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 colla