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

这个项目涉及两大类问题。第一个问题涉及各种交换矩阵和喷射方案，它是由交换矩阵中的一个公开问题的代数几何解释引起的。在第一类问题中，PI，Sthuraman，希望研究由某些矩阵可交换的条件所定义的各种变体，如固定矩阵的中心化子中的交换对的变体，交换三元组的变体，以及交换对的变体。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，自然地产生了交换矩阵对的喷射方案，就像确定性变化的喷射方案一样。在第二类问题中，Sthuraman希望研究某些一般定义的素数指数的符号除法代数，并索引素数的各种幂，以确定它们是否是交叉积。符号代数的类属性质使其成为一个重要的研究对象。在一个不同的方向上，Sthuraman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按阶单位的大子群，以及为适当的迹形式构造数域内的正交格。这里研究的问题有两种：一种是动机来自数学内部，另一种是非常重要的是，动机来自无线通信。PI与工程师进行了成功的合作，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。这种协同效应出人意料地出现，并在该领域产生了非常丰硕的成果。除了与工程师一起工作外，值得注意的是，PI在一个以本科生为主的机构任教，移民人口众多，他在那里指导了几名本科生和硕士水平的学生进行数学研究项目。这些学生中的一些人将继续担任学校教师。他建议继续与学生在研究项目上合作。

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.