L1

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 $G^\vee$.

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.