Past

In this talk we discuss the main results of my PhD thesis. We begin by giving some background on Moufang polygons. This is followed by a short introduction of the basic model theoretic notions related to the thesis, such as asymptotic classes of finite structures, measurable structures, (superstable) supersimple theories and (finite Morley rank) S_1 rank. We also mention the relation between Moufang polygons and the associated little projective groups.

Moufang polygons have been classified by Tits and Weiss, and a complete list is given in their book `Moufang polygons'.

This work is inspired by a paper of Kramer, Tent and van Maldeghem called "Simple groups of finite Morley rank and Tits buildings". The authors work in a superstable context. They show that Moufang polygons of finite Morley rank are exactly Pappian polygons, i.e., projective planes, symplectic quadrangles and split Cayley hexagons, provided that they arise over algebraically closed fields.

We work under the weaker assumption of supersimplicity. Therefore, we expect more examples. Indeed, apart from those already occuring in the finite Morley rank case, there are four further examples, up to duality, of supersimple Moufang polygons; namely, Hermitian quadrangles in projective dimension 3 and 4, the twisted triality hexagon and the (perfect) Ree-Tits octagon, provided that the underlying field (or `difference' field in the last case) is supersimple.

As a result, we obtain the nice characterization that supersimple Moufang polygons are exactly those Moufang polygons belonging to families which also arise over finite fields.

Examples of supersimple Moufang polygons are constructed via asymptoticity

arguments: every class C of finite Moufang polygons forms an asymptotic class, and every non-principal ultraproduct of C gives rise to a measurable structure, thus supersimple (of finite S_1 rank). For the remaining cases one can proceed as follows: let \Gamma be any Moufang polygon belonging to a family which does not arise over finite fields, and call K its underlying field; then K is

(first-order) definable in \Gamma, and by applying some model theoretic facts this definability is inconsistent with supersimplicity".

Layered (or laminated) structures are increasingly used in modern industry (e.g., in microelectronics and aerospace engineering). Integrity of such structures is mainly determined by the quality of their interfaces: poor adhesion or delamination can lead to a catastrophic failure of the whole structure. Can nonlinear waves help us to detect such defects? We study the dynamics of a nonlinear longitudinal bulk strain wave in a split, layered elastic bar, made of nonlinearly hyperelastic Murnaghan material. We consider a symmetric two-layered bar and assume that there is perfect interface for x 0, where the x-axis is directed along the bar. Using matched asymptotic multiple-scales expansions and the integrability theory of the KdV equation by the Inverse Scattering Transform, we examine scattering of solitary waves and show that the defect causes generation of more than one secondary solitary waves from a single incident soliton and, thus, can be used to detect the defect. The theory is supported by experimental results. Experiments have been performed in the Ioffe Institute in St. Petersburg (Russia), using holographic interferometry and laser induced generation of an incident compression solitary wave in two- and three-layered polymethylmethacrylate (PMMA) bars, bonded using ethyl cyanoacrylate-based (CA) adhesive.

Artin formalism gives an equality of certain L-functions of elliptic curves or of zeta-functions of number fields. When combined with the Birch and Swinnerton-Dyer conjecture, this can give interesting results about the Galois module structure of the Selmer group of an elliptic curve. When combined with the analytic class number formula, this can help determine the Galois module structure of the group of units of a number field. In this talk, I will introduce the main technique, which is completely representation theoretic, for extracting such information

Work with N.K. Nichols (Reading), C. Boess & A. Bunse-Gerstner (Bremen)

The Gauss-Newton (GN) method is a well known iterative technique for solving nonlinear least squares problems subject to dynamical system constraints. Such problems arise commonly from applications in optimal control and state estimation. Variational data assimilation systems for weather, ocean and climate prediction currently use approximate GN methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new method for deriving low order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to give a state estimation technique that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.

