Past

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.