Timetable Tuesday, 16:30-18:00, Freihaus of TU Wien, green area, 6th floor, seminar room 107. Tu, 24.07.2007 Andreas H. Hamel University Halle-Wittenberg, on leave ORFE, Princeton University "A duality theory for set-valued convex functions with applications to set-valued convex risk measures" Abstract: Duality for extended real-valued convex functions is a well-studied, even classical subject based on works of Fenchel, Moreau, Rockafellar, among many others. A corresponding satisfying theory for functions mapping into the power set of a partially ordered locally convex space is still missing. Such a theory seems to be very desirable since it has already been observed e.g. by Luc in 1989 that the dual of a convex vector optimization problem 'is set-valued in nature'. Moreover, the concept of convex set-valued risk measures has been defined recently in financial mathematics which asks for a corresponding dual representation theory. We shall present a duality concept that is based on a new notion of affine minorants for set-valued functions and show that almost every concept (e.g. properness, sublinearity, conjugates, inf-convolution) and result (e.g. biconjugation and Fenchel-Rockafellar duality theorems) known in the scalar convex analysis can be established within the new set-valued framework. A special feature of the methodology is that proofs do not rely on the corresponding scalar theory - as in almost every previous duality theory for vector optimization problems. Finally, we shall show the theory at work when applied to set-valued convex risk measures in order to give dual representation results.