Past

The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, we prove that, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss the link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes (joint work with Zhonggen Su).

The McKean stochastic game (MSG) is a two-player version of the perpetual American put option. The MSG consists of two agents and a certain payoff function of an underlying stochastic process. One agent (the seller) is looking for a strategy (stopping time) which minimises the expected pay-off, while the other agent (the buyer) tries to maximise this quantity.

For Brownian motion one can find the value of the MSG and the optimal stopping times by solving a free boundary value problem. For a Lévy process with jumps the corresponding free boundary problem is more difficult to solve directly and instead we use fluctuation theory to find the solution of the MSG driven by a Lévy process with no positive jumps. One interesting aspect is that the optimal stopping region for the minimiser "thickens" from a point to an interval in the presence of jumps. This talk is based on joint work with Andreas Kyprianou (University of Bath).

Trading a financial asset involves a sequence of decisions to buy or sell the asset over time. A traditional trading strategy is to buy low and sell high. However, in practice, identifying these low and high levels is extremely challenging and difficult. In this talk, I will present our ongoing research on characterization of these key levels when the underlying asset price is dictated by a mean-reversion model. Our objective is to buy and sell the asset sequentially in order to maximize the overall profit. Mathematically, this amounts to determining a sequence of stopping times. We establish the associated dynamic programming equations (quasi-variational

inequalities) and show that these differential equations can be converted to algebraic-like equations under certain conditions.

The two threshold (buy and sell) levels can be found by solving these algebraic-like equations. We provide sufficient conditions that guarantee the optimality of our trading strategy.

The lecture will describe two variants of thin film flows, one involving wetting and the other involving evaporation. First, describing the spreading of mostly wetting liquid droplets on surfaces decorated with assemblies of micron-size cylindrical posts arranged in regular arrays. A variety of deterministic final shapes of the spreading droplets are obtained, including octagons, squares, hexagons and cricles. Dynamic considerations provide a "shape" diagram and suggest rules for control. It is then shown how these ideas can be used to explore (and control) splashing and to create polygonal hydraulic jumps. Second, the evaporation of volatile liquid drops is considered. Using experiments and theory it is shown how the sense of the internal circulation depends on the ratio of the liquid and substrate conductivities. The internal motions control the deposition patterns and so may impact various printing processes. These ideas are then applied to colloid deposition porous media.

Recently there has been increasing interest in discrete analogues of classical operators in harmonic analysis. Often the difficulties one encounters in the discrete setting require completely new approaches; the most successful current approaches are motivated by ideas from classical analytic number theory. This talk will describe a menagerie of new results for discrete analogues of operators ranging from twisted singular Radon transforms to fractional integral operators both on R^n and on the Heisenberg group H^n. Although these are genuinely analytic results, key aspects of the methods come from number theory, and this talk will highlight the roles played by theta functions, Waring's problem, the Hypothesis K* of Hardy and Littlewood, and the circle method.

The Jacobi-Davidson method, proposed by Sleijpen and Van der Vorst more than a decade ago, has been successfully used to numerically solve large matrix eigenvalue problems. In this talk we will give an introduction to and an overview of this method, and also point out some recent developments.

We consider the hedging of a claim on a non-traded asset using a correlated traded asset, when the agent does not know the true values of the asset drifts, a partial information scenario. The drifts are taken to be random variables with a Gaussian prior distribution. This is updated via a linear filter. The result is a full information model with random drifts. The utility infdifference price and hedge is characterised via the dual problem, for an exponential utility function. An approximation for the price and hedge is derived, valid for small positions in the claim. The effectiveness of this hedging strategy is examined via simulation experiments, and is shown to yield improved results over the Black-Scholes strategy which assumes perfect correlation.

We propose and analyze a fully discrete Fourier collocation scheme to

solve numerically a nonlinear equation in 2D space derived from a

pattern forming gradient flow. We prove existence and uniqueness of the

numerical solution and show that it converges to a solution of the

initial continuous problem. We also derive some error estimates and

perform numerical experiments to illustrate the theory.

In this talk we will begin by discussing the notion of homogenization as an extension to the continuum assumption and regimes in which it breaks down. We then discuss various approaches to dealing with randomness whilst determining the effective properties of acoustic, thermal and elastic media.  In particular we show how the effective properties depend on the randomness of the microstructure

``The Utility of Wealth,'', Markowitz's ``other'' 1952 paper, explains observed risk-seeking and risk-avoidance behaviour by a utility function which has deviation from customary wealth, rather than wealth itself, as its argument. It also assumes that utility is bounded above and below.

This talk presents a class (GUW) of functions which generalise

utility-of-wealth (UW) functions. Unlike the latter functions, the

class is too broad to have interesting, verifiable implications. Rather, various subclasses have such implications. A recent paper by Gillen and Markowitz presents notations to specify various subclasses, and explores the properties of some of these.

This talk extends this classification of risk-facing behaviour to non-utility-maximising behaviour as described by Allais and Ellsberg, and formalised by Mark Machina.

