Past

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.

We provide both a diagrammatic and logical system to reason about

quantum phenomena. Essential features are entanglement, the flow of

information from the quantum systems into the classical measurement

contexts, and back---these flows are crucial for several quantum informatic

scheme's such as quantum teleportation---, and mutually unbiassed

observables---e.g. position and momentum. The formal structures we use are

kin to those of topological quantum field theories---e.g. monoidal

categories, compact closure, Frobenius objects, coalgebras. We show that

our diagrammatic/logical language is universal. Informal

appetisers can be found in:

* Introducing Categories to the Practicing Physicist

http://web.comlab.ox.ac.uk/oucl/work/bob.coecke/Cats.pdf

* Kindergarten Quantum Mechanics

http://arxiv.org/abs/quant-ph/0510032

Professor Qiang Du will go over some work on modelling interface/microstructures with curvature dependent energies and also the effect of elasticity on critical nuclei morphology.

This is a joint work with Bruno Schapira, and it is a work in progress.

We study recurrence and transience properties of some edge reinforced random walks on the integers: the probability to go from $x$ to $x+1$ at time $n$ is equal to $f(\alpha_n^x)$ where $\alpha_n^x=\frac{1+\sum_{k=1}^n 1_{(X_{k-1},X_k)=(x,x+1)}}{2+\sum_{k=1}^n 1_{X_{k-1}=x}}$. Depending on the shape of $f$, we give some sufficient criteria for recurrence or transience of these walks

The complement of a knot or link is a 3-manifold which admits a geometric

structure. However, given a diagram of a knot or link, it seems to be a

difficult problem to determine geometric information about the link

complement. The volume is one piece of geometric information. For large

classes of knots and links with complement admitting a hyperbolic

structure, we show the volume of the link complement is bounded by the

number of twist regions of a diagram. We prove this result for a large

collection of knots and links using a theorem that estimates the change in

volume under Dehn filling. This is joint work with Effie Kalfagianni and

David Futer

Born in France around 1780, the optimal transport problem has known a scientific explosion in the past two decades, in relation with dynamical systems and partial differential equations. Recently it has found unexpected applications in Riemannian geometry, in particular the encoding of Ricci curvature bounds

Abstract: The moduli space of Calabi-Yau manifolds have a natural geometrical structure that has come to be known as special geometry. This geometry will be reviewed in the complex context and it will be shown that much of the structure persists for p-adic Calabi-Yau manifolds.

We show that the profinite completion (a universal algebraic

construction) and the MacNeille completion (an order-theoretic

construction) of a modal algebra $A$ coincide, precisely when the congruences of finite index of $A$ correspond to principal order filters. Examples of such modal algebras are the free K4-algebra and the free PDL-algebra on finitely many generators.

I will associate, to every pair of smooth transversal

Lagrangian spheres in a symplectic manifold having vanishing first Chern

class, its Floer cohomology groups. Hamiltonian isotopic spheres give

rise to isomorphic groups. In order to define these Floer cohomology

groups, I will make a key use of symplectic field theory.

Radial basis function (RBF) methods have been successfully used to approximate functions in multidimensional complex domains and are increasingly being used in the numerical solution of partial differential equations. These methods are often called meshfree numerical schemes since, in some cases, they are implemented without an underlying grid or mesh.

The focus of this talk is on the class of RBFs that allow exponential convergence for smooth problems. We will explore the dependence of accuracy and stability on node locations of RBF interpolants. Because Gaussian RBFs with equally spaced centers are related to polynomials through a change of variable, a number of precise conclusions about convergence rates based on the smoothness of the target function will be presented. Collocation methods for PDEs will also be considered.

Category theory is used to study structures in various branches of

mathematics, and higher-dimensional category theory is being developed to

study higher-dimensional versions of those structures. Examples include

higher homotopy theory, higher stacks and gerbes, extended TQFTs,

concurrency, type theory, and higher-dimensional representation theory. In

this talk we will present two general methods for "categorifying" things,

that is, for adding extra dimensions: enrichment and internalisation. We

will show how these have been applied to the definition and study of

2-vector spaces, with 2-representation theory in mind. This talk will be

introductory; in particular it should not be necessary to be familiar with

any category theory other than the basic idea of categories and functors.

Phylogenetics is the reconstruction and analysis of 'evolutionary'

trees and graphs in biology (and related areas of classification, such as linguistics). Discrete mathematics plays an important role in the underlying theory. We will describe some of the ways in which concepts from combinatorics (e.g. poset theory, greedoids, cyclic permutations, Menger's theorem, closure operators, chordal graphs) play a central role. As well as providing an overview, we also describe some recent and new results, and outline some open problems.

We provide a realization of Cherednik's double affine Hecke

algebras (for GL_n) as a convolution algebra of functions on moduli spaces

of coherent sheaves on an elliptic curve. As an application we give a

geometric construction of Macdonald polynomials as (traces of) certain

natural perverse sheaves on these moduli spaces. We will discuss the

possible extensions to higher (or lower !) genus curves and the relation

to the Hitchin nilpotent variety. This is (partly) based on joint work

with I. Burban and E. Vasserot.

In this talk I will look at a definition of the energy-momentum for the dynamical horizon of a black hole. The talk will begin by examining the role of a special class of observers at null infinity determined by Bramson's concept of frame alignment. It is shown how this is given in terms of asymptotically constant spinor fields and how this framework may be used together with the Nester-Witten two form to give a definition of the Bondi mass at null infinity.

After reviewing Ashtekar's concept of an isolated horizon we will look at the propagation of spinor fields and show how to introduce spinor fields for the horizon which play the role of the asymptotically constant spinor fields at null infinity, giving a concept of alignment of frames on the horizon. It turns out that the equations satisfied by these spinor fields give precisely the Dougan-Mason holomorphic condition on the cross sections of the horizon, together with a simple propagation equation along the generators. When combined with the Nester-Witten 2-form these equations give a quasi-local definition of the mass and momentum of the black hole, as well as a formula for the flux across the horizon. These ideas are then generalised to the case of a dynamical horizon and the results compared to those obtained by Ashtekar as well as to the known answers for a number of exact solutions.

In the Examination Schools