Abstract.  Recently great progress has been achieved by Alessio Figalli et al. doing a fine analysis of the singular set in the obstacle problem. Much less is known about the behavior of the regular set close to singularities. The structure of this set may be very complicated, for example it may contain infinitely many connected components approaching the singularity, funnel-type singularities etc. The issue is related to the characterization of global solutions of the obstacle problem, that is solutions in the entire space. This characterization in turn is related to conjectures in potential analysis and an open problem since the 1980s. In this talk we will discuss progress in the characterization of global solutions and the application to singularities.

Bio.  Prof. Weiss obtained his PhD from University of Bonn in 1993 and was a research assistant there until 1995. He was a JSPS fellow in Chiba University in Japan before he held faculty position at University of Tokyo. He moved back to Germany in 2010, first as a professor at Heinrich Heine University in Dsseldorf and then at University of Duisburg-Essen. Currently, he is the Dean of the Faculty of Mathematics. Prof. Weiss has made significant contribution to the free boundary problems, in particular, the structure of singular set of the free boundary, regularity for water waves, etc. These results were published in many prestigious journals such as Acta Math., Invention. Math, CPAM, etc.