Past

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k≥1 driven by a d-dimensional spatially homogeneous additive Gaussian noise that is white in time and coloured in space.

Brownian Molecular Motors are crucial for cell motility, muscle contraction or any other mechanical task carried out by proteins. After a short introduction to protein motors, I will talk about a numerical appraoch I worked on during the last months, which should enable us to deduct properties for a broad range of protein motors. A special focus should lie on the calculation of the eigenvalue spectrum, which gives insight to motors' stability.

My talk will be about optimal investment in Kabanov's model of currency exchange with transaction costs. The model is general enough to allow a random, discontinuous bid-offer spread. The investor wishes to maximize their "direct" utility of consumption, which is measured in terms of consumption assets linked to some (but not necessarily all) of the traded currencies. The analysis will centre on two conditions under which the existence of a dual minimiser leads to the existence of an optimal terminal wealth. The first condition is a well known, but rather unintuitive, condition on the utility function. The second weaker, and more natural condition is that of "asymptotic satiability" of the value function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer. This is joint work with Luciano Campi.

About a third of us will have a cancer during our lives, and we all know someone with the disease. Despite enormous progress in recent years, so that being diagnosed with cancer is not the death sentence that it once was, treatment can be aggressive, leading to short and long term reductions in quality of life. Cancer and its treatment is still feared, and rightly so - it is a major health concern. Destroying cancer non-invasively using protons or charged light ions such as carbon (Particle Therapy Cancer Research or PTCR) offers advantages over conventional radiotherapy using x-rays, since far lower radiation dose is delivered to healthy normal tissues. PT is also an alternative to radical cancer surgery. Most radiotherapy uses a small electron linear accelerator to accelerate an electron beams to a few million volts and then to generate hard x-rays, whereas CPT uses cyclotrons or synchrotrons to accelerate protons to a few hundred million volts, which themselves sterilise the tumour. More recently, a new concept in accelerators – the “non-scaling Fixed Field Alternating Gradient” accelerator – has been advanced, which offers the prospect of developing relatively compact, high acceleration rate accelerators for a variety of purposes, from neutrino factories and muon acceleration to cancer therapy. However, there are formidable technical challenges to be overcome, including resonance crossing. We have recently been awarded funding in the UK to construct a demonstrator non-scaling FFAG at the Daresbury laboratory (EMMA, the Electron Model with Many Applications), and to design a prototype machine for proton and carbon ion cancer therapy (PAMELA, the Particle Accelerator for MEdicaL Applications). I will describe some of the motivations for developing this new type of accelerator. Finally, although the physics of CPT says that it should be qualitatively and quantitatively better than conventional radiotherapy, the robust clinical analyses (for example, randomised control trials) have not been done, and the meta-analyses seem to suffer from large sample biases. The Particle Therapy Cancer Research Institute (part of the James Martin 21st Century School in Oxford) will study the clinical effectiveness of charged particle therapy to treat cancer, promoting its use in the UK and elsewhere on the basis of robust clinical evidence and analysis.

I will provide an overview of theoretical models aimed at understanding how self-excited oscillations arise when flow is driven through a finite-length flexible tube or channel. This problem is approached using a hierarchy of models, from one to three spatial dimensions, combining both computational and asymptotic techniques. I will explain how recent work is starting to shed light on the relationship between local and global instabilities, energy balances and the role of intrinsic hydrodynamic instabilities. This is collaborative work with Peter Stewart, Robert Whittaker, Jonathan Boyle, Matthias Heil and Sarah Waters.

We are interested in optimizing the compliance of an elastic structure when the applied forces are partially unknown or submitted to perturbations, the so-called "robust compliance".

For linear elasticity,the compliance is a solution to a minimizing problem of the energy. The robust compliance is then a min-max, the minimum beeing taken amongst the possible displacements and the maximum amongst the perturbations. We show that this problem is well-posed and easy to compute.

We then show that the problem is relatively easy to differentiate with respect to the domain and to compute the steepest direction of descent.

