Wonderful Varieties, Hyperplane Arrangements, and Poisson Representation Theory

Geometric representation theory studies the algebraic structures formed by symmetries of geometric objects. It has connections with many areas of algebra and geometry, including algebraic combinatorics, algebraic geometry, mathematical physics, and symplectic geometry. The present project will explore this rich interplay by developing new representation-theoretic objects in algebraic and symplectic geometry. It will also provide research training opportunities for graduate students. In more detail, the project will focus on three interrelated problems. The first project is to introduce a new class of additive analogues of spherical varieties, constructed using degenerations motivated by the theory of Poisson-Lie groups. The second is to explore matroid Schubert varieties and their connections to toric geometry. The third is to develop new connections between Poisson geometry and symplectic representation theory by studying groupoids associated to symplectic resolutions. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.