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

RUI：各种通勤矩阵、Jet 方案和除法代数中的问题

• 批准号: 0700904
• 负责人: B Sethuraman
• 金额: $ 12.1万
• 依托单位: The University Corporation, Northridge
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700904&HistoricalAwards=false
• 关键词: RUI; Problems; Varieties; Commuting; Matrices

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.

这个项目涉及两大类问题。第一个涉及各种交换矩阵和喷气计划，并从代数几何解释的一个公开问题交换矩阵。 另一个是关于除法代数，它产生于经典的交叉积问题，以及最近发现的除法代数在codingtheory中的应用。在第一类问题中，PI，Sethuraman，希望研究由某些矩阵交换的条件定义的各种各样的问题，例如交换对，交换三元组和交换对在固定矩阵的中心化子中的变化。本文的思想是深入研究三种生成交换矩阵代数。在这些考虑中自然出现交换矩阵对的喷射方案，就像行列式簇的喷射方案一样。 在第二类问题中，Sethuraman希望研究某些一般定义的素指数的符号除代数，并对素数的各种幂进行指标，以确定它们是否是交叉积。 符号代数的一般性质使其成为一个重要的研究对象。 在另一个方向，Sethuraman希望继续研究divisionalgebras在无线通信中的应用，包括确定divisionalgebras中各阶单位的大子群，以及在数域中构造正交格以获得适当的迹形式。 这里研究的问题有两种：一种是来自数学的动机，另一种是，非常重要的是，来自无线通信的动机。 PI与工程师成功合作，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。 这种协同作用似乎非常惊人，并在该领域取得了非常丰硕的成果。 除了这项工作与工程师，值得注意的是，PI任教于一个主要的本科院校，有大量的移民人口，在那里他指导了几个本科和硕士水平的学生在数学研究项目。这些学生中的一些人将继续成为教师。他建议继续与学生一起做研究项目。