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

Topological Optimization and Optimal Transport

In the Applied Sciences