The levelset algorithm is then applied and many examples will explain the different mathematical and technical difficulties one faces when one

tries to tackle this problem.

I shall show how cellularity may be used to obtain presentations of the
endomorphism algebras in question, both in the classical and quantum cases.

Electronic structure calculations are commonly used to understand and predict the electronic, magnetic and optic properties of molecular systems and materials. They are also at the basis of ab initio molecular dynamics, the most reliable technique to investigate the atomic scale behavior of materials undergoing chemical reactions (oxidation, crack propagation, ...).

In the first part of my talk, I will briefly review the foundations of the density functional theory for electronic structure calculations. In the second part, I will present some recent achievements in the field, as well as open problems. I will focus in particular on the mathematical modelling of defects in crystalline materials.

Aside from a few tangential problems, this seminar will include a proof of Ostrowski's Theorem. This states than any norm over the rationals is equivalent to either the Euclidean norm or the $p$-adic norm, for some prime $p$.

Given a foliation of a compact manifold, leaves of which are equipped with a Riemannian metric, one can consider the associated "leafwise"

Brownian motion, and study its asymptotic properties (such as asymptotic distribution, behaviour of holonomy maps, etc.).

Lucy Garnet studied such measures, introducing the notion of a harmonic measure -- stationary measure of this process; the name "harmonic" comes from the fact that a measure is stationary if and only if with respect to it integral of every leafwise Laplacian of a smooth function equals zero (so, the measure is "harmonic" in the sense of distributions).

It turns out that for a transversally conformal foliation, unless it possesses a transversally invariant measure (which is a rather rare case), the associated random dynamics can be described rather precisely. Namely, for every minimal set in the foliation there exists a unique harmonic measure supported on it -- and this gives all the possible ergodic harmonic measures (in particular, there is a finite number of them, and they are always supported on the minimal sets).

Also, the holonomy maps turn out to be (with probability one) exponentially contracting -- so, the Lyapunov exponent of the dynamics is negative. Finally, for any initial point almost every path tends to one of the minimal sets and is asymptotically distributed with respect to the corresponding harmonic measure -- and the functions defining the probabilities of tending to different sets form a base in the space of continuous leafwise harmonic functions.

An interesting effect that is a corollary of this consideration is that for transversally conformal foliations the number of the ergodic harmonic measures does not depend on the choice of Riemannian metric on the leaves. This fails for non-transversally conformal foliations:

there is an example, recently constructed in a joint with S.Petite (following B.Deroin's technique).

There is a well-known relationship between the theory of formal group schemes and stable homotopy theory, with Ravenel's chromatic filtration and the nilpotence theorem of Hopkins, Devinatz and Smith playing a central role. It is also familiar that one can sometimes get a more geometric understanding of homotopical phenomena by examining how they interact with group actions. In this talk we will explore this interaction from the chromatic point of view.

Stochastic geometry gradually becomes a necessary theoretical tool to model and analyse modern telecommunication systems, very much the same way the queuing theory revolutionised studying the circuit switched telephony in the last century. The reason for this is that the spatial structure of most contemporary networks plays crucial role in their functioning and thus it has to be properly accounted for when doing their performance evaluation, optimisation or deciding the best evolution scenarios.  The talk will present some stochastic geometry models and tools currently used in studying modern telecommunications.  We outline specifics of wired, wireless fixed and ad-hoc systems and show how the stochastic geometry modelling helps in their analysis  and optimisation.

I'll discuss my ongoing attempt to modernise the theory of the image of J.
Some features
that I would like to have are as follows:

1) Most of the spectra involved in the story should be E_\infty (or strictly
commutative)
    ring spectra, and most of the maps involved should respect this structure.  New
    machinery for dealing with E_\infty rings should be used systematically.

2) As far as possible the constructions used should not depend on arbitrary choices
     or on gratuitous localisation.

3) The Bernoulli numbers should enter via their primary definition as coefficients of a
     certain power series.

4) The image of J spectrum should be defined as the Bousfield localisation of S^0 with
    respect to KO, and other properties or descriptions should be deduced from this one.

