EXC 2044 - T06: Singularities and PDEs

Basic data for this project

Description

In this topic, our goal is to utilise and further develop the theory of non-linear PDEs to understand singular phenomena arising in geometry and in the description of the physical world. Particular emphasis is put on the interplay of geometry and partial differential equations and also on the connection with theoretical physics. The concrete research projects have been organised in three units and ‒ to give just a few examples ‒ range from problems originating in geometric analysis such as understanding the type of singularities developing along a sequence of four-dimensional Einstein manifolds, to problems in evolutionary PDEs, such as the Einstein equations of general relativity or the Euler equations of fluid mechanics, where one would like to understand the formation and dynamics (in time) of singularities. The third unit is concerned with regularity estimates, which play a crucial role in the recent breakthroughs in theory of stochastic PDEs allowing the formulation of PDEs in extremely singular settings.