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.

本项目旨在从现代新技术的角度研究组合学和代数中的一些经典问题。PI对代数计算感兴趣，比如描述空间的方程，以及它们如何与看似无关的计算相互作用。该项目被分成三个具有不同主题的子项目。这些研究包括对多项式的重根空间的研究，超齐次空间，以及李代数在表示稳定性理论中的应用。它们之间的共性包括从不同的角度查看特定的代数计算，以获得令人惊讶的更易于处理的不同代数计算。这扩展了PI和他的合作者以前的工作，目的是把它带到新的方向。该项目还将为培养研究生提供机会。PI将研究介于表示理论和交换代数之间的三个主要主题，并在两个方向上应用这些主题。第一个主题是计算二进制形式空间中多根轨迹的方程和协同。PI计划使用某些单类型结构，随着二进制形式的程度增长，将它们一起研究。最近，我们成功地利用这一方法，给出了典型嵌入投影曲线上的一般格林猜想的一个新的证明。第二个主题是通过它们与经典格罗滕迪克-施普林格解析的超齐次空间和超类似物的相干上同调的联系来计算类行列式的共合性。这是由于李超代数在这些合子上存在意想不到的作用，事实上，为它们的存在提供了一个概念上的解释。另一方面，它也提供了一种新的策略来寻找Borel-Weil-Bott定理的超级类似物，这是PI计划探索的。第三个主题涉及curry代数；这是PI最近与安德鲁·斯诺登（Andrew Snowden）合作提出的概念。这提供了表征稳定性理论与李理论中的结构（如Bernstein-Gelfand-Gelfand类别o）之间的深层联系。PI计划从李理论中引入这些技术，以发现和证明表征稳定性及其应用中的新结构结果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。