Past

The construction of the asymptotic cone of a metric space which allows one to capture the "large scale geometry" of that space has been introduced by Gromov and refined by van den Dries and Wilkie in the 1980's. Since then asymptotic cones have mainly been used as important invariants for finitely generated groups, regarded as metric spaces using the word metric.

However since the construction of the cone requires non-principal ultrafilters, in many cases the cone itself is very hard to compute and seemingly basic questions about this construction have been open quite some time and only relatively recently been answered.

In this talk I want to review the definition of the cone as well as considering iterated cones of metric spaces. I will show that every proper metric space can arise as asymptotic cone of some other proper space and I will answer a question of Drutu and Sapir regarding slow ultrafilters.

I will give a short introduction to non-standard analysis using Nelson's Internal Set Theory, and attempt to give some interesting examples of what can be done in NSA. If time permits I will look at building models for IST inside the usual ZFC set theory using ultrapowers.

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.