Past

Refining Julia Robinson's 1949 work on the undecidability of the first order theory of Q, we prove that Z is definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points.

The birational classification of varieties is an interesting and ongoing problem in algebraic geometry. This talk aims to give an

overview of the progress made on this problem in the special case where the varieties considered are surfaces in projective space.

Linear operators preserving non-vanishing properties are an important

tool in e.g. combinatorics, the Lee-Yang program on phase transitions, complex analysis, matrix theory. We characterize all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing when the variables are in prescribed open circular domains, which solves the higher dimensional counterpart of a long-standing classification problem going back to P\'olya-Schur. This also leads to a self-contained theory of multivariate stable polynomials and a natural framework for dealing with Lee-Yang and Heilmann-Lieb type problems in a uniform manner. The talk is based on joint work with Petter Brändén.

In this talk I shall review analytical and numerical results on equilibrium configurations of rotating fluid bodies within Einstein's theory of gravitation.

I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin & Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, & Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $u\in L^{6/5}(0,T;L^\infty)$ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).

We consider one-dimensional Brownian motion conditioned (in a suitable

sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.5860... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). I will also describe other conditionings of Brownian motion in which this principle of entropic repulsion manifests itself.

Joint work with Itai Benjamini.

Particle current is the net number of particles that pass an observer who moves with a deterministic velocity V. Its fluctuations in time-stationary interacting particle systems are nontrivial and draw serious attention. It has been known for a while that in most models diffusive scaling and the corresponding Central Limit Theorem hold for this quantity. However, such normal fluctuations disappear for a particular value of V, called the characteristic speed.

For this velocity value, the correct scaling of particle current fluctuations was shown to be t1/3 and the limit distribution was also identified by K. Johansson in 2000 and later by P. L. Ferrari and H. Spohn in 2006. These results use heavy combinatorial and analytic tools, and their application is limited to a few particular models, one of which is the totally asymmetric simple exclusion process (TASEP). I will explain a purely probabilistic, more robust approach that provides the t2/3-scaling of current variance, but not the limit distribution, in (non-totally) asymmetric simple exclusion (ASEP) and some other particle systems. I will also point out a key feature of the models which allows the proof of such universal behaviour.

Joint work with Júlia Komjáthy and Timo Seppälläinen)

Abstract: We discuss an extension of Hitchin's generalised geometry, based on the exceptional groups, that provides a unified geometrical description of supersymmetric flux backgrounds in eleven-dimensional supergravity. We focus on N=1 seven-dimensional compactifications. The background is characterised by an element phi, the analogue of the generalised complex structure, that lies in a particular orbit of the 912 representation of E7. As an application we show that the four-dimensional effective superpotential takes a universal form, that is, a homogeneous E7-invariant functional of phi.

Let E be an elliptic curve defined by a cubic equation with rational coefficients.
The Sato-Tate Conjecture is a statistical assertion about the variation of the number of points of E over finite fields. I review some of the main steps in my proof of this conjecture with Clozel, Shepherd-Barron, and Taylor, in the case when E has non-integral j-invariant. Emphasis will be placed on the steps involving moduli spaces of certain Calabi-Yau hypersurfaces with level structure.

If one admits a version of the stable trace formula that should soon be available, the same techniques imply that, when E and E' are two elliptic curves that are not isogenous, then the numbers of their points over finite fields are statistically independent. For reasons that have everything to do with the current limits to our understanding of the Langlands program, the analogous conjectures for three or more non-isogenous elliptic curves are entirely out of reach.

Fixed-point logics are a class of logics designed for formalising

recursive or inductive definitions. Being initially studied in

generalised recursion theory by Moschovakis and others, they have later

found numerous applications in computer science, in areas

such as database theory, finite model theory, and verification.

A common feature of most fixed-point logics is that they extend a basic

logical formalism such as first-order or modal logic by explicit

constructs to form fixed points of definable operators. The type of

fixed points that can be formed as well as the underlying logic

determine the expressive power and complexity of the resulting logics.

In this talk we will give a brief introduction to the various extensions

of first-order logic by fixed-point constructs and give some examples

for properties definable in the different logics. In the main part of

the talk we will concentrate on extensions of first-order

logic by least and inflationary fixed points. In particular, we

compare the expressive power and complexity of the resulting logics.

The main result will be to show that while the two logics have rather

different properties, they are equivalent in expressive power on the

class of all structures.