CBMS Conference: Ramanujan's Partition Congruences, Mock Theta Functions, and Beyond

This award provides funding for the research conference 'Ramanujan's ranks, mock theta functions, and beyond' organized as part of the Conference Board of the Mathematical Sciences (CBMS) series. The event will take place at University of Texas Rio Grande Valley during May 16-20, 2022. The main goal of this event is to provide an introduction to various topics in mathematical number theory to a broader audience. The conference revolves around ten lectures to be given by Professor Frank Garvan of the University of Florida Gainesville. The discoveries of the mathematician Srinivasa Ramanujan have found applications in disciplines from supercomputing to chemistry, and to physics and the study of black holes. Recent work has been motivated by data obtained from the innovative use of both mathematical theory and the use of computer algebra systems. In accordance with the goals of the NSF-CBMS program, the main purpose of this conference is to stimulate further research by bringing together leading mathematicians and junior researchers interested in q-series, modular forms, and partitions. We expect that many of the scientists attending this CBMS conference to come from all over the Gulf Coast States and Mexico. Such a conference will have immeasurable impact on the new doctoral program in mathematics that is planned at the University of Texas Rio Grande Valley.

Professor Garvan will lecture on the topics of Ramanujan's partition congruences and mock theta functions. The viewpoint of these lectures and topics goes back to the work of the physicist Freeman Dyson. In 1944 Dyson asked for deeper combinatorial reasons for Ramanujan's famous partition congruences. Dyson's question was completed by Garvan in 1987. In his article, A Walk through Ramanujan's Garden, Dyson foresaw that 'Garvan's beautiful discovery is not an end but a beginning.' Dyson proffered the challenge to build a grand synthesis of partition congruences and mock theta functions to develop the group theoretic structure of mock theta functions. This synthesis would be analogous to Hecke's development of modular forms from Jacobi's theory of theta functions and would lead to applications in physics. Professor Garvan will lecture on how modern mathematics has taken up both of Dyson's challenges. This will involve combinatorial techniques and function theoretic methods including the classical theory of basic hypergeometric functions as well the modern theory of weak harmonic Maass forms and p-adic modular forms. Further information about the conference will be available at the website: https://www.cbmsweb.org

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.