The classical expected utility maximisation theory for financial asset allocation is premised on the assumption that human beings when facing risk make rational choices. The theory has been challenged by many observed and repeatable empirical patterns as well as a number of famous paradoxes and puzzles. The prospect theory in behavioural finance use cognitive psychological techniques to incorporate anomalies in human judgement into economic decision making. This lecture explains the interplay between risk and human judgement, and its impact on dynamic asset allocation via mathematically establishing and analysing a behavioural portfolio choice model.

The geometric Langlands program aims at a "spectral decomposition" of certain derived categories, in analogy with the spectral decomposition of function spaces provided by the Fourier transform. I'll explain such a geometrically-defined spectral decomposition of categories for a particular geometry that arises naturally in connection with integrable systems (more precisely, the quantum Calogero-Moser system) and representation theory (of Cherednik algebras). The category in this case comes from the moduli space of vector bundles on a curve equipped with a choice of ``mirabolic'' structure at a point. The spectral decomposition in this setting may be understood as a case of ``tamely ramified geometric Langlands''. In the talk, I won't assume any prior familiarity with the geometric Langlands program, integrable systems or Cherednik algebras.

I will give a characterization of connected Lie groups admitting a quasi-isometric embedding into a CAT(0) metric space. The proof relies on the study of the geometry of their asymptotic cones

In the first part of the talk, we fit the Hua-Pickrell measure (which is a two parameters deformation of the Haar measure) on the unitary group and the Ewens measure on the symmetric group in a same framework. We shall see that in the unitary case, the eigenvalues follow a determinantal point process with explicit hypergeometric kernels. We also study asymptotics of these kernels. The techniques used rely upon splitting of the Haar measure and sampling techniques. In the second part of the talk, we provide a matrix model for the circular Jacobi ensemble, which is the sampling used for the Hua-Pickrell measure but this time on Dyson's circular ensembles. In this case, we use the theory of orthogonal polynomials on the unit circle. In particular we prove that when the parameter of the sampling grows with n, both the spectral measure and the empirical spectral measure converge weakly in probability to a non-trivial measure supported only by one piece of the unit circle.

We consider a critical, measure-valued branching diffusion ξ in Rd, where the branching is continuous and the spatial motion is given by the heat flow. For d ≥ 2 and fixed t > 0, ξt is known to be an a.s. singular random measure of Hasudorff dimension 2. We explain how it can be approximated by Lebesgue measure on ε-neighbourhoods of the support. Next we show how ξt can be approximated in total variation near n points, and how the associated Palm distributions arise in the limit from elementary conditioning. Finally we hope to explan the duality between moment and Palm measures, and to show how the latter can be described in terms of discrete “Palm trees.”

In recent years Schanuel’s Conjecture (SC) has played a fundamental role

in the Theory of Transcendental Numbers and in decidability issues.

Macintyre and Wilkie proved the decidability of the real exponential field,

modulo (SC), solving in this way a problem left open by A. Tarski.

Moreover, Macintyre proved that the exponential subring of R generated

by 1 is free on no generators. In this line of research we obtained that in

the exponential ring $(\mathbb{C}, ex)$, there are no further relations except $i^2 = −1$

and $e^{i\pi} = −1$ modulo SC. Assuming Schanuel’s Conjecture we proved that

the E-subring of $\mathbb{R}$ generated by $\pi$ is isomorphic to the free E-ring on $\pi$.

These results have consequences in decidability issues both on $(\mathbb{C}, ex)$ and

$(\mathbb{R}, ex)$. Moreover, we generalize the previous results obtaining, without

assuming Schanuel’s conjecture, that the E-subring generated by a real

number not definable in the real exponential field is freely generated. We

also obtain a similar result for the complex exponential field.

Many portfolio optimization problems are directly or indirectly concerned with the current maximum of the underlying. For example, loockback or Russian options, optimization with max-drawdown constraint , or indirectly American Put Options, optimization with floor constraints.

The Azema-Yor martingales or max-martingales, introduced in 1979 to solve the Skohorod embedding problem, appear to be remarkably efficient to provide simple solution to some of these problems, written on semi-martingale with continuous running supremum.

The phenomenon of ignition is one with which we are all familiar, but which is remarkably difficult to define and model effectively. My own (description rather than definition) is “initiation of a (high temperature) self-sustaining exothermic process”; it may of course be desirable, as in your car’s engine, or highly undesirable, as the cause of many disastrous fires and explosions Both laboratory experiments and numerical simulations demonstrate its extreme sensitivity to external influences, past history and process (essentially chemical) kinetics, but at the heart of all instances there appears to be some “critical” unstable equilibrium state. Though some analytical modelling has been useful in particular cases, this remains in general virgin territory for applied mathematicians – perhaps there is room for some “knowledge transfer” here.

We recall that an elliptic curve is a curve of genus one with a rational point on it. Certain algorithms for determining the structure of the group of rational points on an elliptic curve produce a whole set of curves of genus one and then require that we determine which of these curves has a rational point.

Unfortunately no algorithm which has been proved to terminate is known for doing this. Such an algorithm or proof would probably have profound implications for the study of elliptic curves and may shed light on the Birch and Swinnerton-Dyer conjecture.

This talk will be about joint work with Samir Siksek (Warwick) on the development of a new algorithmic criterion for determining that a given curve of genus one has no rational points. Both the theory behind the criterion and recent attempts to make the criterion computationally practical, will be detailed.