Past

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.

We develop a methodology to optimally design a financial issue to hedge

non-tradable risk on financial markets.The modeling involves a minimization

of the risk borne by issuer given the constraint imposed by a buyer who

enters the transaction if and only if her risk level remains below a given

threshold. Both agents have also the opportunity to invest all their residual

wealth on financial markets but they do not have the same access to financial

investments. The problem may be reduced to a unique inf-convolution problem

involving some transformation of the initial risk measures.

We calculate an analytic value for the correlation coefficient between a geometric, or exponential, Brownian motion and its time-average, a novelty being our use of divided differences to elucidate formulae. This provides a simple approximation for the value of certain Asian options regarding them as exchange options. We also illustrate that the higher moments of the time-average can be expressed neatly as divided differences of the exponential function via the Hermite-Genocchi integral relation, as well as demonstrating that these expressions agree with those obtained by Oshanin and Yor when the drift term vanishes.

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 construct spaces of manifolds of various dimensions following

Vassiliev's approach to the theory of knots. These are infinite-dimensional

spaces with hypersurface, corresponding to manifolds with Morse singularities.

Connected components of the complement to this discriminant are homotopy

equivalent to the covering spaces of BDiff(M). These spaces appear to be a

natural base over which one can consider parametrised versions of Floer and

Seiberg-Witten theories.