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.
This award supported a twelve-month research fellowship by Dr. Christine Berkesch to work with Dr. Mikael Passare at Stockholm University in Sweden. This project consisted of two independent components. The first circle of problems worked towards an explicit understanding of the parametric behavior of the solution space of a hypergeometric system of partial differential equations. The solutions of these systems occur naturally in mathematics, physics, and engineering. Berkesch, together with Jens Forsgard and Mikael Passare, discovered Euler--Mellin integrals, which provide solutions for every parameter of the hypergeometric system and have simple, explicit domains of integration. This new tool relates to monodromy and local cohomology, providing a link between the complex analysis and commutative algebra involved in the study of hypergeometric systems. Berkesch also extended results from her Ph.D. thesis to compute not only the rank (or dimension of the solution space) but also the full characteristic cycle of a hypergeometric system, which sheds light on the parametric behavior of its higher derived, or Gevrey, solutions. The second independent circle of problems addressed analogs of Boij--Soederberg theory about Betti numbers over the standard graded polynomial ring. The primary tools for this work were sheaf cohomology, graded and local free resolutions, and the combinatorics of posets and polyhedral fans. With several collaborators, Berkesch answered this question completely for three classes of rings and partially for another. In addition, a new interpretation of the poset structure in the original theory was provided by Berkesch, Daniel Erman, Manoj Kummini, and Steven Sam, as well as a generalization and explicit construction of the original proof of the existence of pure free resolutions by Eisenbud and Schreyer. During her year and Stockholm and throughout the following year during which this project was funded, Berkesch mentored Jens Forsgard, a graduate student at Stockholm University. She mentored a larger group graduate students conducting research in Boij--Soederberg theory as a Postdoc TA at the week-long MSRI Workshop: Summer Graduate Workshop in Commutative Algebra in Berkeley, California. She also gave a series of four hour-long lectures on hypergeometric systems to the faculty and graduate students in a variety of research areas at KTH (The Royal Institute of Technology in Stockholm) and Stockholm University.
