Past

Let $X$ be a Poisson point process and $K$ a d-dimensional convex set.
For a point $x \in X$ denote by $v_X(x)$ the Voronoi cell with respect to $X$, and set $$ v_X (K) := \bigcup_{x \in X \cap K } v_X(x) $$ which is the union of all Voronoi cells with center in $K$. We call $v_X(K)$ the Poisson-Voronoi approximation of $K$.
For $K$ a compact convex set the volume difference $V_d(v_X(K))-V_d(K) $ and the symmetric difference $V_d(v_X(K) \triangle K)$ are investigated.
Estimates for the variance and limit theorems are obtained using the chaotic decomposition of these functions in multiple Wiener-Ito integrals

The Harder Narasimhan type (in the sense of Gieseker semistability)

of a pure-dimensional coherent sheaf on a projective scheme is known to vary

semi-continuously in a flat family, which gives the well-known Harder Narasimhan

stratification of the parameter scheme of the family, by locally closed subsets.

We show that each stratum can be endowed with a natural structure of a locally

closed subscheme of the parameter scheme, which enjoys an appropriate universal property.

As an application, we deduce that pure-dimensional coherent sheaves of any given

Harder Narasimhan type form an Artin algebraic stack.

As another application - jointly with L. Brambila-Paz and O. Mata - we describe

moduli schemes for certain rank 2 unstable vector bundles on a smooth projective

curve, fixing some numerical data.

Non-linear backward stochastic differential equations (BSDEs in

short) were firstly introduced by Pardoux and Peng (\cite{PP1990},

1990), who proved the existence and uniqueness of the adapted solution, under smooth square integrability assumptions on the coefficient and the terminal condition, and when the coefficient $g(t,\omega ,y,z)$ is Lipschitz in $(y,z)$ uniformly in $(t,\omega

)$. From then on, the theory of backward stochastic differential equations (BSDE) has been widely and rapidly developed. And many problems in mathematical finance can be treated as BSDEs. The natural connection between BSDE and partial differential equations (PDE) of parabolic and elliptic types is also important applications. In this talk, we study a new developement of BSDE, 

BSDE with contraint and reflecting barrier.

The existence and uniqueness results are presented and we will give some application of this kind of BSDE at last.

We will sketch a new proof of the Theorem of Ax and Kochen that any projective hypersurface over the p-adic numbers has a p-adic rational point, if it is given by a homogeneous polynomial with more variables than the square of its degree d, assuming that p is large enough with respect to the degree d. Our proof is purely algebraic geometric and (unlike all previous ones) does not use methods from mathematical logic. It is based on a (small upgrade of a) theorem of Abramovich and Karu about weak toroidalization of morphisms. Our method also yields a new alternative approach to the model theory of henselian valued fields (including the Ax-Kochen-Ersov transfer principle and quantifier elimination).

Rubbers and biological soft tissues undergo large isochoric motions in service, and can thus be modelled as nonlinear, incompressible elastic solids. It is easy to enforce incompressibility in the finite (exact) theory of nonlinear elasticity, but not so simple in the weakly nonlinear formulation, where the stress is expanded in successive powers of the strain. In linear and second-order elasticity, incompressibility means that Poisson's ratio is 1/2. Here we show how third- and fourth-order elastic constants behave in the incompressible limit. For applications, we turn to the propagation of elastic waves in soft incompressible solids, a topic of crucial importance in medical imaging (joint work with Ray Ogden, University of Aberdeen).

I will talk about a recent paper of Huh, who, building on a wealth of pretty geometry and topology, has given a proof of a conjecture dating back to 1968 regarding the chromatic polynomial (the polynomial that determines how many ways there are of colouring the vertices of a graph with n colours in such a way that no vertices which are joined by an edge have the same colour). I will mainly talk about the way in which a problem that is explicitly a combinatorics problem came to be encoded in algebraic geometry, and give an overview of the geometry and topology that goes into the solution. The talk should be accessible to everyone: no stacks, I promise.

The reconstruction of the early universe amounts to recovering the tiny density fluctuations of the early universe (shortly after the "big bang") from the current observation of the matter distribution in the universe. Following Zeldovich, Peebles and, more recently Frisch and collaboratoirs, we use a newtonian gravitational model with time dependent coefficients taking into accont general relativity effects. Due to the (remarkable) convexity of the corresponding action, the reconstruction problem apparently reduces to a straightforward convex minimization problem. Unfortunately, this approach completely ignores the mass concentration effects due to gravitational instabilities.

In this lecture, we show a way of modifying the action in order to take concentrations into account. This is obtained through a (questionable) modification of the gravitation model,

by substituting the fully nonlinear Monge-Amp`ere equation for the linear Poisson equation. (This is a reasonable approximation in the sense that it makes exact some approximate solutions advocated by Zeldovich for the original gravitational model.) Then the action can be written as a perfect square in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developped by Ghoussoub. A fully discrete algorithm is introduced for the EUR problem in one space dimension.

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).