Interactions between Algebra, Algebraic Geometry and Topology

在这项建议中，研究者和合作者研究代数、代数几何和拓扑学中的问题。其中三个项目研究自然出现在代数各个分支中的非交换代数族。第一个项目是研究高次形式的Clifford代数的有限维不可约表示。研究这些代数的方法是用超曲面上某些自反层的模空间来表示问题。另一个目标是研究这些结果在医学上的应用。第二个项目(与D·Chan联合)是关于复杂的、射影的、代数曲面上的极大阶层的工作的继续。其主要目的是对最大阶数组的各种类型进行分类。这项工作遵循了最近在非对易代数几何中发展起来的思想。第三个项目(与A.Ram联合)是研究有理Cherednik代数是否可以沿着单分次Hecke代数的直线建立Springer对应。另一个目的是研究这些代数是否可以与N.Wallach对Springer对应的方法相关联。最后一个项目(与M.Banagl合作)是研究局部对称空间的约化Borel-Serre紧化是否存在与定义在有理数上的半单代数群相容的交链紧的自对偶层。在过去的几十年里，数学和理论物理的各个分支之间的相互作用已经被证明是对所有领域都特别丰富的。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。代数几何是将多变量多项式方程的解作为几何对象进行研究的数学领域。它不仅在数学上得到了应用，而且在计算机科学、编码理论、机器人学和物理学中的弦理论等领域也得到了应用。该提案的前两个项目使用了这门学科中的工具来研究范围广泛的数学家和物理学家感兴趣的“代数”。将这类代数的整个族描述为一个几何对象是有用的，而本项目的一个目标就是获得这样的描述。这项研究的第三个方面是被称为表征理论的学科的一部分。这里的主要研究对象是“表征”，即编码关于自然界中对称性的信息的过程。这项建议的最后一部分是三门学科的交集：数论(研究数系)、表示论和拓扑学。在拓扑空间中，研究以确定哪些性质在弹性变形(如扭转和拉伸)下不改变。本研究研究的空间对数论信息进行了编码。目的是调查这些空间是否存在本身满足某种对称性的不变量。

In this proposal, the investigator and collaborators study problems inalgebra, algebraic geometry and topology. Three of these projectsstudy families of noncommutative algebras arising naturally in variousbranches of algebra. The first project is to study finite dimensionalirreducible representations of Clifford algebras of forms of higherdegree. The approach in studying these algebras is to work on areformulation of the questions in terms of moduli spaces of certainreflexive sheaves on hypersurfaces. Another goal is to investigatearithmetic applications of these results. The second project (jointwith D. Chan) is continuation of previous work on sheaves of maximalorders on complex, projective, algebraic surfaces. The main goal is toclassify various classes of such sheaves of maximal orders. This workfollows recently developed ideas in noncommutative algebraicgeometry. The third project (joint with A. Ram) is to investigatewhether the Springer correspondence can be set up for rationalCherednik algebras along the lines of single graded Heckealgebras. Another goal is to investigate whether these algebras can berelated to N. Wallach's approach to the Springer correspondence. Thelast project (joint with M. Banagl) is to investigate whether thereexist self-dual sheaves compatible with the intersection chain sheaveson reductive Borel-Serre compactifications of locally symmetric spacesassociated to semisimple algebraic groups defined over rationalnumbers. In the last few decades, the interaction between various branches ofmathematics and theoretical physics has proved to be especiallyenriching for all the fields. One important thread in theseconnections has been algebraic geometry, a very old subject that datesback at least to ancient Greece. Algebraic geometry is the area of mathematics that studies solutions to multi-variable polynomial equations as geometric objects. It has found applications not only inmathematics, but also in computer science, coding theory, robotics andstring theory in physics to name a few areas impacted by it. The firsttwo projects of this proposal use tools from this subject to study"algebras" that are of interest to a wide range of mathematicians andphysicists. It is useful to describe the whole family of such classesof algebras as a geometric object, and a goal of this project is toobtain such descriptions. The third aspect of this research is partof a subject called representation theory. The main objects of studyhere are "representations", processes which encode information aboutsymmetry in nature. The last part of this proposal stands at theintersection of three subjects: number theory (where number systemsare studied), representation theory, and topology. In topology spacesare studied to determine which properties do not change under elasticdeformations such as twisting and stretching. The spaces to bestudied in this research encode number theoretic information. The goalis to investigate whether there are invariants of these spaces whichthemselves satisfy some symmetry.