Homological Commutative Algebra and Group Actions in Geometry

该项目关注代数结构中的对称类型，这些对称类型在广泛的应用领域中出现，包括生物、化学、几何、优化和物理。这项研究涉及到一系列新工具的开发，以代数方式表达几何性质。该项目的教育和更广泛的影响部分通过与研究生的工作与研究相结合。PI还将继续参与几项指导计划，包括增强研究生教育多样性（EDGE）计划、数学联盟，以及与明尼苏达州女性数学计划相关的数学研究生专业发展研讨会；组织区域和国际会议组织；以及开源计算机代数系统Macaulay2的软件开发和分发。这个项目的研究组成部分试图为以下每一个问题建立一个基本框架：(1)光滑环上束的线束分辨率对应的复合体，(2)无限设置中等变理想的自由分辨率，例如，可数无限维空间上对称群作用的情况，以及(3)球面上Koszul同调的d模变体。具体包括：(1)建立和应用射影空间基本同调结果的光滑环面变体的类似物；(2)确定如何在具有合适的单形或群作用的无限设置中跟踪和计算不变协同；(3)将偏微分方程超几何系统的同调框架从环面作用的设置推广到约化群的设置。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PDE不变量，来揭示潜在的几何和群体行为。

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.