Past

Einstein bequeathed many things to differential geometry — a
global viewpoint and the urge to find new structures beyond Riemannian
geometry in particular. Nevertheless, his gravitational equations and
the role of the Ricci tensor remain the ones most closely associated
with his name and the subject of much current research. In the
Riemannian context they make contact in specific instances with a wide
range of mathematics both analytical and geometrical. The talk will
attempt to show how diverse parts of mathematics, past and present,
have contributed to solving the Einstein equations.

The satisfiability (SAT) problem is a central problem in mathematical

logic, computing theory, and artificial intelligence. We consider

instances of SAT specified by a set of boolean variables and a

propositional formula in conjunctive normal form. Given such an instance,

the SAT problem asks whether there is a truth assignment to the variables

such that the formula is satisfied. It is well known that SAT is in

general NP-complete, although several important special cases can be

solved in polynomial time. Extending the work of de Klerk, Warners and van

Maaren, we present new linearly sized semidefinite programming (SDP)

relaxations arising from a recently introduced paradigm of higher

semidefinite liftings for discrete optimization problems. These

relaxations yield truth assignments satisfying the SAT instance if a

feasible matrix of sufficiently low rank is computed. The sufficient rank

values differ between relaxations and can be viewed as a measure of the

relative strength of each relaxation. The SDP relaxations also have the

ability to prove that a given SAT formula is unsatisfiable. Computational

results on hard instances from the SAT Competition 2003 show that the SDP

approach has the potential to complement existing techniques for SAT.

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.