Interactions between noncommutative algebra, algebraic geometry and representation theory

提出的研究是在非交换代数，代数几何和表示理论的交叉。第一个方向的项目涉及代数曲面的非交换类似物，称为曲面上的极大阶。它们是曲面上的相干非交换代数捆，其一般柄是一个中心简单代数。关于极大阶的课题包括研究端阶的粗模空间、3折上的阶、某些端阶的模和衍生范畴以及特征p上某些K3曲面的Brauer群。课题的第二个方向是研究高次形式的Clifford代数及其表示。在这个方向上的项目包括利用算术Cohen-Macaulay束构造三元三次形式Clifford代数的不可约表示的模空间，以及利用二元形式Clifford代数构造曲面上的极大阶。提出的项目的第三个方向涉及计算某些Shimura品种的Hecke对应的Lefschetz数。在过去的几十年里，数学和理论物理的各个领域之间的互动越来越多，成果也越来越多。这些联系中的一条重要线索是代数几何，这是一门非常古老的学科，至少可以追溯到古希腊。代数几何是将许多变量多项式方程的解作为几何对象来研究的一门学科。建议的项目使用代数几何中的工具来解决非交换代数中的问题。在非交换代数中，研究的主要对象是xy与yx不相同的多项式。物理学家对这种非交换代数也很感兴趣。在过去的15年里，在理解这些代数方面取得了很大的进展，本提案中的项目将促进这方面的知识。在另一个方向上提出的项目位于数论（研究数字系统），表示理论（研究对称性）和拓扑学的交叉点。在拓扑学中，研究了在扭曲和拉伸等弹性变形下不发生变化的空间性质。本课题研究的空间编码数论信息，具有各种对称性。

The proposed research is at the intersection of noncommutative algebra, algebraic geometry and representation theory. The projects in the first direction concern noncommutative analogs of algebraic surfaces called maximal orders on surfaces. These are coherent sheaves of noncommutative algebras on surfaces whose generic stalk is a central simple algebra. The proposed projects related to maximal orders include studying coarse moduli spaces of terminal orders, orders on 3-folds, module and derived categories of certain terminal orders and Brauer groups of some K3 surfaces in characteristic p. The second direction of proposed projects concerns Clifford algebras of higher degree forms and their representations. The projects in this direction include using arithmetically Cohen-Macaulay bundles to construct moduli spaces of irreducible representations of Clifford algebras of ternary cubic forms, and using Clifford algebras of binary forms to construct maximal orders on surfaces. The third direction of proposed projects concerns computations of Lefschetz numbers for Hecke correspondences of certain Shimura varieties. Over the past few decades there have been increasing and fruitful interactions between various areas of mathematics and theoretical physics. One important thread in these connections has been algebraic geometry, a very old subject that dates back at least to ancient Greece. Algebraic geometry is a subject in which solutions of many variable polynomial equations are studied as geometric objects. The proposed projects use tools from algebraic geometry to solve problems in noncommutative algebra. In noncommutative algebra, the main objects of study are polynomials in which the xy is not the same as yx. Such noncommutative algebras are of interest to physicists as well. A great deal of progress has been achieved in the past 15 years in understanding such algebras and the projects in this proposal will advance this knowledge. The proposed projects in another direction lie at the intersection of number theory (where number systems are studied), representation theory (where symmetries are studied), and topology. In topology, properties of spaces are studied which do not change under elastic deformations such as twisting and stretching. The spaces to be studied in this project encode number theoretic information and possess various kinds of symmetries.