Enumeration Problems in Algebraic Geometry and Representation Theory

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This research project concerns both algebraic geometry, which seeks to characterize solutions to algebraic equations with geometry, and representation theory, which is a systematic investigation of symmetry. Despite their abstract nature, the answers to many questions in both of these subjects boil down to being able to compute certain numbers. The purpose of this research project is to use new techniques from algebraic geometry and commutative algebra to recast these numbers as combinatorial quantities, making them easier to understand with a computer. By utilizing conceptual connections to other scientific fields, this research will also advance the understanding of several questions in mathematical physics and mathematical biology. Undergraduate students are involved directly as collaborators in the project, providing them with training in advanced mathematical topics and the use of 3D printing techniques in mathematical research.The algebraic geometry of moduli spaces of principal bundles and branching varieties naturally produces two interesting enumeration problems: counting branching multiplicities of a map of reductive groups, and finding the dimension of the spaces of conformal blocks from the Wess-Zumino-Novikov-Witten model of conformal field theory. This research aims to further understanding of these quantities using the theory of Newton-Okounkov bodies and the quickly evolving field of Berkovich geometry. These theories will be used to provide new polyhedral descriptions of conformal blocks and branching multiplicities, as well as further the understanding of the topology and symplectic geometry of the spaces under consideration.