Partial differential equations are equations from calculus that are used to model phenomena in the physical world, for example, when studying the motion of interacting water waves. A fundamental question is to find the quantities which are conserved by the waves, and this search lies at the heart of the study of integrable systems. The classical integrable Hamiltonian systems, like the KdV equation and the non-linear Schrodinger system, have been playing a fundamental role in physical theories, like the theory of wave interactions, plasma physics, and fiber optics, to name a few. The main goal of the first part of this project is to build up a rigorous theory of integrable systems of Hamiltonian partial differential equations, based on algebraic structures that were inspired by physics. It is expected that the integrable systems discovered as a result of this project will play an important role in the study of various physical phenomena. The second part of this project has its origins in the last letter that Ramanujan wrote to G. H. Hardy in the early 1920's, where he wrote down 17 functions that had never been studied before (and which he named mock theta functions). The second direction of the project is to study the connections of mock theta functions to representation theory. It is expected that this will lead to new types of mock theta functions and to applications in number theory.
Some of the problems to be explored related to the first direction include the following: (a) develop a theory of non-local Hamiltonian structures and corresponding theory of integrable systems, based on the notion of a ''non-local'' Poisson vertex algebra; (b) compute the variational Poisson cohomology for not necessarily quasi-constant coefficient Poisson differential operators, developing the methods of a 2013 paper with De Sole; (c) develop further the theory of the generalized Drinfeld-Sokolov hierarchies and their Dirac reductions and establish their connection to the Kac-Wakimoto hierarchies. Some of the problems to be explored related to the second direction include the following: (a) compute character formulas for admissible representations of affine Lie superalgebras; (b) find explicit transformation formulas for the corresponding modified non-holomorphic theta-functions, in the spirit of Zwegers; (c) using quantum Hamiltonian reduction, compute modular transformation formulas and fusion rules of new representations of superconformal algebras.
