## C Singular and geometric PDEs

Project group C consists of projects that explore PDEs coming from problems in pure and applied mathematics to obtain, represent, and analyze their solutions. The notion of sparsity runs through the entire project group: It is the lower dimensionality of the attracting submanifolds and leading-order approximations that makes it so appealing and potentially powerful to characterize and exploit their role. As a concrete and illustrative example, we point to the use of singular structures as moduli in the task of representing solutions.

Singular structures appear in various forms. At times one might look at a singular structure in the domain of the solutions, for instance singular loci of harmonic maps, shock waves in hyperbolic systems, or bubbling-off points in sequences of minimal surfaces, to name a few. At other times, a singular structure might be a structure in phase space, such as the critical set of a Lyapunov functional for a given evolution, which then turns out to consist of stationary points. Singular structures might also be present in the equation itself, as singularities of the coefficients of the PDE. Yet another notion of singular structure relevant to projects grouped here refers to methods: sharp interfaces are used both for computing (numerical methods) and for existence proofs (free boundary methods). Projects in Group C explore one or more of these notions of singular structure to tackle various theoretical and applied problems.

Group C contains the following projects.

C01 Singularity formation in dissipative harmonic flows (Christof Melcher, Arnold Reusken)

C02 Intrinsic convexity in the Mullins-Sekerka evolution (Michael Westdickenberg, Maria G. Westdickenberg)

C04 Mathematical analysis of domain decomposition methods for the efficient solution of continuum solvation models (Arnold Reusken, Benjamin Stamm)

C05 Numerical approximation of the Gross-Pitaevskii equation via vortex tracking (Christof Melcher, Benjamin Stamm)

C06 Elimination theory for deformed differential calculus (Daniel Robertz)