Past

I will show that one can (at least in theory) guarantee the "validity" of a numerical approximation of a solution of the 3D Navier-Stokes equations using an explicit a posteriori test, despite the fact that the existence of a unique solution is not known for arbitrary initial data.

The argument relies on the fact that if a regular solution exists for some given initial condition, a regular solution also exists for nearby initial data ("robustness of regularity"); I will outline the proof of robustness of regularity for initial data in $H^{1/2}$.

I will also show how this can be used to prove that one can verify numerically (at least in theory) the following statement, for any fixed R > 0: every initial condition $u_0\in H^1$ with $\|u\|_{H^1}\le R$ gives rise to a solution of the unforced equation that remains regular for all $t\ge 0$.

This is based on joint work with Sergei Chernysehnko (Imperial), Peter Constantin (Chicago), Masoumeh Dashti (Warwick), Pedro Marín-Rubio (Seville), Witold Sadowski (Warsaw/Warwick), and Edriss Titi (UC Irivine/Weizmann).

We consider a process $X_t\in\R^d$, $t\ge0$, introduced by Durrett and Rogers in 1992 in order to model the shape of a growing polymer; it undergoes a drift which depends on its past trajectory, and a Brownian increment. Our work concerns two conjectures by these authors (1992), concerning repulsive interaction functions $f$ in dimension $1$ ($\forall x\in\R$, $xf(x)\ge0$).

We showed the first one with T. Mountford (AIHP, 2008, AIHP Prize 2009), for certain functions $f$ with heavy tails, leading to transience to $+\infty$ or $-\infty$ with probability $1/2$. We partially proved the second one with B. T\'oth and B. Valk\'o (to appear in Ann. Prob. 2011), for rapidly decreasing functions $f$, through a study of the local time environment viewed from the

particule: we explicitly display an associated invariant measure, which enables us to prove under certain initial conditions that $X_t/t\to_{t\to\infty}0$ a.s., that the process is at least diffusive asymptotically and superdiffusive under certain assumptions.

Thanks to a Last Passage Percolation model, 3 colored sources are in competition to fill all the positive quadrant N2. There is coexistence when the 3 souces have infected an infinite number of sites.
A coupling between the percolation model and a particle system -namely, the TASEP- allows us to compute the coexistence probability.

There are several different approaches to noncommutative algebraic geometry. I will present one of these approaches. A noncommutative space will be an (abelian) category. I will show how to associate a ringed space to a category. In the case of the category of quasi-coherent sheaves on a scheme this construction will recover the scheme back. I will also give examples coming from quantum groups.

• Laura Gallimore - Modelling Cell Motility
• Y. M. Lai - Stochastic Synchronization of Neural Populations
• Jay Newby - Quasi-steady State Analysis of Motor-driven Transport on a 2D Microtubular Network

Games are ubiquitous in set theory and in particular can be used to build models (often using some large cardinal property to justify the existence of strategies). As a reversal one can define large cardinal properties in terms of such games.

We look at some such that build models through indiscernibles, and that have recently had some effect on structures at aleph_2.

I will introduce the notion of a nilsequence, which is a kind of

"higher" analogue of the exponentials used in classical Fourier analysis. I

will summarise the current state of our understanding of these objects. Then

I will discuss a variety of applications: to solving linear equations in

primes (joint with T. Tao), to a version of Waring's problem for so-called

generalised polynomials (joint with V. Neale and Trevor Wooley) and to

solving certain pairs of diagonal quadratic equations in eight variables

(joint work with L. Matthiesen). Some of the work to be described is a

little preliminary!

Predicting the dynamics of foams requires input from geometry and both Newtonian and non-Newtonian fluid mechanics, among many other fields. I will attempt to give a flavour of this richness by discussing the static structure of a foam and how it allows the derivation of dynamic properties, at least to leading order. The latter includes coarsening due to gas diffusion, liquid drainage under gravity, and the flow of the bubbles themselves.

Based on an MPI library written over 10 years ago, OP2 is a new open-source library which is aimed at application developers using unstructured grids. Using a single API, it targets a variety of HPC architectures, including both manycore GPUs and multicore CPUs with vector units. The talk will cover the API design, key aspects of the parallel implementation on the different platforms, and preliminary performance results on a small but representative CFD test code.

Project homepage: http://people.maths.ox.ac.uk/gilesm/op2/

Many classes of polarised projective algebraic varieties can be constructed via explicit constructions of corresponding graded rings. In this talk we will discuss two methods, namely Basket data method and Key varieties method, which are often used in such constructions. In the first method we will construct graded rings corresponding to some topological data of the polarised varieties. The second method is based on the notion of weighted flag variety, which is the weighted projective analogue of a flag variety. We will describe this notion and show how one can use their graded rings to construct interesting classes of polarised varieties.

In portfolio management, there are specific strategies for trading between two assets that are cointegrated. These are commonly referred to as pairs-trading or spread-trading strategies. In this paper, we provide a theoretical framework for portfolio choice that justifies the choice of such strategies. For this, we consider a continuous-time error correction model to model the cointegrated price processes and analyze the problem of maximizing the expected utility of terminal wealth, for logarithmic and power utilities. We obtain and justify an extra no-arbitrage condition on the market parameters with which one obtains decomposition results for the optimal pairs-trading portfolio strategies.

After a quick-and-dirty introduction to nonstandard analysis, we will

define the asymptotic cones of a metric space and we will play around

with nonstandard tools to show some results about them.

For example, we will hopefully prove that any separable asymptotic cone

is proper and we will classify real trees appearing as asymptotic cones

of groups.

This talk will be divided into three parts. In the first part we will recall basic notions and facts of differential geometry and the Ricci flow equation. In the second part we will talk about singularities for the Ricci flow and Ricci flow on homogeneous spaces. Finally, in the third part

of the talk, we will focus on the case of Ricci flow on compact homogeneous spaces with two isotropy summands.

Abstract:

this is joint work with Eli Aljadeff.

Let G be a group, H a finite index subgroup. Moore's conjecture says that under a certain condition on G and H (which we call the Moore's condition), a G-module M which is projective over H is projective over G. In other words- if we know that a module is ``almost projective'', then it is projective. In this talk we will survey cases in which the conjecture is known to be true. This includes the case in which the group G is finite and the case in which the group G has finite cohomological dimension.

As a generalization of these two cases, we shall present Kropholler's hierarchy LHF, and discuss the conjecture for groups in this hierarchy. In the case of finite groups and in the case of finite cohomological dimension groups, the conjecture is proved by the same finiteness argument. This argument is straightforward in the finite cohomological dimension case, and is a result of a theorem of Serre in case the group is finite. We will show that inside Kropholler's hierarchy the conjecture holds even though this finiteness condition might fail to hold.

We will also discuss some other cases in which the conjecture is known to be true (e.g. Thompson's group F).