Multigraded Methods for Syzygies, Arrangements, and Differential Operators

该项目研究的核心是具有基团作用的同源对象，这些对象出现在广泛的应用领域，包括生物、化学、代数拓扑和几何、最优化和物理等。该项目包括开发一系列新的工具，通过某些具有多项式项的矩阵族，以代数方式表示几何性质。教育和更广泛的影响部分通过与本科生和研究生以及博士后的合作与这些项目相结合。PI还将继续参与几个指导活动，包括增强研究生教育(EDGE)计划、数学联盟和明尼苏达州妇女数学计划；组织区域和国际会议组织；以及开源计算机代数系统Macaulay 2的软件开发和分发。该项目的研究部分寻求为以下每一个建立一个基本框架：(1)与光滑环面上的轮的线束分解相对应的自由复合体；(2)通过光滑环簇上的矢量丛来分解某些等变层的复合体；(3)超平面的排列；以及(4)环面环的微分算子环。更具体地说，研究内容包括：(1)构造光滑环簇的对角线的显式向量丛分解，这将得到著名的Hilbert Syzygy定理的类似结果；(2)通过光滑环簇上的向量丛构造某些等变层的分解；(3)素特征的A-超几何多项式；(4)环面环的微分算子环。这项研究将产生一系列新的工具，通过帮助计算重要的代数几何和偏微分方程不变量，揭示潜在的几何和群体行为。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

At the heart of the research components of this project are homological objects with group actions, which arise in a wide range of areas of application, including biology, chemistry, algebraic topology and geometry, optimization, and physics, among others. The project involves the development of a host of new tools to express geometric properties algebraically, via certain families of matrices with polynomial entries. The education and broader impacts portions integrate with these projects through work with undergraduate and graduate students, as well as postdocs. The PI will also continue involvement in several mentoring initiatives, including the Enhancing Diversity in Graduate Education (EDGE) Program, Math Alliance, and 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) free complexes corresponding to line bundle resolutions of sheaves on smooth toric varieties, (2) resolutions of certain equivariant sheaves via vector bundles over smooth toric varieties, (3) arrangements of hyperplanes, and (4) rings of differential operators for toric face rings. More specifically the research includes (respectively): (1) constructing an explicit vector bundle resolution of the diagonal for smooth toric varieties, which will yield an analogue of the celebrated Hilbert Syzygy Theorem, (2) resolutions of certain equivariant sheaves via vector bundles over smooth toric varieties, (3) A-hypergeometric polynomials in prime characteristic, and (4) rings of differential operators for toric face rings. The research will yield a host of new tools that will shed light on the underlying geometry and group actions by aiding the computation of important algebro-geometric and PDE invariants.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.