RUI: Problems in Varieties of Commuting Matrices, Jet Schemes, and Division Algebras

这个项目涉及两大类问题。第一部分涉及可交换矩阵和射流格式的变化，它是由一个可交换矩阵开放问题的代数-几何解释引起的。另一个涉及除法代数，它源于交叉积的经典问题，以及最近发现的除法代数在编码理论中的应用。在第一类问题中，PI Sethuraman希望研究由某些矩阵可交换条件所定义的各种变体，例如固定矩阵中心中心中的可交换对、可交换三元组和可交换对的变体。这个想法是为了深入了解三个生成的交换矩阵代数。在这些考虑中，交换矩阵对的射流格式和行列式的射流格式自然出现。在第二类问题中，Sethuraman希望研究素数指数的某些一般定义符号除法代数和素数的各种幂指数，以确定它们是否为交叉积。符号代数的一般性质使它成为一个重要的研究对象。在另一个方向上，Sethuraman希望继续研究除法代数在无线通信中的应用，包括确定除法代数中有序单位的大子群，以及在适当的迹形式下在数域内构造正交格。本建议所考虑的问题来自代数和代数几何。这里研究了两类问题：一类问题的动机来自数学内部，另一类问题的动机来自无线通信，这一点非常重要。PI与工程师们有过成功的合作，在那里他能够利用某些抽象定义的数学对象（“除法代数”）来解决一个非常实际的电信问题。这种协同作用出乎意料地出现，并在该领域取得了丰硕的成果。除了与工程师一起工作外，值得注意的是，PI在一个主要的本科院校任教，那里有大量的移民人口，他在数学方面指导了几名本科生和硕士生的研究项目。这些学生中有些人将来会成为教师。他建议继续在研究项目上与学生合作。

This project deals with two broad classes of problems. The firstconcerns varieties of commuting matrices and jet schemes, and arisesfrom algebro-geometric interpretations of an open problem incommuting matrices. The other concerns division algebras, andarises from the classical question of crossed products, and as well,from recently discovered applications of division algebras to codingtheory.In the first class of problems, the PI, Sethuraman, wishes to studyvarious varieties defined by the condition that certain matricescommute, such as varieties of commuting pairs, of commuting triples,and of commuting pairs in the centralizers of fixed matrices. Theidea is to gain insight into three generated commutative matrixalgebras. Jet schemes of commuting pairs of matrices arise naturallyin these considerations, as do jet schemes of determinantalvarieties. In the second class of problems, Sethuraman wishes tostudy certain generically defined symbol division algebras of primeexponent and index various powers of the prime to determine if theseare crossed products. The generic nature of the symbol algebramakes this a significant object to study. In a different direction,Sethuraman wishes to continue to study applications of divisionalgebras to wireless communication, including determining largesubgroups of units in orders in division algebras, and theconstruction of orthogonal lattices inside number fields forappropriate trace forms.The problems considered in this proposal arise from algebra andalgebraic geometry. There are two kinds of problems studied here:one where the motivation comes from within mathematics, and theother where, very significantly, the motivation comes fromwireless communication. The PI has had a successful collaborationwith engineers where he was able to utilize certain abstractlydefined mathematical objects (``division algebras'') to solve a verypractical telecommunications problem. The synergy appeared mostunexpectedly, and has led to a very fruitful body of results in thefield. In addition to this work with engineers, it is to be notedthat the PI teaches in a primarily undergraduate institution, with alarge immigrant population, where he has guided severalundergraduate and masters level students on research projects inmathematics. Some of these students will go on to be schoolteachers. He proposes to continue working with students on researchprojects.