Universal centralizers, Morita abelianization, and wonderful models in Lie theory

Lie theory is a uniform framework for contextualizing symmetry in algebra and geometry. It is thereby central to mathematically rigorous realizations of symmetry reduction: the notion that the symmetries of a mathematical object should give rise to a quotient of that object. Successful realizations of this philosophy include geometric invariant theory in algebraic geometry, and Hamiltonian reduction in symplectic geometry. In this project the PI will harness the theory of centralizers, generalized Hamiltonian reduction, and Morita abelianization, with a view toward creating new avenues of research on algebraic symmetry reduction and adjacent subjects. The project will also provide research training opportunities for students. In more detail, this project has three broad objectives. The first is to develop a theory of universal abelianized centralizers for sheets in semisimple Lie algebras, in part based on generalized Hamiltonian reduction; one specific goal is to find new instances of mirror symmetry and computable examples of Coulomb branches. The second objective, the PI will use a principle of Morita abelianization to construct new topological quantum field theories adjacent to the long-standing Moore-Tachikawa conjecture; one impetus is to generalize and ultimately prove the Moore-Tachikawa conjecture. Finally, the third objective is to develop new, Lie-theoretic versions of de Concini-Procesi wonderful models. 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.