Past

We construct a market of bonds with jumps driven by a general marked point

process as well as by an Rn-valued Wiener process, in which there exists at least one equivalent

martingale measure Q0. In this market we consider the mean-variance hedging of a contingent

claim H 2 L2(FT0) based on the self-financing portfolios on the given maturities T1, · · · , Tn

with T0 T. We introduce the concept of variance-optimal martingale

(VOM) and describe the VOM by a backward semimartingale equation (BSE). We derive an

explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by

the solutions of two BSEs.

The setting of this problem is a bit unrealistic as we restrict the available bonds to those

with a a pregiven finite number of maturities. So we extend the model to a bond market with

jumps and a continuum of maturities and strategies which are Radon measure valued processes.

To describe the market we consider the cylindrical and normalized martingales introduced by

Mikulevicius et al.. In this market we the consider the exp-utility problem and derive some

results on dynamic indifference valuation.

The talk bases on recent common work with Dewen Xiong.

Traditional Faraday waves appear in a layer of liquid that is shaken vertically. These patterns can take the form of horizontal stripes, close-packed hexagons, or even squares or quasiperiodic patterns. Faraday waves are commonly observed as fine stripes on the surface of wine in a wineglass that is ringing like a bell when periodically forced.

Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed gases subject to periodic modulation of the strength of the transverse confinement's trap.

We offer a fully analytical explanation of the observed parametric resonance yielding the pattern periodicity versus the driving frequency. These results, corroborated by numerical simulations, match extremely well with the experimental observations.

I shall describe joint work with Gritsenko and Hulek in which we study the moduli spaces of polarised holomorphic symplectic manifolds via their periods. There are strong similarities with moduli spaces of K3 surfaces, but also some important differences, notably that global Torelli fails. I shall explain (conjecturally) why and show how the techniques used to obtain general type results for K3 moduli can be modified to give similar, and quite strong, results in this case. Mainly I shall concentrate on the case of deformations of Hilbert schemes of K3 surfaces.

During the last two years, sparsity has become a key concept in various areas

of applied mathematics, computer science, and electrical engineering. Sparsity

methodologies explore the fundamental fact that many types of data/signals can

be represented by only a few non-vanishing coefficients when choosing a suitable

basis or, more generally, a frame. If signals possess such a sparse representation,

they can in general be recovered from few measurements using $\ell_1$ minimization

techniques.

One application of this novel methodology is the geometric separation of data,

which is composed of two (or more) geometrically distinct constituents -- for

instance, pointlike and curvelike structures in astronomical imaging of galaxies.

Although it seems impossible to extract those components -- as there are two

unknowns for every datum -- suggestive empirical results using sparsity

considerations have already been obtained.

In this talk we will first give an introduction into the concept of sparse

representations and sparse recovery. Then we will develop a very general

theoretical approach to the problem of geometric separation based on these

methodologies by introducing novel ideas such as geometric clustering of

coefficients. Finally, we will apply our results to the situation of separation

of pointlike and curvelike structures in astronomical imaging of galaxies,

where a deliberately overcomplete representation made of wavelets (suited

to pointlike structures) and curvelets/shearlets (suited to curvelike

structures) will be chosen. The decomposition principle is to minimize the

$\ell_1$ norm of the frame coefficients. Our theoretical results, which

are based on microlocal analysis considerations, show that at all sufficiently

fine scales, nearly-perfect separation is indeed achieved.

This is joint work with David Donoho (Stanford University).

We give a three dimensional generalization of the classical McKay correspondence construction by Gonzales-Sprinberg and Verdier. This boils down to computing for the Bridgeland-King-Reid derived category equivalence the images of twists of the point sheaf at the origin of C^3 by irreducible representations of G. For abelian G the answer turns out to be closely linked to a piece of toric combinatorics known as Reid's recipe.

Most financial models introduced for the purpose of pricing and hedging derivatives concentrate

on the dynamics of the underlying stocks, or underlying instruments on which the derivatives

are written. However, as certain types of derivatives became liquid, it appeared reasonable to model

their prices directly and use these market models to price or hedge exotic derivatives. This framework

was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets.

We discuss the characterization of arbitrage free dynamic stochastic models for the markets with

infinite number of European Call options as the liquid derivatives. Subject to our assumptions on the

presence of jumps in the underlying, the option prices are represented either through local volatility or

through local L´evy measure. Each of the latter ones is then given dynamics through an Itˆo stochastic

process in infinite dimensional space. The main thrust of our work is to characterize absence of arbitrage

in this framework and address the issue of construction of the arbitrage-free models.

Associated to a pencil of algebraic curves with singular fibres is a bundle of Jacobians (which are abelian varieties off the discriminant locus of the family and semiabelian varieties over it). Normal functions, which are holomorphic sections of such a Jacobian bundle, were introduced by Poincare and used by Lefschetz to prove the Hodge Conjecture (HC) on algebraic surfaces. By a recent result of Griffiths and Green, an appropriate generalization of these normal functions remains at the center of efforts to establish the HC more generally and understand its implications. (Furthermore, the nature of the zero-loci of these normal functions is related to the Bloch-Beilinson conjectures on filtrations on Chow groups.)

Abel-Jacobi maps give the connection between algebraic cycles and normal functions. In this talk, we shall discuss the limits and singularities of Abel-Jacobi maps for cycles on degenerating families of algebraic varieties. These two features are strongly connected with the issue of graphing admissible normal functions in a Neron model, properly generalizing Poincare's notion of normal functions. Some of these issues will be passed over rather lightly; our main intention is to give some simple examples of limits of AJ maps and stress their connection with higher algebraic K-theory.

