Optimal mass transport (OMT) is a rich area of research with applications to numerous disciplines including econometrics, fluid dynamics, statistical physics, shape optimization, expert systems, and meteorology. The problem was originally formulated on the space of scalar probability densities. In the present talk, we describe a non-commutative generalization of OMT, to the space of Hermitian matrices with trace one, and to the space of matrix-valued probability densities. Our approach follows a fluid dynamics formulation of OMT, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. The non-commutative OMT introduces a natural distance metric for matrix-valued distributions. Our original motivation aimed at developing tools for multivariate time series modeling and matrix-valued power spectral analysis. However, the emergent theory turned out to have immediate applications in diffusion tensor imaging (DTI) where images are now tensor fields representing orientation and shape of brain white-matter. Thus, the framework we have developed allows us to compare, interpolate and fuse DTI images in a disciplined manner and, thereby, may lead to high resolution advances that in turn promise improved in vivo imaging of important brain structures. In addition, our formulation allows determining the gradient flow for the quantum entropy relative to the matricial Wasserstein metric induced by the non-commutative OMT. This may have implications to some key issues in quantum control and information theory.

*Yongxin Chen received his BSc in Mechanical Engineering from Shanghai Jiao Tong University, China, in 2011. He obtained his Ph.D. in Mechanical Engineering from University of Minnesota in 2016 under the supervision of Tryphon Georgiou, with a Ph.D. minor in Mathematics. He is now a postdoctoral researcher in the Department of Medical Physics at Memorial Sloan Kettering Cancer Center. He is interested in the application of mathematics in engineering, physics, economics and biology. His current research focuses on control theory, stochastic processes and optimal mass transport theory. *