Past

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.

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.