5) There should be a clear conceptual explanation for the parallel appearance of
    Bernoulli numbers in the homotopy groups of J, K(Z) and in spectra related to
    surgery theory.

This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross (CIR) process,  without any restriction on its
parameters. At the same time, it gives a general   recursive
construction method to get weak second-order schemes that extends the one introduced by Ninomiya and Victoir. Combining these both results, this allows to propose a second-order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models

Let T be a (one-sorted first order) geometric theory (so T

has infinite models, T eliminates "there exist infinitely many" and

algebraic closure gives a pregeometry). I shall present some results

about T_P, the theory of lovely pairs of models of T as defined by

Berenstein and Vassiliev following earlier work of Ben-Yaacov, Pillay

and Vassiliev, of van den Dries and of Poizat. I shall present

results concerning superrosiness, the independence property and

imaginaries. As far as the independence property is concerned, I

shall discuss the relationship with recent work of Gunaydin and

Hieronymi and of Berenstein, Dolich and Onshuus. I shall also discuss

an application to Belegradek and Zilber's theory of the real field

with a subgroup of the unit circle. As far as imaginaries are

concerned, I shall discuss an application of one of the general

results to imaginaries in pairs of algebraically closed fields,

adding to Pillay's work on that subject.

The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We will introduce the model and discuss some of its mathematical properties. In particular, we will focus on the possibility that singularities may develop.

The rate at which singularities develop is investigated in settings with certain symmetries. We use the method of matched asymptotic expansions and identify different scenarios for singularity formation. More specifically, we distinguish between singularities that develop in finite time and those that need infinite time to form.

Finally, we discuss which results can be proven rigorously, as well as some open problems, and we address stability issues (ongoing work with JF Williams).

We discuss numerical methods for the solution of the palindromic eigenvalue problem Ax=λ A^Tx, where A is a complex matrix. Such eigenvalue problems occur, for example, in the vibration analysis of rail tracks.

The structure of palindromic eigenvalue problems leads to a symmetry in the spectrum: all eigenvalues occur in reciprocal pairs. The need for preservation of this symmetry in finite precision arithmetic requires the use of structure-preserving numerical methods. In this talk, we explain how such methods can be derived.

In this paper we employ the quantile formulation to solve the SP/A portfolio choice model in continuous time. We show that the original version of the SP/A model proposed by Lopes is ill-posed in the continuous-time setting. We then generalise the SP/A model to one where a utility function is included, while the probability weighting

(distortion) function is still present. The feasibility and well-posedness of the model are addressed and an explicit solution is derived. Finally, we study how the aspiration level and the probability weighting function affect the optimal solution

I will briefly discuss the construction of the moduli spaces of (semi)stable bundles on a given curve. The main aim of the talk will be to describe various features of the geometry and topology of these moduli spaces, with emphasis on methods as much as on results. Topics may include irreducibility, cohomology, Verlinde numbers, Torelli theorems.

The subgroup growth of finitely generated groups was seen last term, in a lecture of Dan Segal. This time, we see representation growth, and how it is similar to, and different from, subgroup growth.

Khovanov homology is an invariant of knots in $S^3$. In its original form,

it is a "homological version of the Jones polynomial"; Khovanov and

Rozansky have generalized it to other knot polynomials, including the

HOMFLY polynomial.

In the second talk, I'll discuss how Khovanov homology and its generalizations lead to a relation between the HOMFLY polynomial and the topology of flag varieties.

Khovanov homology is an invariant of knots in $S^3$. In its original form,

it is a "homological version of the Jones polynomial"; Khovanov and

Rozansky have generalized it to other knot polynomials, including the

HOMFLY polynomial.

The first talk will be an introduction to Khovanov homology and its generalizations.

We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface

into $R^n$, $n = 3, 4$, if there is one conformal immersion with Willmore energy smaller than a certain bound $W_{n,p}$ depending on codimension and genus $p$ of the Riemann surface. For tori in codimension $1$, we know $W_{3,1} = 8\pi$ . Joint work with Enrst Kuwert.