Global cohomological approaches to L-functions

This award concerns Number theory, the analysis of equations involving integers and their solutions, which is one of the oldest branches of mathematics. As such, it has a long history of being driven by empirical observations; such important results as the law of quadratic reciprocity and the prime number theorem originated from numerical experiments. With an eye on the ongoing revolution in artificial intelligence, the PI will combine the latest theoretical developments in number theory with a big data approach to uncover hidden structures in the theory of L-functions. The PI will also promulgate this work through mentoring of PhD students, dissemination of advanced course materials, organization of workshops, and nonprofit governance, all with a view towards broadening participation. The PI will study Hasse-Weil L-functions associated to algebraic varieties over number fields through a mix of theoretical and computational techniques. On the theoretical side, the PI is investigating recent evidence pointing towards a global cohomological interpretation of these L-functions, using as a test case the families of motives parametrized by hypergeometric differential equations. On the computational side, the PI is developing streamlined algorithms to compute hypergeometric L-functions, partially informed by q-de Rham cohomology; this yields a rich data set for investigating Frobenius distributions, special values, murmurations, and other phenomena. 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.