Vector Bundles of Conformal Blocks on Moduli Spaces

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This project studies the relationship between the families of algebraic curves, a central topic in algebraic geometry, and related notions arising in mathematical physics. In algebraic geometry, families of curves are studied by the use of moduli spaces, which parametrize the different curves in the family. Besides providing deep insights about curves, moduli spaces exhibit important behaviors which help shape our understanding of how geometric objects may be parametrized, and therein serve as envoys of a larger mathematical world, giving valuable insights into the development of higher dimensional theory. Recent work has revealed that certain aspects of moduli spaces of curves reflect underlying geometric structures. This research project investigates open questions related to these discoveries. Graduate students are involved in the project, and the investigator co-organizes the Mathematics Department's MathCamp high-school program.The projects are organized along four general themes. For smooth curves, conformal blocks are identified with global sections of certain ample line bundles on a projective variety. Project One aims to study to what extent this description may or may not hold for non-smooth curves. Project Two proposes birational models of the moduli space as configurations of weighted points supported on curves embedded in homogeneous varieties, generalizing those given by pointed rational normal curves in projective space. To answer open problems about the multiplicative eigenpolyhedra from representation theory, and cones spanned by certain sets of first Chern classes of vector bundles of conformal blocks, the goal of Project Three is to leverage information gathered from maps between subsets of the two. Project Four aims to understand positive cycles of higher codimension given by vector bundles of conformal blocks.