Past

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.

The stability of our Solar System has been debated since Newton devised

the laws of gravitation to explain planetary motion. Newton himself

doubted the long-term stability of the Solar System, and the question

has remained unanswered despite centuries of intense study by

generations of illustrious names such as Laplace, Langrange, Gauss, and

Poincare. Finally, in the 1990s, with the advent of computers fast

enough to accurately integrate the equations of motion of the planets

for billions of years, the question has finally been settled: for the

next 5 billion years, and barring interlopers, the shapes of the

planetary orbits will remain roughly as they are now. This is called

"practical stability": none of the known planets will collide with each

other, fall into the Sun, or be ejected from the Solar System, for the

next 5 billion years.

Although the Solar System is now known to be practically stable, it may

still be "chaotic". This means that we may---or may not---be able

precisely to predict the positions of the planets within their orbits,

for the next 5 billion years. The precise positions of the planets

effects the tilt of each planet's axis, and so can have a measurable

effect on the Earth's climate. Although the inner Solar System is

almost certainly chaotic, for the past 15 years, there has been

some debate about whether the outer Solar System exhibits chaos or not.

In particular, when performing numerical integrations of the orbits of

the outer planets, some astronomers observe chaos, and some do not. This

is particularly disturbing since it is known that inaccurate integration

can inject chaos into a numerical solution whose exact solution is known

to be stable.

In this talk I will demonstrate how I closed that 15-year debate on

chaos in the outer solar system by performing the most carefully justified

high precision integrations of the orbits of the outer planets that has

yet been done. The answer surprised even the astronomical community,

and was published in _Nature Physics_.

I will also show lots of pretty pictures demonstrating the fractal nature

of the boundary between chaos and regularity in the outer Solar System.

The talk is about the homotopy type of configuration spaces. Once upon a time there was a conjecture that it is a homotopy invariant of closed manifolds. I will discuss the strong evidence supporting this claim, together with its recent disproof by a counterexample. Then I will talk about the corrected version of the original conjecture.

Much of group theory is concerned with whether one property entails another. When such a question is answered in the negative it is often via a pathological example. We will examine the Rips construction, an important tool for producing such pathologies, and touch upon a recent refinement of the construction and some applications. In the course of this we will introduce and consider the profinite topology on a group, various separability conditions, and decidability questions in groups.

I'll explain how the `Auslander philosophy' from finite dimensional algebras gives new methods to tackle problems in higher-dimensional birational geometry. The geometry tells us what we want to be true in the algebra and conversely the algebra gives us methods of establishing derived equivalences (and other phenomenon) in geometry. Algebraically two of the main consequences are a version of AR duality that covers non-isolated singularities and also a theory of mutation which applies to quivers that have both loops and two-cycles.

In this talk I will present the rigorous construction of

singular solutions for two kinetic models, namely, the Uehling-Uhlenbeck

equation (also known as the quantum Boltzmann equation), and a class of

homogeneous coagulation equations. The solutions obtained behave as

power laws in some regions of the space of variables characterizing the

particles. These solutions can be interpreted as describing particle

fluxes towards or some regions from this space of variables.

The construction of the solutions is made by means of a perturbative

argument with respect to the linear problem. A key point in this

construction is the analysis of the fundamental solution of a linearized

problem that can be made by means of Wiener-Hopf transformation methods.

Condition supercritical percolation so that the origin is enclosed by a dual circuit whose interior traps an area of n^2.

The Wulff problem concerns the shape of the circuit. We study the circuit's fluctuation. A well-known measure of this fluctuation is maximum local roughness (MLR), which is the greatest distance from a point on the circuit to the boundary of circuit's convex hull. Another is maximum facet length (MFL), the length of the longest line segment of which this convex hull is comprised.

In a forthcoming article, I will prove that

for various models including supercritical percolation, under the conditioned measure,

MLR = \Theta(n^{1/3}\log n)^{2/3}) and MFL = \Theta(n^{2/3}(log n)^{1/3}).

An important tool is a result establishing the profusion of regeneration sites in the circuit boundary. The talk will focus on deriving the main results with this tool

Dislocation channel-veins and Persist Slip Band (PSB) structures are characteristic configurations in material science. To find out the formation of these structures, the law of motion of a single dislocation should be first examined. Analogous to the local expansion in electromagnetism, the self induced stress is obtained. Then combining the empirical observations, we give a smooth mobility law of a single dislocation. The stability analysis is carried our asymptotically based on the methodology in superconducting vortices. Then numerical results are presented to validate linear stability analysis. Finally, based on the evidence given by the linear stability analysis, numerical experiments on the non-linear evolution are carried out.

