Interactions between noncommutative algebra, algebraic geometry and representation theory

这项提案中提议的项目有三个方向。第一个项目研究代数曲面的非交换类比。在前人关于射影非奇异曲面上极大阶分类的工作之后，本课题研究了剩余的极大阶类。这个项目的另一个目标是给出曲面上的序的显式构造，然后用它们来研究它们的模和派生范畴。这个方向的最后一个目标是开始研究特征p中曲面的最大阶。第二个项目继续了先前关于局部对称空间的约化Borel-Serre紧化上与交链层相容的自对偶层的工作。我们的目的是看这样的层是否存在用于某些么正下村簇的紧致化，以及这些层在多大程度上可以用于分析Hecke对应的Lefschetz数。最后一个项目通过研究期望的多项式现象来研究亏零曲线的双Hurwitz数的结构。在过去的几十年里，数学和理论物理的各个领域之间进行了非常卓有成效的互动。这些联系中的一条重要线索是代数几何，这是一个非常古老的学科，至少可以追溯到古希腊。代数几何是一门把许多变量多项式方程的解作为几何对象来研究的学科。建议的项目使用代数几何中的工具来解决非对易代数中的问题。在非交换代数中，主要的研究对象是XY与YX不同的多项式。这样的非交换代数也引起了物理学家的兴趣。这项建议的第二个项目是三门学科的交叉：数论(研究数系)、表示论(研究对称性)和拓扑学。在拓扑学中，研究了空间在扭转和拉伸等弹性变形作用下不变的性质。本项目要研究的空间是数论信息的编码。这些空间有一些自然的对称性。我们的目标是研究这些对称的不动点。最后一个项目是组合学和代数几何的交叉点。我们的目标是了解球体可以被类似的几何对象覆盖的方式的数量，这些几何对象称为具有特定限制的代数曲线。

The proposed projects in this proposal are in three directions. The first project studies noncommutative analogs of algebraic surfaces. Following previous work on classification of maximal orders on projective, nonsingular surfaces, this project studies remaining classes of maximal orders. Another goal for this project is to give explicit constructions of orders on surfaces and then use them to study their module and derived categories. The last goal in this direction is to initiate study of maximal orders on surfaces in characteristic p. The second project continues previous work on self dual sheaves compatible with the intersection chain sheaves on reductive Borel-Serre compactifications of locally symmetric spaces. The goal is to see if such sheaves exist for compactifications of certain unitary Shimura varieties and to see to what extent these sheaves can be used in analysis of the Lefschetz numbers of Hecke correspondences. The last project investigates structure of double Hurwitz numbers of genus zero curves by studying the expected polynomiality phenomenon.In the last few decades, there have been very fruitful interactionsbetween various areas of mathematics and theoretical physics. Oneimportant 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. The second project of this proposal is at the intersection of three subjects: number theory (where number systems are studied), representation theory (where symmetries are studied), and topology. In topology those 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. These spaces have some natural symmetries. The goal is to investigate fixed points of these symmetries. The last project is at the intersection of combinatorics and algebraic geometry. The goal is to understand the number of ways in which a sphere can be covered by similar geometric objects called algebraic curves with specified restrictions.