Past

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.