Homological Commutative Algebra and Group Actions in Geometry

这个项目涉及代数结构中的对称性类型，这些对称性出现在广泛的应用领域，包括生物、化学、几何、优化和物理。这项研究涉及开发一系列新的工具来表示几何性质的代数。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。国际数学联合会还将继续参与若干辅导活动，包括促进研究生教育多样性方案、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妇女数学方案提供咨询；组织区域和国际会议；为开放源码计算机代数系统Macaulay2开发和分发软件。这个项目的研究部分试图为以下每一个建立一个基本的框架：(1)对应于光滑环簇上的层的线丛分解的复形，(2)无限环境中等变理想的自由分解，例如，在可数无限维空间上的对称群作用的情况，以及(3)球面簇上的Koszul同调的D-模变体。具体内容包括：(1)发展和应用射影空间基本同调结果的光滑环面变种的类似物；(2)确定如何在无限环境中用合适的么半群或群作用跟踪和计算不变同调；(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偏微分方程组不变量，来揭示底层几何和存在的群动作。

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.