CAREER: Cluster Algebras in Representation Theory, Geometry, and Physics

University Of Southern California, Los Angeles CA

Investigators

Abstract

This project pursues fundamental advances connecting a range of mathematical disciplines, specifically representation theory (the study of symmetry), algebraic geometry (the study of polynomial equations), and symplectic geometry (the geometry of classical mechanics). The projects are connected by their use cluster algebras to enrich these subjects and develop new connections among them. The research will contribute to the fundamental goal of developing the mathematical foundations of quantum field theory by providing new, mathematically rigorous definitions of certain notions studied in modern theoretical physics. This project provides research training opportunities for graduate students. In more detail, the PI will develop the theory of a class of coherent sheaves inaugurated in his prior work. Key conjectures to be proved include that categories of such Kp-sheaves are monoidal cluster categorifications in wide generality, that in the adjoint case they obey a coherent variant of the geometric Satake equivalence, and that in the context of quiver gauge theories they admit a type of Schur-Weyl duality with affine Khovanov-Lauda-Rouquier algebras. The PI will continue his work using cluster combinatorics to develop new ideas in symplectic geometry. Finally, the PI will establish new cases of a conjecture describing acyclic cluster algebras in terms of the representation theory of Kac-Moody Lie groups. 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.