17:00
17:00
14:15
The solutions to a class of non-linear stochastic partial
differential equations
Abstract
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.
16:30
Representation theory and combinatorics, from Young tableaux to the loop Grassmannian
Abstract
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.
14:30
14:15
From wetting to filling and back again: wedge covariance and non-local interfacial models
17:00
LS-galleries and MV-cycles
Abstract
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 $G^\vee$.
16:30
16:30
3D surface-tension-driven instabilities of liquid-lined elastic tubes - a model for pulmonary airway closure
16:15
Jacobians and Hessians are scarcely matrices!!
Abstract
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.
17:00
17:00
15:00
12:00
17:00
Geometry and physics of packing and unpacking, DNA to origami
(Alan Tayler Lecture)
16:30
14:15