International Research Fellowship Program: The Solution Space of a Hypergeometric System

The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twelve-month research fellowship by Dr. Christine Berkesch to work with Dr. Mikael Passare at Stockholm University in Sweden. This project consists of two independent components. The main goal of the first circle of problems is to obtain an explicit understanding of the parametric behavior of the solution space of a hypergeometric system of partial differential equations. Other goals include gaining a better understanding of the complexity of the combinatorics that impact its dimension, as well as making steps towards explaining the parametric behavior of all derived solutions (Gevrey and regular) of a hypergeometric system. This research is continuing work from the PI's dissertation, where homological and combinatorial viewpoints were used to study this dimension. It also requires techniques from the host's area of expertise, complex analysis. The second project, based on a question of D. Eisenbud (U.C. Berkeley), establishes a multigraded generalization of the new and powerful Boij-Söderberg theory. The primary tools for this work are sheaf cohomology, multigraded free resolutions, and the combinatorics of posets and polyhedral fans. This project fosters interaction between several areas of mathematics, including algebraic geometry, combinatorics, commutative algebra, and complex analysis. An explicit understanding of the parametric behavior of hypergeometric systems will have an impact across mathematics and physics. The Boij-Soderberg component will not only shed light on the mysteries surrounding the standard graded case, but will also develop a new theory with which to study enumerative properties of foundational objects in algebraic geometry and commutative algebra, namely vector bundles on products of projective spaces and multigraded modules over Cox rings.