Optimality in analysis and geometry of probability measures

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The main goal of the proposed research is to establish versatile theories on fundamental questions that arise when probability measures are coupled with optimization. Such situations are ubiquitous in science and engineering as probability measures model randomness, datasets, and mass distributions. We view these through the lens of optimal transport theory, which considers the phenomena when mass distributions are matched in such a way to minimize a certain transport cost of moving mass from one location to another. This is a rapidly growing area with fundamental contributions to problems in geometry, probability, and partial differential equations, which is celebrated by two Fields medals in the last 10 years. My objectives in the proposal are: (1) To develop mathematical methods for analyzing the structure of optimal transport when there are additional probabilistic constraints which require the mass to be moved by a specified stochastic process, such as a martingale or Brownian motion. These additional constraints give more fruitful but nontrivial structures to optimal transport, and a successful theory will make new connections between probability, partial differential equations, and geometry. It will also have applications to pricing theory in finance as well as to understanding the motion of large numbers of particles in crowds or swarms. (2) To develop mathematical methods for understanding and utilizing the geometric average between probability measures, called the Wasserstein barycentre. I will focus on revealing analytical features of Wasserstein barycentres, and this will involve some of the outstanding open problems in the area, in particular, fundamental questions on the statistics of random shapes. Recent progress in computational methods is enabling optimal transport theory to effectively handle a variety of applications, to areas including fluid mechanics, economics, computer graphics, and even to machine learning. These developments enhance the theoretical investigations that I emphasize in this proposal, by providing an abundance of examples and related questions. On the other hand, theoretical progress in the proposed research program will contribute to making innovative methods for wider applications. For example, the remarkable recent development of the Wasserstein GAN in machine learning originated from the theory of Wasserstein distances on the space of probability measures. This proposal also contains applications to biological problems concerning plant roots, where understanding their shapes may support agricultural research into managing and improving food production.