L1

The talk will discuss the variationnal problem on finite

dimensional normed spaces and Finsler manifolds.

We first review different notions of ellipticity (convexity) for

parametric integrands (densities) on normed spaces and compare them with

different minimality properties of affine subspaces. Special attention will

be given to Busemann and Holmes-Thompson k-area. If time permits, we will

then present the first variation formula on Finsler manifolds and exhibit a

class of minimal submanifolds.

Integral currents were introduced by H. Federer and W. H. Fleming in 1960

as a suitable generalization of surfaces in connection with the study of area

minimization problems in Euclidean space. L. Ambrosio and B. Kirchheim have

recently extended the theory of currents to arbitrary metric spaces. The new

theory provides a suitable framework to formulate and study area minimization

and isoperimetric problems in metric spaces.

The aim of the talk is to discuss such problems for Banach spaces and for

spaces with an upper curvature bound in the sense of Alexandrov. We present

some techniques which lead to isoperimetric inequalities, solutions to

Plateau's problem, and to other results such as the equivalence of flat and

weak convergence for integral currents.

Galois groups of p-class towers of number fields have long been a mystery,

but recent calculations have led to glimpses of a rich theory behind them,

involving Galois actions on trees, families of groups whose derived series

have finite index, families of deficiency zero p-groups approximated by

p-adic analytic groups, and so on.

We consider semilinear Sturm-Liouville and elliptic problems with jumping

nonlinearities. We show how `half-eigenvalues' can be used to describe the

solvability of such problems and consider the structure of the set of

half-eigenvalues. It will be seen that for Sturm-Liouville problems the

structure of this set can be considerably more complicated for periodic than

for separated boundary conditions, while for elliptic partial differential

operators only partial results are known about the structure in general.