Multigraded Methods for Syzygies, Arrangements, and Differential Operators

该项目的研究组成部分的核心是具有群作用的同调对象，这些对象出现在广泛的应用领域，包括生物学，化学，代数拓扑和几何，优化和物理等。该项目涉及开发一系列新的工具，通过某些具有多项式项的矩阵族，以代数方式表达几何特性。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还将继续参与几项指导活动，包括加强研究生教育多样性（EDGE）计划、数学联盟和明尼苏达州妇女参与数学计划;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。该项目的研究内容旨在为每个以下内容：（1）对应于光滑复曲面簇上层的线丛分解的自由复形，（2）光滑复曲面簇上某些等变层通过向量丛的分解，（3）超平面的排列，（4）复曲面面环的微分算子环。更具体地说，研究包括（分别）：（1）构造光滑复曲面簇的对角线的显式向量丛分解，它将产生著名的Hilbert合之定理的类似物，（2）某些等变层通过光滑复曲面簇上的向量丛的分解，（3）素特征的A-超几何多项式，（4）复曲面环的微分算子环。该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PDE不变量，揭示潜在的几何和群体作用。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.