Analysis of Discrete Structures and Applications

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This project treats a broad range of problems in the analysis of discrete structures ranging from classical problems in analytic number theory concerning the gap distribution between primes and the shape of number fields, to the mixing behavior of random walks on groups. A central theme is the interaction between techniques from analytic number theory, probability, and combinatorics. The principal investigator recently made a breakthrough by using probabilistic techniques to solve a famous problem of Paul Erdos in combinatorial number theory, and part of the investigation will develop applications of this result. The problems studied have applications to cryptography and quantum computation. Funding is to be used to support the research of a group of undergraduate and graduate students and to support outreach efforts aimed at high school and undergraduate students interested in problem solving and beginning research.The project includes problems on automorphic forms, sieve methods, and the analysis of random walks on groups. The projects are split between topics in pure analytic number theory, such as the use of automorphic forms to study the lattice shape of the ring of integers of number fields, projects which use techniques common in analytic number theory, like the theory of exponential sums and oscillatory integrals, to study the asymptotic mixing of Markov chains and other limit problems in probability, and projects which use more purely probabilistic techniques. A significant part of the project extends the theory of zeta functions of prehomogeneous vector spaces in studying low degree number fields. A variety of modern analytic techniques are to be employed, including the use of automorphic forms on higher rank groups, partition functions from statistical mechanics, the distribution of points on nilmanifolds, and the theory of products of random matrices.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.