Enumeration Problems in Algebraic Geometry and Representation Theory

Overview: The PI proposes to study two classes of enumeration problems from algebraic geometry and representation theory, counting conformal blocks, and counting branching multiplicities of a map of reductive groups. These programs are addressed by connecting them to combinatorial features of interesting algebraic varieties: moduli spaces of algebraic and topological principal bundles, and branching varieties. The research program utilizes techniques from two emerging fields from algebraic geometry: Berkovich analytification and Newton-Okounkov bodies, both of which transform algebraic and representation theoretic information into polyhedral geometry. These techniques are used to conjecture a new description of conformal blocks. Along the way, solutions to the enumeration problem are linked to interesting open problems in the algebraic, symplectic, and tropical geometry of moduli spaces of principal bundles. In particular, the PI proposes to relate the topology and symplectic geometry of the moduli spaces being considered to polytopes used in the solution to the enumeration problem. Intellectual Merit : The problem addressed in this proposal has important applications to representation theory and mathematical physics. The spaces under consideration are central objects in algebraic geometry, therefore the computations of this program are of interest to geometers and representation theorists. The wealth of information already known about these spaces makes them optimal as test-beds for the new geometric techniques used in this program, and the theories behind these constructions will benefit from new examples. Broader Impacts : The PI describes two projects which utilize 3D printing/fast prototyping technology for education, research, outreach and community engagement. The first project is a redesign of the calculus curriculum based around labs, with 3D printing activities as a centerpiece. The second project is the Mason Experimental Geometry Lab, a new institution at George Mason University which acts as a base of operations for research and community engagement activities.