An overview of the early history of the soliton (1960-1970) and equipartition in nonlinear 1D lattices : From Fermi-Pasta-Ulam to Korteweg de Vries, to Nonlinear Schrodinger*…., and recent developments .

For a logarithmic utility function we extend our rezult with Xu and Zhou for case of the geometrical Brownian motion with drift term which depends of the some hidden parameter.

Anthony Lock will speak on "A Column Model of Moist Convection".

I will speak about the CBP (canonical base property) for types of finite SU-rank. This property first appears in a paper by Pillay and Ziegler, who show that it holds for types of finite rank in differentially closed fields of characteristic 0, as well as in existentially closed difference fields. It is unknown whether this property holds for all finite rank types in supersimple theories. I will first recall the definition of a canonical base, and give some natural examples. I will then  talk about a reduction of the problem (which allows one to extend the Pillay-Ziegler result to existentially closed fields of any characteristic), and finally derive some consequences of the CBP, in particular the UCBP, thus answering a question of Moosa and Pillay.  If time permits, I will show an application of these results to difference

fields.

An overview of the experiments of Steinbergs group, Theory-and-models and comparison of the applicability of recent reduced models.

Self adjusted meshes have important benefits approximating PDEs with solutions that exhibit nontrivial characteristics. When appropriately chosen, they lead to efficient, accurate and robust algorithms. Error control is also important, since appropriate analysis can provide guarantees on how accurate the approximate solution is through a posteriori estimates. Error control may lead to appropriate adaptive algorithms by identifying areas of large errors and adjusting the mesh accordingly. Error control and associated adaptive algorithms for important equations in Mathematical Physics is an open problem.

In this talk we consider the main structure of an algorithm which permits mesh redistribution with time and the nontrivial characteristics associated with it. We present improved algorithms and we discuss successful approaches towards error control for model problems (linear and nonlinear) of parabolic or hyperbolic type.

A -polarised K3 surface admits an embedding into weighted projective space defined by its polarisation. Let X be a family of such surfaces, then one can construct a projective model W of X such that the map from X to W realises this embedding on the general fibre. This talk considers what happens to W when we allow the fibres of the family X to degenerate.

The semisimplicity problem is the long-standing conjecture that the group algebra $KG$ of a $p'$-group $G$ over a field $K$ of characteristic $p\geqslant 0$ has zero Jacobson radical. We will discuss recent advances in connection with this problem.

We will discuss a connection between small cancellation conditions and isoperimetric inequalities. Additionally we shall look at a useful construction connecting small cancellation complexes and cube complexes.

I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.

I will talk about joint work with Dimca, respectively Behrend and Bryan, in which we refine the numerical DT-Behrend invariants of Hilbert schemes of threefolds by using vanishing cycle motives (a la Kontsevich-Soibelman) or mixed Hodge modules, leading to deformed MacMahon formulae.

Two classes of solutions of the Einstein equations with symmetry which

are frequently studied are the Bianchi and Gowdy models. The aim of this

talk is to explain certain relations between these two classes of

spacetimes which can provide insights into the dynamics of both. In

particular it is explained that the special case of the Gowdy models known as circular loop spacetimes are Bianchi models in disguise. Generalizations of Gowdy spacetimes which can be thought of as inhomogeneous perturbations of some of the Bianchi models are introduced.

Results concerning their dynamics are presented.

We study the homogenization and singular perturbation of the

wave equation in a periodic media for long times of the order

of the inverse of the period. We consider inital data that are

Bloch wave packets, i.e., that are the product of a fast

oscillating Bloch wave and of a smooth envelope function.

We prove that the solution is approximately equal to two waves

propagating in opposite directions at a high group velocity with

envelope functions which obey a Schr\"{o}dinger type equation.

Our analysis extends the usual WKB approximation by adding a

dispersive, or diffractive, effect due to the non uniformity

of the group velocity which yields the dispersion tensor of

the homogenized Schr\"{o}dinger equation. This is a joint

work with M. Palombaro and J. Rauch.