Discrete Convexity, Curvature, and Implications

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Convexity in continuous domains has been known to be a powerful property and an important tool for more than a century. Thanks to several independent recent developments in the topics of optimal transport, partial differential equations, geometric and functional analysis, the interplay of convexity and curvature has further enriched the understanding of several topics in probability and analysis. While convexity has been developed through the important notion of submodularity in discrete settings, curvature remains elusive. In the past five years, the investigator and various collaborators have introduced discrete notions of the so-called Ricci curvature in Markov chains and graphs and derived several interesting consequences. However, much remains to be understood and developed in terms of convexity and curvature in discrete spaces. Identifying suitable applications in mathematics and computer science first will be an important guiding principle in developing any new theory. In discrete probability, the Ahlswede-Daykin Four Functions Theorem has had numerous applications (by way of the FKG inequality and other consequences) and similarly the Prekopa-Leindler inequality has been instrumental in deepening the understanding of the geometric and functional inequalities, while expanding the applications of the (equivalent) Brunn-Minkowski inequality to branches of probability, analysis and PDEs. A main technical objective of this project is to relate these two seminal results, using optimal transport-based couplings, and derive refined versions of each inequality. Other aspects include deriving higher-order Buser inequalities relating the k-th eigenvalue of the Laplacian to the higher-order (multipartite) Cheeger isoperimetric constant, under additional assumptions on volume growth and/or discrete curvature. The project also involves several specific problems for investigation with postdoctoral fellows and students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.