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

国际研究奖学金项目使美国科学家和工程师能够在国外进行9至24个月的研究。该计划的奖励为联合研究提供了机会，并利用独特或互补的设施、专业知识和国外的实验条件。该奖项将支持Christine Berkesch博士与瑞典斯德哥尔摩大学的Mikael Passare博士进行为期12个月的研究。这个项目由两个独立的部分组成。第一组问题的主要目标是获得对超几何偏微分方程组解空间的参数行为的明确理解。其他目标包括更好地理解影响其维度的组合学的复杂性，以及朝着解释超几何系统的所有派生解（Gevrey和正则）的参数行为迈出一步。这项研究是从PI的论文继续工作，其中同源和组合的观点被用来研究这个维度。它还需要主持人专业领域的技术，复杂的分析。第二个项目，基于D. Eisenbud（加州大学伯克利分校）的一个问题，建立了一个新的和强大的Boij-Söderberg理论的多级推广。这项工作的主要工具是轴上同调、多阶自由分辨率、偏置集和多面体扇的组合。该项目促进了数学多个领域之间的互动，包括代数几何、组合学、交换代数和复分析。对超几何系统的参数行为的明确理解将对整个数学和物理产生影响。Boij-Soderberg分量不仅将揭示围绕标准梯度情况的奥秘，而且还将发展一种新的理论，用于研究代数几何和交换代数中基本对象的枚举性质，即投影空间乘积上的向量束和Cox环上的多重梯度模块。

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.