Duisburg-Essen University

 An "open de Rham space" refers to a moduli space of meromorphic connections on the projective line with underlying trivial bundle.  In the case where the connections have simple poles, it is well-known that these spaces exhibit hyperkähler metrics and can be realized as quiver varieties.  This story can in fact be extended to the case of higher order poles, at least in the "untwisted" case.  The "twisted" spaces, introduced by Bremer and Sage, refer to those which have normal forms diagonalizable only after passing to a ramified cover.  These spaces often arise as quotients by unipotent groups and in some low-dimensional examples one finds some well-known hyperkähler manifolds, such as the moduli of magnetic monopoles.  This is a report on ongoing work with Tamás Hausel and Dimitri Wyss.

In this talk we consider optimal stopping problems under a class of coherent risk measures which includes such well known risk measures as weighted AV@R or absolute semi-deviation risk measures. As a matter of fact, the dynamic versions of these risk measures do not have the so-called time-consistency property necessary for the dynamic programming approach. So the standard approaches are not applicable to optimal stopping problems under coherent risk measures. In this paper, we prove a novel representation, which relates the solution of an optimal stopping problem under a coherent risk measure to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming-based approaches possible. In particular, we derive the analogue of the dual representation of Rogers and Haugh and Kogan. Several numerical examples showing the usefulness of the new representation in applications are presented as well.