By discussing topics such as shape representations relaxation theory and optimal transport
trends and synergies of mathematical tools required for optimization of geometry and topology
of shapes are explored. Furthermore applications in science and engineering including
economics social sciences biology physics and image processing are covered. ContentsPart I
Geometric issues in PDE problems related to the infinity Laplace operator Solution of free
boundary problems in the presence of geometric uncertainties Distributed and boundary control
problems for the semidiscrete Cahn Hilliard Navier Stokes system with nonsmooth Ginzburg Landau
energies High-order topological expansions for Helmholtz problems in 2D On a new phase field
model for the approximation of interfacial energies of multiphase systems Optimization of
eigenvalues and eigenmodes by using the adjoint method Discrete varifolds and surface
approximation Part II WeakMonge Ampere solutions of the semi-discrete optimal transportation
problem Optimal transportation theory with repulsive costs Wardrop equilibria: long-term
variant degenerate anisotropic PDEs and numerical approximations On the Lagrangian branched
transport model and the equivalence with its Eulerian formulation On some nonlinear evolution
systems which are perturbations of Wasserstein gradient flows Pressureless Euler equations with
maximal density constraint: a time-splitting scheme Convergence of a fully discrete variational
scheme for a thin-film equatio Interpretation of finite volume discretization schemes for the
Fokker Planck equation as gradient flows for the discrete Wasserstein distance
