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.

Contents Part 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

Weak Monge-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