Multiple Dirichlet Series and Probabilistic Methods in Number Theory

This project studies two central topics in number theory. One is the study of Galois groups, which describe the hidden symmetries of solutions of equations. The other is the study of L-functions, which are powerful tools to count particular types of numbers, including prime numbers. The Principal Investigator will work on these with two different approaches. One approach is the construction of probabilistic models. Probabilistic models allow mathematicians to develop models that predict the properties a mathematical object will likely have, even when it is not possible to know with certainty what properties it has. Another approach is by multiple Dirichlet series, which mathematicians create by modifying a number-theoretic problem until its solution is as symmetrical as possible, and then use these symmetries to find a solution to the original problem. The Principal Investigator will develop new probabilistic models and new multiple Dirichlet series, and use them to make progress on fundamental number-theoretic problems. Furthermore, the project also develops connections with other areas of mathematics - the Principal Investigator's work on probabilistic models will lead to new results in probability theory, while the work on multiple Dirichlet series will demonstrate connections between these series, topology, and quantum algebra, leading to new results in those areas. The Principal Investigator will also train future mathematicians. The Principal Investigator and collaborators will work in several interrelated directions. One will construct new multiple Dirichlet series, with clear potential applications to moments of L-functions, by a novel unifying perspective on multiple Dirichlet series. The second will construct probabilistic models for the Galois group of the maximal unramified extension of a random number field, and then probabilistic models for the entire absolute Galois group. The third will give probabilistic models for statistics of L-functions that fit them more closely than the previously considered probabilistic models. The probabilistic perspective will also lead to new evaluations of moments of L-functions, by better understanding the main term in problems where separating the main term from the error term is difficult. The fourth project will construct new multiple Dirichlet series relevant to L-functions of fields with nonabelian Galois groups. 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.