## Program

 10:00 - 10:05 Welcoming address Prof. Martin Egelhaaf (Vice-rector for research, young researchers and equal opportunities, Bielefeld University) 10:05 - 10:15 Welcoming address Tian Qiru (Vice Consul General, Chinese Consulate General in Düsseldorf) 10:15 - 10:45 Short presentation of the CRC Prof. Michael Röckner (Spokesperson of the CRC) 11:00 - 11:45 Inaugural talk "Taming the uncertainty" Prof. Halil Mete Soner (ETH Zürich) All applications of quantitative methods are subject to incomplete and uncertain inputs. Despite these ambiguities one has to complete the tasks at hand. In decision theory, this is a classical paradigm of decisions under uncertainty and has clear applications in all of sciences. Indeed modern probability theory has an exciting range of applications ranging from uncertainty quantification in numerical analysis, policy decisions, turbulence to risk management. In this talk, I will outline and discuss the basic notions and approaches related to uncertainty, risk and preferences. 12:00 - 14:00 Reception/Lunch 14:00 - 14:45 Talk related to project area A "Nonlocal minimal surfaces" Prof. Enrico Valdinoci (University of Milan) Nonlocal minimal surfaces are objects recently introduced by Caffarelli, Roquejoffre and Savin. They are minimizers of a functional depending on a fractional parameter, say sigma in (0,1). When sigma approaches 1, this functional recovers the usual notion of perimeter (and, in some sense, so do the corresponding nonlocal minimal surfaces), but when sigma approaches 0 the energy contributions coming from infinity become predominant and the energy is related to a volume-type functional. The regularity of the nonlocal minimal surfaces seems to be a rather challenging topic: till now, a complete regularity theory is available only either in the plane or up to dimension 7 when sigma is sufficiently close to 1. The boundary behavior of the minimal surfaces shows also very interesting features, such as stickiness phenomena that are special for the nonlocal setting and have no classical counterpart. Critical points of the fractional perimeter satisfy a suitable equation of nonlocal mean curvature type, and objects with constant nonlocal mean curvature have been recently investigated in terms of existence and classification. The nonlocal mean curvature also produces an interesting geometric flow, which is also related to problems of cellular automata. We plan to discuss some recent developments related to these topics and present some open problems. 15:00 - 15:45 Talk related to project area B "On the 2D Euler equations with random initial conditions" Prof. Franco Flandoli (University of Pisa) In recent years, PDEs with random initial conditions attracted much attention for the possibility to extend the range of deterministic results, usually in the direction of less smooth initial conditions; the most extensive successes have been made for dispersive equations, like nonlinear Schrodinger and wave equations and KDV equations. However, a result of this nature was know since an older work of S. Albeverio and A.-B. Cruzeiro devoted to the 2D Euler equations. Their work is revisited, following the approach of weak vorticity formulation. A solution is constructed as a limit of random point vortices. This allows to prove that it is also limit of $L^\infty$-vorticity solutions. The result for Gaussian initial conditions is generalized to initial measures that have a continuous bounded density with respect to the original Gaussian measure and preliminary results on the continuity equation in Hilbert spaces associated to 2D Euler equations are given. 16:00 - 16:45 Talk related to project area C "Financial market equilibria under Knightian uncertainty" Prof. Frank Riedel (Bielefeld University) In diffusion models, few suitably chosen financial securities allow to complete the market. As a consequence, the efficient allocations of static Arrow-Debreu equilibria can be attained in Radner equilibria by dynamic trading. We show that this celebrated result generically fails if there is Knightian uncertainty about volatility. A Radner equilibrium with the same efficient allocation as in an Arrow-Debreu equilibrium exists if and only if the discounted net trades of the equilibrium allocation display no ambiguity in the mean. This property is violated generically in endowments, and thus Arrow-Debreu equilibrium allocations are generically unattainable by dynamically trading few long-lived assets.

## Contact

Dr. Claudia Köhler
+49 (0)521 106-4767
ckoehler@math.uni-bielefeld.de