A very new theme in homological mirror symmetry concerns what the mirror of a normal function should be; in work of Morrison and Walcher, the mirror is related to counting holomorphic disks in a CY 3-fold bounding on a Lagrangian. Along slightly different lines, we shall briefly describe a surprising application of "higher" normal functions to growth of enumerative (Gromov-Witten) invariants in the context of local mirror symmetry.

Let $G=(V,E)$ be a graph with weights $\{w_e : e \in E\}$, and assume all weights are distinct. If $G$ is finite, then the well-known Prim's algorithm constructs its minimum spanning tree in the following manner. Starting from a single vertex $v$, add the smallest weight edge connecting $v$ to any other vertex. More generally, at each step add the smallest weight edge joining some vertex that has already been "explored" (connected by an edge) to some unexplored vertex.

If $G$ is infinite, however, Prim's algorithm does not necessarily construct a spanning tree (consider, for example, the case when the underlying graph is the two-dimensional lattice ${\mathbb Z}^2$, all weights on horizontal edges are strictly less than $1/2$ and all weights on vertical edges are strictly greater than $1/2$).

The behavior of Prim's algorithm for *random* edge weights is an interesting and challenging object of study, even
when the underlying graph is extremely simple. This line of research was initiated by McDiarmid, Johnson and Stone (1996), in the case when the underlying graph is the complete graph $K_n$. Recently Angel et. al. (2006) have studied Prim's algorithm on regular trees with uniform random edge weights. We study Prim's algorithm on $K_n$ and on its infinitary analogue Aldous' Poisson-weighted infinite tree. Along the way, we uncover two new descriptions of the Poisson IIC, the critical Poisson Galton-Watson tree conditioned to survive forever.

Joint work with Simon Griffiths and Ross Kang.

Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by C. Darwin. In a very simple, general and idealized description, their environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to use the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born individuals undergo small variation of the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait?

We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'fittest trait' and eventually to compute various forms of branching points which represent the cohabitation of two different populations.

The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that desribe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed.

I'll explain (following Kontsevich and Soibelman) how cluster transformations intertwine non-commutative DT invariants for CY3 algebras related by a tilt.

We have recently found a way to describe large-scale neural

activity in terms of non-equilibrium statistical mechanics.

This allows us to calculate perturbatively the effects of

fluctuations and correlations on neural activity. Major results

of this formulation include a role for critical branching, and

the demonstration that there exist non-equilibrium phase

transitions in neocortical activity which are in the same

universality class as directed percolation. This result leads

to explanations for the origin of many of the scaling laws

found in LFP, EEG, fMRI, and in ISI distributions, and

provides a possible explanation for the origin of various brain

waves. It also leads to ways of calculating how correlations

can affect neocortical activity, and therefore provides a new

tool for investigating the connections between neural

dynamics, cognition and behavior

In this talk, Professor Lions will first present several examples of numerical simulations of complex industrial systems. All these simulations rely upon some mathematical models involving partial differential equations and he will briefly explain the nature, history and role of such equations. Examples showing the importance of the mathematical analysis (i.e. ‘understanding’) of those models will be presented, concluding with a few trends and perspectives.

Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and has himself become a renowned mathematician, making numerous important contributions to the theory of non-linear partial differential equations. He was awarded a Fields Medal in 1994, in particular for his work with Ron DiPerna giving the first general proof that the Boltzmann equation of the kinetic theory of gases has solutions. Other awards Lions has received include the IBM Prize in 1987 and the Philip Morris Prize in 1991. Currently he holds the position of Chair of Partial Differential Equations and their Applications at the prestigious Collège de France in Paris.

This lecture is given as part of the 7th ISAAC Congress • www.isaac2009.org

Clore Lecture Theatre, Huxley Building, Imperial College London,
South Kensington Campus, London SW7 2AZ

RSVP: Attendance is free, but with registration in advance
Michael Ruzhansky • @email

I will describe some recent joint work with Davide Gaiotto and Greg Moore, in which we explain the origin of the wall-crossing formula of Kontsevich and Soibelman, in the context of N=2 supersymmetric field theories in four dimensions. The wall-crossing formula gives a recipe for constructing the smooth hyperkahler metric on the moduli space of the field theory reduced on a circle to 3 dimensions. In certain examples this moduli space is actually a moduli space of ramified Higgs bundles, so we obtain a new description of the hyperkahler structure on that space.

When two inclusions (in a composite) get closer and their conductivities degenerate

to zero or infinity, the gradient of the solution to the

conductivity equation blows up in general. We show

that the solution to the conductivity equation can be decomposed

into two parts in an explicit form: one of them has a bounded

gradient and the gradient of the other part blows up. Using the

decomposition, we derive the best possible estimates for the blow-up

of the gradient. The decomposition theorem and estimates have an

important implication in computation of electrical field in

the presence of closely located inclusions.

This short course runs from Monday 29th June to Friday 3rd July. For details of the course and how to register, please visit http://www2.maths.ox.ac.uk/oxmos/meetings/moms/.

Preview of problems to be solved at the study Group in Limerick taking place in the following week.

I will explain what a perfect obstruction theory is, and how it gives rise to a "virtual" fundamental class of the right expected dimension, even when the dimension of the moduli space is wrong. These virtual fundamental classes are one of the main preoccupations of "modern" moduli theory, being the central object of study in Gromov-Witten and Donaldson-Thomas theory. The purpose of the talk is to remove the black-box status of these objects. If there is time I will do some cheer-leading for dg-schemes, and try to convince the audience that virtual fundamental classes are most happily defined to live in the dg-world.