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.

该项目的目的是从新的现代技术的角度研究组合和代数的一些古典主题。 PI对代数计算感兴趣，例如表征空间的方程式以及它们如何与看似无关的计算相互作用。该项目分为具有不同主题的三个子弹。其中包括研究具有重复根，超均匀空间的多项式空间，以及在表示稳定理论中使用Lie代数。它们之间的共同点涉及从不同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，这些计算更容易进行。这扩展了PI及其合作者的先前工作，其目的是将其带入新的方向。该项目还将为培训研究生提供机会。PI将在代表理论和交换代数之间进行三个主要主题，并在这两个方向上进行这些主题之间的应用。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式和共融。 PI计划随着二元形式的程度使用某些单型型结构增长。最近，这被成功地用于给出新的绿色猜想的新证明。第二个主题涉及确定性品种的共同体通过与超同质空间的连贯共同体和经典Grothendieck-Springer分辨率的超级类似物的连贯性进行计算。这是由于在这些共同体上谎言超级骨的意外行动的存在，实际上为它们的存在提供了概念上的解释。另一方面，它还提供了一种新的策略，可以找到PI计划探索的Borel-Weil-Bott Theorem的超级类似物。第三个主题涉及咖喱代数； PI最近与安德鲁·斯诺登（Andrew Snowden）合作提出的一个概念。这提供了表示稳定理论与谎言理论（例如伯恩斯坦 - 吉尔夫德 - 吉尔福德）类别O之间的深厚联系。PI计划将这些技术从谎言理论中导入这些技术，以在表示稳定性及其应用中找到并证明了新的结构性结果。该奖项反映了NSF的法定任务，并通过评估了基金会的范围，并通过评估了基金会的范围。