external signal concentrations, a process referred to as chemotaxis. This chemotactic behaviour may of either a chemoattractive or a chemorepulsive nature.

Understanding such behaviour at the single cell level in terms of the underlying signal transduction networks is highly challenging for various reasons, including the strong non-linearity of the signal processing as well as other complicating factors.

In this talk we will discuss modelling approaches which are aimed at trying to understand how signal transduction in the networks of eukaryotic cells can lead to appropriate internal signals to guide the cell motion either up-gradient or down-gradient. One part of the talk will focus on system-specific mechanistic modelling. This will be complemented by simplified models to address how signal transduction is organized in cells so that they may exhibit both attractive and repulsive gradient sensing.

This is joint with with Mark Berman, Uri Onn, and Pirita Paajanen.

Let K be a local field with valuation ring O and residue field of size q, and G a Chevalley group. We study counting problems associated with the group G(O). Such counting problems are encoded in certain zeta functions defined as Poincare series in q^{-s}. It turns out that these zeta functions are bounded sums of rational functions and depend only on q for all local fields of sufficiently large residue characteristic. We apply this to zeta functions counting conjugacy classes or dimensions of Hecke modules of interwining operators in congruence quotients of G(O). To prove this we use model-theoretic cell decomposition and quantifier-elimination to get a theorem on the values of 'definable' integrals over local fields as the local field varies.

Human behaviour can show surprising properties when looked at from a collective point of view. Data on collective behaviour can be gleaned from a number of sources, and mobile phone data are increasingly becoming used. A major challenge is combining behavioural data with health data. In this talk I will describe our approach to understanding behaviour change related to change in health status at a collective level.

Please note that this is a short notice change from the originally advertised talk by Dr Shahrokh Shahpar (Rolls-Royce plc.)

The new talk "Multilevel Monte Carlo method" is given by Mike Giles, Oxford-Man Institute of Quantitative Finance, Mathematical Institute, University of Oxford.

Joint work with Rob Scheichl, Aretha Teckentrup (Bath) and Andrew Cliffe (Nottingham)

In this talk we will present results concerning the large scale long time behaviour of particles moving in a periodic (random) velocity field subject to molecular diffusion. The particle can be considered massless (passive tracer) or not (inertial particle). Under appropriate assumptions for the velocity field the large scale long time behavior of the particle is described by a Brownian motion with an effective diffusivity matrix K.

We then present some numerical algorithms concerning the calculation of the effective diffusivity in the limit of vanishing molecular diffusion (stochastic geometric integrators). Time permitting we will discuss the case where the driving noise is no longer white but colored and study the effects of this change to the effective diffusivity matrix.

A factorability structure on a group G is a specification of normal forms of group elements as words over a fixed generating set. There is a chain complex computing the (co)homology of G. In contrast to the well-known bar resolution, there are much less generators in each dimension of the chain complex. Although it is often difficult to understand the differential, there are examples where the differential is particularly simple, allowing computations by hand. This leads to the cohomology ring of hv-groups, which I define at the end of the talk in terms of so called "horizontal" and "vertical" generators.

I will describe recent work with Charles Fefferman on a

construction of families of analytic almost-sharp fronts for SQG. These

are special solutions of SQG which have a very sharp transition in a

very thin layer. One of the main difficulties of the construction is the

fact that there is no formal limit for the family of equations. I will

show how to overcome this difficulty, linking the result to joint work

with C. Fefferman and Kevin Luli on the existence of a "spine" for

almost-sharp fronts. This is a curve, defined for every time slice by a

measure-theoretic construction, that describes the evolution of the

almost-sharp front.

A topological space $(X,\tau)$ is submaximal if $\tau$ is the maximal element of $[{\tau}_{s}]$. Submaximality was first defined and characterized by Bourbaki. Since then, some mathematicians presented several characterizations of submaximal spaces.

In this paper, we will attempt to develop the concept of submaximality and offer some new results. Furthermore, some results concerning $\alpha$-scattered space will be obtained.

A factorability structure on a group G is a specification of normal forms

of group elements as words over a fixed generating set. There is a chain

complex computing the (co)homology of G. In contrast to the well-known bar

resolution, there are much less generators in each dimension of the chain

complex. Although it is often difficult to understand the differential,

there are examples where the differential is particularly simple, allowing

computations by hand. This leads to the cohomology ring of hv-groups,

which I define at the end of the talk in terms of so called "horizontal"

and "vertical" generators.

Word maps on groups were studied extensively in the past few years, in connection to various conjectures on profinite groups, finite groups, finite simple groups, etc. I will provide background, as well as very recent works (joint with Larsen, Larsen-Tiep,

Liebeck-O'Brien-Tiep) on word maps with relations to representations (e.g. Gowers' method and character ratios), geometry and probability.

Recent applications, e.g. to subgroup growth and representation varieties, will also be described.

I will conclude with a list of problems and conjectures which are still very much open.  The talk should be accessible to a wide audience.