Multigraded commutative algebra

多级交换代数

• 批准号: 2409776
• 负责人: Daniel Erman
• 金额: $ 26.5万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-11-15 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2409776&HistoricalAwards=false
• 关键词: Multigraded; commutative; algebra

Polynomial equations are some of the most fundamental objects in mathematics and science, and they arise in a huge array of examples: Fermat’s Last Theorem, physical models of motion, the ideal gas law, and much more. Understanding the solutions of polynomial equations is thus a fundamental challenge that cuts across all quantitative fields. The PI’s work aims to develop techniques—both theoretical and algorithmic—for better understanding the solutions of polynomial equations that are endowed with extra symmetries. As such equations often arise in mathematical or scientific applications, the work has the potential for wide impact.The PI aims to expand the literature on multigraded polynomials, building on his prior work of using virtual resolutions as a foundation for geometric applications of multigraded syzygies. One project would provide novel multigraded analogues of the Hilbert Syzygy Theorem and of Beilinson's resolution of the diagonal, thus establishing two foundational results. A second project on a multigraded version of Green's Linear Syzygy Theorem would have more direct geometric applications. And a third project would have computational applications, yielding a new and potentially much faster algorithm for computing sheaf cohomology on a toric variety. The project will also have broader impacts through the PI’s mentoring of PhD students.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.

多项式方程是数学和科学中一些最基本的对象，它们在大量示例中：费马特的最后定理，运动的物理模型，理想的气体定律等等。因此，了解多项式方程的解决方案是削减所有定量领域的基本挑战。 PI的工作旨在开发技术（包括理论和算法）的技术，以更好地理解具有超对称性的多项式方程的解决方案。由于这种股票经常在数学或科学应用中产生，因此这项工作具有广泛影响的潜力。PI旨在扩大有关多元多项式的文献，其基础是他先前使用虚拟分辨率作为多面式Syzygies几何应用的基础。一个项目将提供希尔伯特·赛齐（Hilbert Syzygy）定理和贝林森（Beilinson）对角线的新型多层类似物，从而确立了两个基础结果。 Green线性Syzygy定理的多面版本的第二个项目将具有更直接的几何应用。第三个项目将具有计算应用程序，从而产生一种新的且可能更快的算法，用于计算复合品种的捆捆同学。该项目还将通过PI对博士学位学生的心理作用产生更大的影响。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估诚实的支持。