Department of

Mathematics


Seminar Calendar
for MATRIX events the year of Sunday, July 3, 2022.

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Monday, February 21, 2022

5:00 pm in 343 Altgeld Hall,Monday, February 21, 2022

Curvature-dimension conditions for symmetric quantum Markov semigroups

Haonan Zhang (IST )

Abstract: The curvature-dimension condition consists of the lower Ricci curvature bound and upper dimension bound of the Riemannian manifold, which has a number of geometric consequences and is very helpful in proving many functional inequalities. The Bakry--Émery theory and Lott--Sturm--Villani theory allow to extend this notion beyond the Riemannian manifold setting and have seen great progress in the past decades. In this talk, I will first review several notions around lower Ricci curvature bounds in the noncommutative setting and present our work on gradient estimates. Then I will speak about two noncommutative versions of curvature-dimension conditions for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet--Myers theorem, and concavity of entropy power in the noncommutative setting. I will also give some examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group algebras, and depolarizing semigroups. Our work also gives new proofs and new results in the discrete setting. This is based on arXiv:2007.13506 and arXiv:2105.08303, joint work with Melchior Wirth (IST Austria).