Research in Geometric Representation Theory
The mathematical field of representation theory studies algebraic models of symmetry. Beginning in the late 1970s, the geometric construction of algebraic models of symmetry resulted in a fundamentally new subject known as geometric representation theory. The leading role in this developing area has been played by the algebraic theory of differential equations, as mediated by the structure of D-modules.
Much of Professor Nevins’ recent work has aimed at expanding the toolkit of D-modules to more general contexts, with applications to a broad new range of problems in geometric representation theory. Jointly with Professor Kevin McGerty (Mathematical Institute, Oxford University), he extended the Beilinson-Bernstein localization theorem for D-modules from its original context to a much more general setting. They established precise relationships between a piece of the category of (twisted) G-equivariant D-modules on a variety X with an action of a reductive group G and modules over a quantum Hamiltonian reduction of X. During his Center appointment, Professor Nevins plans to continue this collaboration and establish a general structure theory for the entire category of G-equivariant D-modules, and then apply it.
Also during his Center appointment, Professor Nevins plans to continue a wider collaboration to prove a now-standard expectation in symplectic topology, i.e., that the Fukaya categories of certain real symplectic manifolds should be realized by categories of deformation-quantization modules. The next steps in this area are to develop the deformation theory of cell categories and to construct a period map. Accomplishing these steps will provide a concrete characterization of Fukaya categories.
The two projects are closely related: each makes manifest a structure indicated by Morse theory, and, thus, together they express a satisfying underlying unity to emergent phenomena in the study of D-modules.