Interactions between noncommutative algebra, algebraic geometry and representation theory

本提案中的拟议项目有三个方向。第一个项目研究代数曲面的非交换类似物。继以前的工作分类的最大订单的投影，非奇异表面，这个项目研究剩余类的最大订单。这个项目的另一个目标是给出曲面上阶的显式构造，然后用它们来研究它们的模和派生范畴。在这个方向的最后一个目标是启动研究的最大订单表面的特征p.第二个项目继续以前的工作自对偶层兼容的交叉链层约化Borel-塞尔紧化的局部对称空间。我们的目标是要看看如果这样的层存在的某些酉志村品种的紧化，并在何种程度上这些层可以用于分析莱夫谢茨数的赫克对应。最后一个项目通过研究亏格零曲线的期望多项式现象来研究亏格零曲线的双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有着非常富有成果的相互作用。这些联系中的一个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。代数几何是以多元多项式方程的解为几何对象来研究的一门学科。拟议的项目使用代数几何的工具来解决非交换代数中的问题。在非交换代数中，主要的研究对象是xy与yx不相同的多项式。这种非交换代数也是物理学家感兴趣的。这个建议的第二个项目是在三个学科的交叉点：数论（研究数字系统），表示论（研究对称性）和拓扑学。在拓扑学中，研究的是在弹性变形（如扭曲和拉伸）下不发生变化的空间的性质。在这个项目中要研究的空间编码数论信息。这些空间具有一些自然的对称性。我们的目标是调查这些对称的不动点。最后一个项目是在组合学和代数几何的交叉点。目标是了解球体可以被类似的几何对象（称为具有指定限制的代数曲线）覆盖的方式的数量。

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.