Past

One knows, for example by proving well-posedness for an initial value problem with data at the singularity, that there exist many cosmological solutions of the Einstein equations with an initial curvature singularity but for which the conformal metric can be extended through the singularity. Here we consider a converse, a local extension problem for the conformal structure: given an incomplete causal curve terminating at a curvature singularity, when can one extend the conformal structure to a set containing a neighbourhood of a final segment of the curve?

We obtain necessary and sufficient conditions based on boundedness of tractor curvature components. (Based on work with Christian Luebbe: arXiv:0710.5552, arXiv:0710.5723.)

We study Onsager’s model of isotropic–nematic phase transition with orientation parameter on a circle and sphere. We show the axial symmetry and derive explicit formulae for all critical points. Using the information about critical points we investigate a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the free energy is a functional of spatially-dependent orientation distribution, however, the spatial variation effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. As a particular example we consider a simplified model with orientation parameter on a circle and illustrate its relation to complex Ginzburg-Landau theory.

The arc complex is a combinatorial moduli space, very similar to the curve complex. Using the techniques of Masur and Minsky, as well as new ideas, I'll sketch the theorem of the title. (Joint work with Howard

Masur.) If time permits, I'll discuss an application to the cusp shapes of fibred hyperbolic three-manifolds. (Joint work with David Futer.)

We are planning to have dinner at Chiang Mai afterwards.

If anyone would like to join us, please can you let me know today, as I plan to make a booking this evening. (Chiang Mai can be very busy even on a Monday.)

The Bakry Emery criterion asserts that a probability measure with a strictly positive generalized curvature satisfies a logarithmic Sobolev inequality, and by results of Bakry and Ledoux an isoperimetric inequality of Gaussian type. These results were complemented by a theorem of Wang: if the curvature is bounded from below by a negative number, then under an additional Gaussian integrability assumption, the log-Sobolev inequality is still valid.

The goal of this joint work with A. Kolesnikov is to provide an extension of Wang's theorem to other integrability assumptions. Our results also encompass a theorem of Bobkov on log-concave measures on normed spaces and allows us to deal with non-convex potentials when the convexity defect is balanced by integrability conditions. The arguments rely on optimal transportation and its connection to the entropy functional

Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented.

The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations.

Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.

Computational Mechanics (CM) has become

a scientific discipline in itself, being High Perfomance Computational

Mechanics (HPCM) a key sub-discipline. The effort for the most efficient use of

distributed memory machines provides a different perspective to CM scientists

relative to a wide range of topics, from the very physics of the problem to

solve to the numerical method used. Marenostrum supercomputer is the largest

facility in Europe and the 5th in the world (top500.org - Spring 2007). This

talk describes the research lines in the CASE Dpt. of the BSC applied to

Aerospace, Bio-mechanics, Geophysics or Environment, through the development of

Alya, the in-house HPCM code for complex coupled problems capable of running

efficiently in large distributed memory facilities.

Abstract: I show that the effective action of string compactifications has astructure that can naturally solve the supersymmetric flavour and CP problems. At leading order in the $g_s$ and $\alpha'$ expansions, the hidden sector factorises. The moduli space splits into two mirror parts that depend on K\"ahler and complex structure moduli. Holomorphy implies the flavour structure of the Yukawa couplings arises in only one part. In type IIA string theory flavour arises through the K\"ahler moduli sector and in type IIB flavour arises through the complex structure moduli sector. This factorisation gives a simple solution to the supersymmetric flavour and CP problems: flavour physics is generated in one sector while supersymmetry is broken in the mirror sector. This mechanism does not require the presence of gauge, gaugino or anomaly mediation and is explicitly realised by phenomenological models of IIB flux compactifications.

Some recent joint results with N. V. Krylov on the convergence of solutions of finite difference schemes are presented.

The finite difference schemes, considered in the talk correspond to discretizations (in the space variable) of second order parabolic and of second order elliptic (possibly degenerate) equations.

Space derivatives of the solutions to the finite difference schemes are estimated, and these estimates are applied to show that the convergence of finite difference approximations for equations in the whole space can be accelerated to any given rate. This result can be applied to stochastic PDEs, in particular to the Zakai equation of nonlinear filtering, when the signal and observation noises are independent.

