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

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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.