I am principal investigator of the project Gradient flow techniques for quantum Markov semigroups funded by the Austrian Science Fund (FWF) under the Esprit Programme, with Jan Maas as my mentor. Below you can find the project description.
Duration: July 1, 2022 – June 30, 2025
Project ID: ESP 156 ESPRIT-Programm
Total Funding Sum: 294.015,98 €
Gradient flow techniques for quantum Markov semigroups
Classical and quantum Markov semigroups feature in a variety of mathematical disciplines from probability over geometry to analysis and mathematical physics. A prominent example is the heat semigroup on a Riemannian manifold, and deep insights into the structure and long-term behavior of Markov semigroups can be gained by viewing them as abstract heat semigroups and applying methods inspired by Riemannian geometry.
One particularly powerful geometric notion in this context are lower Ricci curvature bounds and their generalizations to abstract diffusion semigroups by Bakry—Émery and Lott—Sturm—Villani. In recent years, these theories have been extended to cover also Markov semigroups with non-local generators and semigroups acting on quantum systems, with fruitful applications to functional inequalities for these Markov semigroups.
I will develop a research program that allows us to significantly enlarge the scope of these ideas in noncommutative analysis. Firstly, I will construct a noncommutative analog of the Wasserstein metric for a given GNS-symmetric quantum Markov semigroup, which allows to view the semigroup as gradient flow of the relative entropy, and study the resulting notion of noncommutative lower Ricci curvature bounds. Secondly, I will develop new gradient flow techniques that apply not only to the modified logarithmic Sobolev inequality, but other noncommutative functional inequalities such as exponential decay estimates for the sandwiched Rényi entropies and the logarithmic Sobolev inequality.
Extending recent work by myself and the mentor, the construction in the first part will rely on a differential calculus for GNS-symmetric quantum Markov semigroups on infinite-dimensional von Neumann algebras newly to be developed. The second part will use a modified gradient estimate for sandwiched Rényi entropies and gradient flow methods for nonlinear noncommutative flows.
This project will expand the scope of quantum optimal transport methods significantly to cover numerous new examples in noncommutative geometry, mathematical physics, the theory of quantum groups and other fields, and will thus make a fundamental contribution to the interplay between noncommutative notions of Ricci curvature bounds and functional inequalities for quantum Markov semigroups.