Scalar balance laws with monostable reaction, possibly non-convex flux, and

viscosity $\varepsilon$ are known to admit so-called entropy travelling fronts for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both when $\varepsilon$ is positive, when all fronts are smooth, and for $\varepsilon =0$, when the possibly non-convex flux results in fronts of speed close to the minimal value typically having discontinuities where jump conditions hold.

I will discuss the vanishing-viscosity limit of these fronts.

I shall discuss joint work with Angus Macintyre on the model theory of the ring of adeles of a number field

Macroscopic properties of solids are inherently connected to their micro- and nano-scale details. For example, the microstructure and defect distribution influence the elastic and plastic properties of a crystal while the details of a defect are determined by its elastic far-field. The goal of multi-scale modelling is to understand such connections between microscopic and macroscopic material behaviour. This workshop brings together researchers working on different aspects of multi-scale modelling of solids: mathematical modelling, analysis, numerical computations, and engineering applications.

Let $A$ be a finite set in some ambient group. We say that $A$ is a $K$-approximate group if $A$ is symmetric and if the set $A.A$ (the set of all $xy$, where $x$, $y$ lie in $A$) is covered by $K$ translates of $A$. I will illustrate this notion by example, and will go on to discuss progress on the "rough classification" of approximate groups in various settings: abelian groups, nilpotent groups and matrix groups of fixed dimension. Joint work with E. Breuillard.

I will discuss recent joint work with A. Ionescu and S.
Klainerman on the black hole uniqueness problem. A classical result of
Hawking (building on earlier work of Carter and Robinson) asserts that any
vacuum, stationary black hole exterior region must be isometric to the
Kerr exterior, under the restrictive assumption that the space-time metric
should be analytic in the entire exterior region.
We prove that Hawking's theorem remains valid without the assumption of
analyticity, for black hole exteriors which are apriori assumed to be "close"
to the Kerr exterior solution in a very precise sense. Our method of proof
relies on certain geometric Carleman-type estimates for the wave operator.

In the previous talks we have seen the formulation of the Birch--Swinnerton-Dyer conjecture. This talk will focus on a fundamental question in diophantine geometry. Namely, given an algebraic curve \textit{C} defined over $\mathbb{Q}$ possessing at least one rational point, what is

the probability that \textit{C} has infinitely many rational points?

For curves of genus 0, the answer has been known ever since the ancient Greeks roamed the earth, and for genus > 1 the answer is also known (albeit for a much shorter time). The remaining case is genus 1, and this question has a history filled with tension and

conflict between data and conjecture.

I shall describe the heuristics behind the conjectures, taking into account the Birch--Swinnerton-Dyer Conjecture and the Parity Conjecture. I shall go on to outline the contrary numeric data, both in families of elliptic curves and for all elliptic curves of increasing conductor.

If one instead considers elliptic curves over function fields $\mathbb{F}_{q} (t)$, then, via a conjecture of Artin and Tate, one can compute the rank (and more) of elliptic curves of extremely large discriminant degree. I shall briefly describe the interplay between Artin--Tate and

Birch--Swinnerton-Dyer, and give new evidence finally supporting the conjecture.

Joint work with Soren Galatius. We study categories C of d-dimensional cobordisms, from the perspective of Galatius, Madsen, Tillmann and Weiss. Their main result is the determination of the homotopy type of the classifying-space of such cobordism categories, as the infinite loop space of a certain Thom spectrum. One can investigate subcategories D of C having the property that the classifying-space BD is equivalent to BC, the smaller such D one can find the better.

We prove that in may cases of interest, D can be taken to be a homotopy commutative monoid. As a consequence, the stable cohomology of many moduli spaces of surfaces can be identified with that of the infinite loop space of certain Thom spectra.

My goal is to estimate unknown parameters in the vector field of a rough differential equation, when the expected signature for the driving force is known and we estimate the expected signature of the response by Monte Carlo averages.

I will introduce the "expected signature matching estimator" which extends the moment matching estimator and I will prove its consistency and asymptomatic normality, under the assumption that the vector field is polynomial.  Finally, I will describe the polynomial system one needs to solve in order to compute this estimatior.