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

国际研究奖学金计划：超几何系统的解空间

• 批准号: 0964985
• 负责人: Christine Berkesch
• 金额: $ 9.06万
• 依托单位: Berkesch Christine
• 依托单位国家: 美国
• 项目类别: Fellowship Award
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0964985&HistoricalAwards=false
• 关键词: International; Research; Fellowship; Program; Solution

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.

国际研究奖学金计划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。 该计划的奖项提供了联合研究的机会，以及使用国外独特或互补的设施，专业知识和实验条件。 该奖项将支持为期12个月的研究奖学金博士克莉丝汀伯克施工作与博士迈克尔帕萨雷在斯德哥尔摩大学在瑞典。 该项目由两个独立的部分组成。第一圈问题的主要目标是获得一个明确的理解超几何系统的偏微分方程的解空间的参数行为。其他目标包括更好地理解影响其维度的组合数学的复杂性，以及采取步骤解释超几何系统的所有导出解（Gevrey和正则）的参数行为。这项研究是继续工作，从PI的论文，其中同源和组合的观点被用来研究这个维度。它还需要从主持人的专业领域的技术，复杂的分析。第二个项目是基于D.艾森巴德（U.C. Berkeley），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层上同调，多级自由决议，以及偏序集和多面体风扇的组合。 该项目促进数学的几个领域之间的互动，包括代数几何，组合数学，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数学和物理产生影响。 Boij-Soderberg分支不仅将揭示围绕标准分次情形的奥秘，而且还将发展一个新的理论，用于研究代数几何和交换代数中基础对象的枚举性质，即考克斯环上投射空间和多重分次模的乘积上的向量丛。