Past

We will present different ideas leading to and evolving around geometry over the field with one element. After a brief summary of the so-called numbers-functions correspondence we will discuss some aspects of Weil's proof of the Riemann hypothesis for function fields. We will see then how lambda geometry can be thought of as a model for geometry over $\mathbb{F}_\mathrm{un}$ and what some familiar objects should look like there. If time permits, we will

explain a link with stable homotopy theory.

One of the main aims in the theory of percolation is to find the `critical probability' above which long range connections emerge from random local connections with a given pattern and certain individual probabilities. The quintessential example is Kesten's result from 1980 that if the edges of the square lattice are selected independently with probability $p$, then long range connections appear if and only if $p>1/2$.  The starting point is a certain self-duality property, observed already in the early 60s; the difficulty is not in this observation, but in proving that self-duality does imply criticality in this setting.

Since Kesten's result, more complicated duality properties have been used to determine a variety of other critical probabilities. Recently, Scullard and Ziff have described a very general class of self-dual percolation models; we show that for the entire class (in fact, a larger class), self-duality does imply criticality.

We present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the

space $\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is the first instance in which this method has been applied to a parabolic equation. This is joint work with Carlos Kenig.

We investigate the minimization problem for the variational integral

$$\int_\Omega\sqrt{1+|Dw|^2}\,dx$$

in Dirichlet classes of vector-valued functions $w$. It is well known that

the existence of minimizers can be established if the problem is formulated

in a generalized way in the space of functions of bounded variation. In

this talk we will discuss a uniqueness theorem for these generalized

minimizers. Actually, the theorem holds for a larger class of variational

integrals with linear growth and was obtained in collaboration with Lisa

Beck (SNS Pisa).

Joint work with C. Donati-Martin and D. Feral

This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion 'factor' process. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance.) By using a change of measure technique introduced by Kuroda and Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, which has no jumps. The main result of the paper is that the Hamilton-Jacobi-Bellman equation for this problem has a classical solution. The proof uses Bellman's "policy improvement"

method together with results on linear parabolic PDEs due to Ladyzhenskaya et al. This is joint work with Sebastien Lleo.

VC dimension and density are properties of a collection of sets which come from probability theory.  It was observed by Laskowski that there is a close tie between these notions and the model-theoretic property called NIP. This tie results in many examples of collections of sets that have finite VC dimension. In general, it is difficult to find upper bounds for the VC dimension, and known bounds are mostly very large. However, the VC density seems to be more accessible. In this talk, I will explain all of the above acronyms, and present a theorem which gives an upper bound (in some cases optimal) on the VC density of formulae in some examples of NIP theories. This represents joint work of myself with M. Aschenbrenner, A. Dolich, D. Macpherson and S.

Starchenko.

 

 

 

Frank-Read sources are among the most important mechanisms of dislocation multiplication,

and their operation signals the onset of yield in crystals. We show that the critical

stress required to initiate dislocation production falls dramatically at high elastic

anisotropy, irrespective of the mean shear modulus. We hence predict the yield stress of

crystals to fall dramatically as their anisotropy increases. This behaviour is consistent

with the severe plastic softening observed in alpha-iron and ferritic steels as the

alpha − gamma martensitic phase transition is approached, a temperature regime of crucial

importance for structural steels designed for future nuclear applications.