The research is divided into three directions: (A) geometric aspects of optimal transport, (B) applications to stochastic dynamics, and (C) optimal transport in non-commutative probability. Major results have been obtained in all of these directions.
A) One of the highlights is an investigation of the geodesics of discrete optimal transport metrics using discrete Hamilton–Jacobi equations. Another key contribution is a trajectorial approach to entropy dissipation and discrete gradient flows.
B) A main line of research deals with the large-scale behaviour of dynamical optimal transport on graphs. Several major results have been obtained, including convergence theorems for optimal transport on finite-volume meshes and homogenisation results for optimal transport on periodic graphs.
C) We developed a rich theory of dynamical optimal transport for density matrices with applications to entropy inequalities in non-commutative probability and variational structures for quantum Markov semigroups.
Other highlights include a study of evolutionary Gamma-convergence using discrete transport metrics, a gradient flow formulations of dissipative PDE on metric graphs, and a treatment of discrete and continuous models for chemical reaction networks using optimal transport methods.
The primary impact of the project is in mathematics, but several of the obtained results may have future applications in other sciences, including physics, chemistry, biology, and computer science.
