New perspectives in combinatorial algebra

本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课题。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无关的计算相互作用。该项目分为三个不同主题的子项目。这些包括研究空间的多项式与重复的根源，超齐次空间，并使用李代数在代表性的稳定性理论。它们之间的共同点是从不同的角度观察特定的代数计算，以获得不同的代数计算，令人惊讶的是，它更容易处理。这扩展了PI及其合作者以前的工作，目的是将其带入新的方向。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元形式空间中多重根轨迹的方程和合冲。PI计划使用某些单子类型的结构来研究它们，因为二元形式的程度越来越高。最近，这被成功地用于给出关于正则嵌入射影曲线的一般绿色猜想的一个新证明。第二个主题是通过它们与超齐性空间的相干上同调和经典Grothendieck-Springer分解的超类似物的联系来计算类行列式簇的合冲。这是由于李超代数在这些合点上存在意想不到的作用，事实上，这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，PI计划探索这一点。第三个主题涉及curried代数;最近由PI与Andrew Snowden合作引入的概念。这提供了表示稳定性理论和李理论中的构造（如Bernstein-Gelfand-Gelfand范畴O）之间的深层联系。PI计划从Lie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NSF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。

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.