New perspectives in combinatorial algebra

组合代数的新视角

• 批准号: 2302149
• 负责人: Steven Sam
• 金额: $ 34.45万
• 依托单位: University of California-San Diego
• 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: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302149&HistoricalAwards=false
• 关键词: New; perspectives; combinatorial; algebra

The purpose of this project is to study some classical topics in combinatorics and algebra from the perspective of new modern techniques. The PI is interested in algebraic computations such as the equations that characterize a space, and how they interact with seemingly unrelated computations. The project is broken into three subprojects with a different theme. These include the study of the space of polynomials with repeated roots, super homogeneous spaces, and the use of Lie algebras in representation stability theory. The commonality among them involves viewing specific algebraic computations from a different angle to get a different algebraic computation that surprisingly is much more tractable. This expands on previous work of the PI and his collaborators, and the intention is to take it into new directions. The project will also provide opportunities for training graduate students.The PI will work on three main topics sitting between representation theory and commutative algebra, with applications flowing between these subjects in both directions. The first topic concerns computing the equations and syzygies of multiple root loci in spaces of binary forms. The PI plans to study them all together as the degree of the binary forms grows using certain monad-type constructions. Recently, this was successfully used to give a new proof of the generic Green conjecture on canonically embedded projective curves. The second topic concerns the calculation of syzygies of determinantal-like varieties via their connection to the coherent cohomology of super homogeneous spaces and super analogues of the classical Grothendieck-Springer resolution. This is motivated by the existence of unexpected actions of Lie superalgebras on these syzygies and, in fact, offers a conceptual explanation for their existence. On the other hand, it also offers a new strategy to find super analogues of the Borel-Weil-Bott theorem, which the PI plans to explore. The third topic concerns curried algebras; a concept recently introduced by the PI in collaboration with Andrew Snowden. This provides a deep connection between representation stability theory and constructions in Lie theory such as the Bernstein-Gelfand-Gelfand category O. The PI plans to import these techniques from Lie theory to find and prove new structural results in representation stability and its applications.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.

该项目的目的是从现代新技术的角度研究组合学和代数中的一些经典主题。 PI 对代数计算感兴趣，例如表征空间的方程，以及它们如何与看似不相关的计算相互作用。该项目分为三个具有不同主题的子项目。其中包括具有重复根的多项式空间、超齐次空间的研究以及李代数在表示稳定性理论中的使用。它们之间的共同点包括从不同的角度查看特定的代数计算，以获得不同的代数计算，令人惊讶的是，这种计算更容易处理。这扩展了 PI 及其合作者之前的工作，其目的是将其带入新的方向。该项目还将提供培训研究生的机会。PI 将致力于表示论和交换代数之间的三个主要主题，并在这些主题之间双向流动应用。第一个主题涉及计算二进制形式空间中多个根基因座的方程和齐齐性。 PI 计划随着使用某些单子类型结构的二进制形式的程度的增长来一起研究它们。最近，这被成功地用于给出关于正则嵌入射影曲线的通用格林猜想的新证明。第二个主题涉及通过与超齐次空间的相干上同调和经典 Grothendieck-Springer 解析的超类似物的联系来计算类行列式簇的 syzygies。这是由李超代数对这些 syzygies 的意外行为的存在所激发的，事实上，为它们的存在提供了概念上的解释。另一方面，它还提供了一种新策略来寻找 PI 计划探索的 Borel-Weil-Bott 定理的超级类似物。第三个主题涉及柯里化代数；这是 PI 最近与安德鲁·斯诺登 (Andrew Snowden) 合作提出的概念。这在表示稳定性理论和李理论的构造（例如 Bernstein-Gelfand-Gelfand O 类）之间提供了深刻的联系。PI 计划从李理论中引入这些技术，以发现和证明表示稳定性及其应用的新结构结果。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。