Research Interests

Algebraic Structures of Quantum Dynamics

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.

Quantum and Discrete Optimal Transport
Analysis on Graphs

Peer-reviewed Articles

  1. (with R. Kumar R) Operator-Valued Twisted Araki-Woods Algebras
    Communications in Mathematical Physics, accepted.
    arXiv:2406.06179
  2. Modular Completely Dirichlet forms as Squares of Derivations
    International Mathematics Research Notices. IMRN, 2024.
    https://doi.org/10.1093/imrn/rnae092, arXiv:2307.04502
  3. Christensen-Evans theorem and extensions of GNS-symmetric quantum Markov semigroups,
    Journal of Functional Analysis, 2024.
    https://doi.org/10.1016/j.jfa.2024.110475, arXiv:2203.00341
  4. (with C. Rouzé, H. Zhang) Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions
    Communications in Mathematical Physics, 2024.
    https://doi.org/10.1007/s00220-024-04981-0, arXiv:2209.07279
  5. (with M. Vernooij) Derivations and KMS-Symmetric Quantum Markov Semigroups
    Communications in Mathematical Physics, 2023.
    https://doi.org/10.1007/s00220-023-04795-6, arXiv:2303.15949
  6. (with B. Hua, M. Keller, M. Schwarz) Sobolev-Type Inequalities and Eigenvalue Growth on Graphs with Finite Measure
    Proceedings of the American Mathematical Society, 2023.
    https://doi.org/10.1090/proc/14361, arXiv:1804.08353
  7. (with L. Dello Schiavo) Ergodic Decompositions of Dirichlet Forms under Order Isomorphisms
    Journal of Evolution Equations, 2023.
    https://doi.org/10.1007/s00028-022-00859-7, arXiv:2109.00615
  8. (with H. Zhang) Curvature-dimension conditions for symmetric quantum Markov semigroups
    Annales Henri Poincaré, 2022.
    https://doi.org/10.1007/s00023-022-01220-x, arXiv:2105.08303
  9. Stability of Kac regularity under domination of quadratic forms
    Advances in Operator Theory, 2022.
    https://doi.org/10.1007/s43036-022-00199-w, arXiv:1709.04164
  10. A Dual Formula for the Noncommutative Transport Distance
    Journal of Statistical Physics, 2022.
    https://doi.org/10.1007/s10955-022-02911-9, arXiv:2104.11923
  11. (with H. Zhang) Complete Gradient Estimates of Quantum Markov Semigroups
    Communications in Mathematical Physics, 2021.
    https://doi.org/10.1007/s00220-021-04199-4, arXiv:2007.13506
  12. (with D. Lenz, T. Weinmann) Self-Adjoint Extensions of Bipartite Hamiltonians
    Proceedings of the Edinburgh Mathematical Society, 2021.
    https://doi.org/10.1017/S0013091521000080, arXiv:1912.03670
  13. (with D. Lenz, M. Schmidt) Uniqueness of form extensions and domination of semigroups
    Journal of Functional Analysis, 2021.
    https://doi.org/10.1016/j.jfa.2020.108848, arXiv:1608.06798
  14. (with C. Richter) Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles
    Discrete and Computational Geometry, 2020.
    https://doi.org/10.1007/s00454-019-00061-6, arXiv:1711.08903
  15. (with D. Lenz, M. Schmidt) Domination of quadratic forms
    Mathematische Zeitschrift, 2020.
    https://doi.org/10.1007/s00209-019-02440-4, arXiv:1711.07225
  16. (with M. Erbar, J. Maas) On the geometry of geodesics in discrete optimal transport
    Calculus of Variations and Partial Differential Equations, 2019.
    https://doi.org/10.1007/s00526-018-1456-1, arXiv:1805.06040
  17. (with M. Keller, D. Lenz, M. Schmidt) Diffusion determines the recurrent graph
    Advances in Mathematics, 2015.
    https://doi.org/10.1016/j.aim.2014.10.003, arXiv:1405.3256

Preprints

  1. (with F. Münch, H. Zhang) Intertwining Curvature Bounds for Graphs and Quantum Markov Semigroups, arXiv:2401.05179
  2. (with M. Keller, D. Lenz, M. Schmidt and M. Schwarz) Boundary representations of intermediate forms between a regular Dirichlet form and its active main part, arXiv:2301.01035.
  3. The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups, arXiv:2207.09247 .
  4. A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy, arXiv:1808.05419.
  5. (with D. Lenz, M. Schmidt) Geometric properties of Dirichlet forms under order isomorphisms, arXiv:1801.08326.