Past

The seminar will address issues related to numerical simulation

of non-linear behaviour of solid materials to impact loading.

The kinematic and constitutive aspects of the transition from

continuum to discontinuum will be presented as utilised

within an explicit finite element development framework.

Material softening, mesh sensitivity and regularisation of

solutions will be discussed.

At the leading order, the low-energy effective field equations in string

theory admit solutions of the form of products of Minkowski spacetime and a

Ricci-flat Calabi-Yau space. The equations of motion receive corrections at

higher orders in \alpha', which imply that the Ricci-flat Calabi-Yau space is

modified. In an appropriate choice of scheme, the Calabi-Yau space remains

Kahler, but is no longer Ricci-flat. We discuss the nature of these

corrections at order {\alpha'}^3, and consider the deformations of all the

known cohomogeneity one non-compact Kahler metrics in six and eight

dimensions. We do this by deriving the first-order equations associated with

the modified Killing-spinor conditions, and we thereby obtain the modified

supersymmetric solutions. We also give a detailed discussion of the boundary

terms for the Euler complex in six and eight dimensions, and apply the

results to all the cohomogeneity one examples. Additional material will be

presented concerning the case of holonomy G_2.

In this talk, we consider a class of non-linear stochastic partial

differential equations. We represent its solutions as the weighted

empirical measures of interacting particle systems. As a consequence,

a simulation scheme for this class of SPDEs is proposed. There are two

sources of error in the scheme, one due to finite sampling of the

infinite collection of particles and the other due to the Euler scheme

used in the simulation of the individual particle motions. The error

bound, taking into account both sources of error, is derived. A

functional limit theorem is also derived. The results are applied to

nonlinear filtering problems.

This talk is based on joint research with Kurtz.

A little more than 100 years ago, Issai Schur published his pioneering PhD
thesis on the representations of the group of invertible complex n x n -
matrices. At the same time, Alfred Young introduced what later came to be
known as the Young tableau. The tableaux turned out to be an extremely useful
combinatorial tool (not only in representation theory). This talk will
explore a few of these appearances of the ubiquitous Young tableaux and also
discuss some more recent generalizations of the tableaux and the connection
with the geometry of the loop grassmannian.

Let $G$ be a complex semisimple algebraic group. We give an interpretation

of the path model of a representation in terms of the geometry of the affine

Grassmannian for $G$.

In this setting, the paths are replaced by LS--galleries in the affine

Coxeter complex associated to the Weyl group of $G$.

The connection with geometry is obtained as follows: consider a

Bott--Samelson desingularization of the closure of an orbit

$G(\bc[[t]]).\lam$ in the affine Grassmannian. The points of this variety can

be viewed as galleries of a fixed type in the affine Tits building associated

to $G$. The retraction of the Tits building onto the affine Coxeter complex

induces in this way, a stratification of the $G(\bc[[t]])$--orbit, indexed by

certain folded galleries in the Coxeter complex.

The connection with representation theory is given by the fact that the

closures of the strata associated to LS-galleries are the

Mirkovic-Vilonen--cycles, which form a basis of the representation $V(\lam)$

for the Langland's dual group $G^\vee$.

To numerical analysts and other applied mathematicians Jacobians and Hessians

are matrices, i.e. rectangular arrays of numbers or algebraic expressions.

Possibly taking account of their sparsity such arrays are frequently passed

into library routines for performing various computational tasks.

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A central goal of an activity called automatic differentiation has been the

accumulation of all nonzero entries from elementary partial derivatives

according to some variant of the chainrule. The elementary partials arise

in the user-supplied procedure for evaluating the underlying vector- or

scalar-valued function at a given argument.

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We observe here that in this process a certain kind of structure that we

call "Jacobian scarcity" might be lost. This loss will make the subsequent

calculation of Jacobian vector-products unnecessarily expensive.

Instead we advocate the representation of the Jacobian as a linear computational

graph of minimal complexity. Many theoretical and practical questions remain unresolved.

Condition numbers are a central concept in numerical analysis.

They provide a natural parameter for studying the behavior of

algorithms, as well as sensitivity and geometric properties of a problem.

The condition number of a problem instance is usually a measure

of the distance to the set of ill-posed instances. For instance, the

classical Eckart and Young identity characterizes the condition

number of a square matrix as the reciprocal of its relative distance

to singularity.

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We present concepts of conditioning for optimization problems and

for more general variational problems. We show that the Eckart and

Young identity has natural extension to much wider contexts. We also

discuss conditioning under the presence of block-structure, such as

that determined by a sparsity pattern. The latter has interesting

connections with the mu-number in robust control and with the sign-real

spectral radius.

I present a calculation of the moduli Kahler potential for M-theory

on a G_2 manifold in a large radius approximation. The result is used to

analyze moduli dynamics and moduli stabilization in the context of the

associated four-dimensional effective theory.

Take any region omega and let function u defined inside omega be the

distance from the boundary, u solves the iconal equation \lt|Du\rt|=1 with

boundary condition zero. Functional u is also conjectured (in some cases

proved) to be the "limiting minimiser" of various functionals that

arise models of blistering and micro magnetics. The precise formulation of

these problems involves the notion of gamma convergence. The Aviles Giga

functional is a natural "second order" generalisation of the Cahn

Hilliard model which was one of the early success of the theory of gamma

convergence. These problems turn out to be surprisingly rich with connections

to a number of areas of pdes. We will survey some of the more elementary

results, describe in detail of one main problems in field and state some

partial results.

We construct and study different surface measures on the space of

paths in a compact Riemannian manifold embedded into the Euclidean

space. The idea of the constructions is to force a Brownian particle

in the ambient space to stay in a small neighbourhood of the manifold

and then to pass to the limit. Finally, we compare these surface

measures with the Wiener measure on the space of paths in the

manifold.

I discuss gauge theories for commutative but non-associative algebras

related to the SO(2k+1) covariant finite dimensional fuzzy 2k-sphere

algebras. A consequence of non-associativity is that gauge fields and

gauge parameters have to be generalized to be functions of coordinates as

well as derivatives. The usual gauge fields depending on coordinates only

are recovered after a partial gauge fixing.The deformation parameter for

these commutative but non-associative algebras is a scalar of the rotation

group. This suggests interesting string-inspired algebraic deformations of

spacetime which preserve Lorentz-invariance.

The talk will be based on hepth/0310153