Most of my recent research has revolved around algebraic structures arising in the description of the dynamics of open quantum systems. This theme can be traced back at least to the work of Lindblad, who showed that derivations occur naturally in the master equation of Markovian open quantum systems. The propagator of this Lindblad master equation is called a quantum Markov semigroup. The interplay between quantum Markov semigroups, derivations and bimodules becomes particularly rich if one moves beyond the realm of of matrix algebras to general von Neumann algebras. Far from being restricted to applications in mathematical physics, quantum Markov semigroups in this general setting touch upon a variety of purely mathematical fields, from von Neumann algebra theory and free probability to unitary representations and group cocycles. Another algebraic structure that has begun to interest me recently is the connection between Lie algebra representations and quantum Markov semigroups. One fascinating aspect of this circle of ideas is that they yield not only qualitative structural insights, but have also been instrumental in the quantitative analysis of the decay to equilibrium of open quantum systems in recent years.
The theory of optimal transport goes back to the work of Gaspard Monge in the 18th century, but it has only recently come to full bloom as field of mathematical interest with rich connections between various disciplines of pure and applied mathematics. The optimal transport problem is a classical optimization problem, seeking to minimize the cost to transport goods, modelled mathematically by probability measures, from a given source to a given destination. What makes this optimization problem stand out is the particularly rich structure exemplified by the Benamou–Brenier formulation, which draws connections to hydrodynamics, and Brenier's theorem, which asserts that optimizers are gradients of convex functions. Of the mathematical applications that have surfaced in recent decades, the connection to curvature and entropy is of particular interest to me.
One recent trend in optimal transport is to study the transportation problem on spaces that are discrete or ‘noncommutative’. In these settings, many of the equivalences known from the continuous world break down and a whole zoo of transport distances arises, each suitable for different applications. As it turns out, many new features of quantum optimal transport are anticipated in discrete optimal transport, and transport distances in either setting have become a useful tool in the analysis of the long-time behavior of Markov chains and quantum Markov semigroups.
In addition to Laplace–Beltrami operators on Riemannian manifolds, graph Laplacians are the prime examples of generators of Markov semigroups/Dirichlet forms.