Homological Commutative Algebra and Group Actions in Geometry

几何中的同调交换代数和群作用

• 批准号: 1661962
• 负责人: Christine Berkesch
• 金额: $ 16.15万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1661962&HistoricalAwards=false
• 关键词: Homological; Commutative; Algebra; Group; Actions

This project concerns types of symmetry in algebraic structures which arise in a wide range of areas of application, including biology, chemistry, geometry, optimization, and physics. The research involves the development of a host of new tools to express geometric properties algebraically. The education and broader impacts portion of the project integrates with the research through work with graduate students. The PI will also continue involvement in several mentoring initiatives, including the Enhancing Diversity in Graduate Education (EDGE) Program, Math Alliance, and a professional development seminar for mathematics graduate students in conjunction with advising Minnesota's Women in Math program; organization of regional and international conferences organization; and software development and distribution for the open source computer algebra system Macaulay2. The research components of this project seek to establish a foundational framework for each of the following: (1) complexes corresponding to line bundle resolutions of sheaves on smooth toric varieties, (2) free resolutions of equivariant ideals in an infinite setting, for instance, the case of a symmetric group action on a countably infinite-dimensional space, and (3) a D-module variant of Koszul homology over spherical varieties. The specific parts include (respectively): (1) developing and applying analogues for smooth toric varieties of foundational homological results for projective space, (2) determining how to track and compute invariant syzygies in infinite settings with a suitable monoid or group action, and (3) generalizing a homological framework for hypergeometric systems of PDEs from the setting of a torus action to that of a reductive group. The project will yield new sets of tools for shedding light on the underlying geometry and group actions present, by aiding in the computation of important algebro-geometric and PDE invariants.

该项目涉及代数结构中的对称类型，这些类型出现在广泛的应用领域，包括生物学、化学、几何、优化和物理学。该研究涉及开发一系列新工具来用代数表达几何性质。该项目的教育和更广泛的影响部分通过与研究生的合作与研究相结合。 PI 还将继续参与多项指导计划，包括增强研究生教育多样性 (EDGE) 计划、数学联盟以及为数学研究生举办的专业发展研讨会，同时为明尼苏达州的女性数学计划提供咨询；组织区域和国际会议；开源计算机代数系统Macaulay2的软件开发和分发。该项目的研究部分旨在为以下各项建立一个基础框架：（1）与光滑复曲面品种上滑轮的线束分辨率相对应的复合体，（2）无限环境中等变理想的自由分辨率，例如，可数无限维空间上的对称群作用的情况，以及 (3) 球簇上 Koszul 同调的 D 模变体。具体部分（分别）包括：（1）开发和应用射影空间基础同调结果的平滑环面变体的类比，（2）确定如何使用合适的幺半群或群动作在无限设置中跟踪和计算不变的syzygies，以及(3) 将偏微分方程的超几何系统的同调框架从环面作用的设置推广到还原群的设置。该项目将产生一套新的工具，通过帮助计算重要的代数几何和偏微分方程不变量，揭示底层几何和存在的群行为。

项目成果

On normalized Horn systems

关于归一化喇叭系统

• DOI: 10.1007/s13348-019-00259-0
• 发表时间: 2020-05
• 期刊: Collectanea Mathematica
• 影响因子: 1.1
• 作者: Berkesch, Christine;Matusevich, Laura Felicia;Walther, Uli
• 通讯作者: Walther, Uli

Characteristic cycles and Gevrey series solutionsof A-hypergeometric systems

A-超几何系统的特征循环和Gevrey级数解

• DOI: 10.2140/ant.2020.14.323
• 发表时间: 2020-01
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Berkesch, Christine;Fernández
• 通讯作者: Fernández

Virtual resolutions for a product of projective spaces

射影空间乘积的虚拟分辨率

• DOI: 10.14231/ag-2020-013
• 发表时间: 2020-07
• 期刊: Algebraic Geometry
• 影响因子: 1.5
• 作者: Erman; Daniel
• 通讯作者: Daniel

Torus equivariant D-modules and hypergeometric systems

环面等变 D 模和超几何系统

• DOI: 10.1016/j.aim.2019.04.050
• 发表时间: 2019-07
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Berkesch, Christine;Matusevich, Laura Felicia;Walther, Uli
• 通讯作者: Walther, Uli

On the parametric behavior of $A$-hypergeometric series

关于$A$-超几何级数的参数行为

• DOI: 10.1090/tran/7071
• 发表时间: 2018-06
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Berkesch, Christine;Forsgård, Jens;Matusevich, Laura Felicia
• 通讯作者: Matusevich, Laura Felicia