Representation theory of affine Lie algebras and W-algebras and local geometric Langlands correspondence

Symmetry is a fundamental form of order in nature and, for this reason, the systematic study of symmetry plays a basic role in how scientists understand nature. Namely, for a given natural phenomenon, its symmetries constrain the possible laws it can obey, which in turn helps scientists determine the laws and their consequences. Over the past century, these methods have played a transformative role in many areas of science, notably chemistry, number theory, and physics. Representation theory is the mathematical study of symmetries and the constraints they dictate. The local geometric Langlands program is a vast web of conjectures in representation theory that has emerged over the past thirty-five years, with deep ties to, and applications in, both number theory and physics. During the period of support, the PI will tackle some of the basic problems in the local geometric Langlands program and develop some of its applications in nearby areas. In addition, the project provides research training opportunities for graduate students. In more detail, on one front, the PI and coauthors will apply new tools from local geometric Langlands to a set of problems originating from the kinematics of two-dimensional conformal field theories in physics, specifically the complete determination of the simple highest weight characters for affine Lie algebras and W-algebras in characteristic zero. On a second front, the PI and coauthors will verify the local geometric Langlands conjecture with restricted variation and pursue its applications in number theory. Finally, the PI and coauthors will develop the local geometric Langlands program in positive characteristic and investigate its implications in modular representation theory. 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.