H. Jerome Keisler suggested to associate to each classical structure M a family of "random" structures consisting of random variables with values in M . Viewing the random structures as structures in continuous logic one is able to prove preservation results of various "good" model theoretic properties e.g., stability and dependence, from the original structure to its randomisation. On the other hand, simplicity is not preserved by this construction. The work discussed is mostly due to H.

Jerome Keisler and myself (given enough time I might discuss some applications obtains in joint work with Alex Usvyatsov).

We discuss recent advances in the probabilistic analysis of financial risk and uncertainty, including risk measures and their dynamics, robust portfolio choice, and some asymptotic results involving large deviations

Consider a system of linear algebraic equations $Ax=b$ where $A$ is an $n$ by $n$ real matrix and $b$ a real vector of length $n$. Unlike in the linear iterative methods based on the idea of splitting of $A$, the Krylov subspace methods, which are used in computational kernels of various optimization techniques, look for some optimal approximate solution $x^n$ in the subspaces ${\cal K}_n (A, b) = \mbox{span} \{ b, Ab, \dots, A^{n-1}b\}, n = 1, 2, \dots$ (here we assume, with no loss of generality, $x^0 = 0$). As a consequence, though the problem $Ax = b$ is linear, Krylov subspace methods are not. Their convergence behaviour cannot be viewed as an (unimportant) initial transient stage followed by the subsequent convergence stage. Apart from very simple, and from the point of view of Krylov subspace methods uninteresting cases, it cannot be meaningfully characterized by an asymptotic rate of convergence. In Krylov subspace methods such as the conjugate gradient method (CG) or the generalized minimal residual method (GMRES), the optimality at each step over Krylov subspaces of increasing dimensionality makes any linearized description inadequate. CG applied to $Ax = b$ with a symmetric positive definite $A$ can be viewed as a method for numerical minimization the quadratic functional $1/2 (Ax, x) - (b,x)$. In order to reveal its nonlinear character, we consider CG a matrix formulation of the Gauss-Christoffel quadrature, and show that it essentially solves the classical Stieltjes moment problem. Moreover, though the CG behaviour is fully determined by the spectral decomposition of the problem, the relationship between convergence and spectral information is nothing but simple. We will explain several phenomena where an intuitive commonly used argumentation can lead to wrong conclusions, which can be found in the literature. We also show that rounding error analysis of CG brings fundamental understanding of seemingly unrelated problems in convergence analysis and in theory of the Gauss-Christoffel quadrature. In remaining time we demonstrate that in the unsymmetric case the spectral information is not generally sufficient for description of behaviour of Krylov subspace methods. In particular, given an arbitrary prescribed convergence history of GMRES and an arbitrary prescribed spectrum of the system matrix, there is always a system $Ax=b$ such that GMRES follows the prescribed convergence while $A$ has the prescribed spectrum.

The representation theory of symmetric groups starts with

the permutation modules. It turns out that the annihilator of a

permutation module can be described explicitly in terms of the

combinatorics of Murphy's cellular basis of the group algebra of the

symmetric group in question. In fact, we will show that the

annihilator is always a cell ideal. This is recent joint work with K.

Nyman.

The setting is a lognormal basis risk model. We study the optimal hedging of a claim on a non-traded asset using a correlated traded asset in a partial information framework, in which trading strategies are required to be adapted to the filtration generated by the asset prices. Assuming continuous observations, we take the assets' volatilites and the correlation as known, but the drift parameters are not known with certainty.

We assume the drifts are random variables with a Gaussian prior distribution, derived from data prior to the hedging timeframe. This distribution is updated via a Kalman-Bucy filter. The result is a basis risk model with random drift parameters.

Using exponsntial utility, the optimal hedging problem is attacked via the dual to the primal problem, leading to a representation for the hedging strategy in terms of derivatives of the indifference price. This representation contains additional terms reflecting uncertainty in the assets' drifts, compared with the classical full information model.

An analytic approximation for the indifference price and hedge is developed, for small positions in the claim, using elementary ideas of Malliavin calculus. This is used to simulate the hedging of the claim over many histories, and the terminal hedging error distribution is computed to determine if learning can counteract the effect of drift parameter uncertainty.