Geometry and Randomness: Counting, Partitions, Stochastics, Shape

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This project supports a research program that bridges mathematics to data science. The techniques developed in this project center on randomized constructions that illuminate geometry at large scale, and which can be useful in the study of networks. Example applications include epidemiology, social networks, and geospatial networks. Many ideas for these applications can be mined from the study of infinite groups; however, most of the technology developed in geometric group theory is asymptotic, and requires passing to an infinite limit. In practical applications, it is essential to have medium-scale techniques that are neither infinitesimal nor asymptotic.This research project will focus on 4 areas, (1) Counting and statistical geometry: methods for studying the precise geometry of geodesics open up applications like rational growth, statistical hyperbolicity, and macro Ricci curvature. (2) Nilpotent geometry: nilpotent groups such as the 3D Heisenberg group are of central interest in geometric analysis, Lie theory, and even the part of control theory that centers on sub-Riemannian geometry. In the 1980s, they also opened up a new vista on geometric group theory, through Gromov's remarkable polynomial growth theorem, which is still being explored for insights. New directions of inquiry explored here bring the geometric analysis together with the combinatorial group theory. (3) Teichmuller geometry and billiards: From the geometry of random triangles to rigidity theorems in symbolic dynamics, the proposal describes an active research program built from an interplay of flat and hyperbolic geometry. (4) Markov chains on graph partitions: How can we efficiently sample from the balanced, connected k-partitions of a graph? And how about preferentially sampling from partitions with a short boundary? This is a question of fundamental interest across many application domains, and it lends itself well to exploration on large datasets. The PI and his/her collaborators have implemented a recombination ("ReCom") Markov chain and research program for understanding its dynamics and geometry. This provides a rich application for ideas from ergodic theory, isoperimetry, and combinatorial models for moduli space.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.