Interactions between noncommutative algebra, algebraic geometry and representation theory

本研究由非对易代数和代数几何交叉的三个相关课题组成。第一个项目涉及被称为极大阶的代数簇的非对易类似；这是簇上的非对易代数的凝聚层，其通用柄是中心单代数。该项目研究了一些极大阶上的模，开发了3-折叠上最大阶的极小模型程序，并研究了特征p中的超奇异K3曲面的Brauer群。第二个项目主要研究了簇上的Clifford代数和Ulrich层。这个项目使用Clifford代数表示来构造代数簇上的Ulrich层，并考察了簇上Ulrich层范畴的稳定性。第三个项目通过研究除法代数的循环性与Clifford代数的表示之间的关系，以及利用加权Clifford代数来研究相对的Brauer曲线群，将Clifford代数与域的Brauer群联系起来。在过去的几十年里，数学和理论物理之间卓有成效的相互作用引起了人们对非对易代数，特别是Clifford代数的极大兴趣。非对易代数类似于多项式，其中乘法的顺序很重要。Clifford代数出现在狄拉克方程的定义中，是量子力学和量子场论的组成部分。非对易代数可以用代数几何的复杂方法来研究，反过来，也可以用来回答问题的是代数几何。代数几何得到了发展，并被证明是解决代数中一些老问题的有力工具。非对易代数几何的研究已经取得了很大的进展，该计划进一步发展了非对易代数(特别是Clifford代数)和相关的代数几何的一些深层次的几何和代数方面的内容。

The proposed research consists of three related projects at the intersection of noncommutative algebra and algebraic geometry. The first project concerns noncommutative analogs of algebraic varieties called maximal orders; these are coherent sheaves of noncommutative algebras on varieties whose generic stalk is a central simple algebra. The project involves studying modules over some maximal orders, developing a minimal model program for maximal orders on 3-folds, and studying Brauer groups of supersingular K3 surfaces in characteristic p. The second project focuses on Clifford algebras and Ulrich sheaves on varieties. This project uses Clifford algebra representations to construct Ulrich sheaves on algebraic varieties and examines stability in the category of Ulrich sheaves on varieties. The third project links Clifford algebras and Brauer groups of fields by investigating the relationship between cyclicity of division algebras and representations of Clifford algebras, and by using weighted Clifford algebras to study relative Brauer groups of curves. The fruitful interactions between mathematics and theoretical physics in the past several decades have resulted in great interest in noncommutative algebras and Clifford algebras in particular. Noncommutative algebras are similar to polynomials in which the order of multiplication matters. Clifford algebras occur in the definition of the Dirac equation and are an integral part of quantum mechanics and quantum field theory. Noncommutative algebras can be studied using the sophisticated methods of algebraic geometry and, conversely, have been used to answer questions is algebraic geometry. Algebraic geometry was developed and has proved to be a powerful tool for addressing some old problems in algebra. The subject of noncommutative algebraic geometry has been progressing rapidly, and the projects in this proposal further develop some deep geometric and algebraic aspects of noncommutative algebras (and Clifford algebras in particular) and related algebraic geometry.