Interactions between Algebra, Algebraic Geometry and Topology

代数、代数几何和拓扑之间的相互作用

• 批准号: 0202295
• 负责人: Rajesh Kulkarni
• 金额: $ 8.36万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-05-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202295&HistoricalAwards=false
• 关键词: Interactions; between; Algebra; Algebraic; Geometry

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.

在这个提案中，研究者和合作者研究代数、代数几何和拓扑中的问题。其中三个项目研究代数各个分支中自然产生的非交换代数族。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根据超曲面上某些自反滑轮的模空间来表述问题。另一个目标是研究这些结果的算术应用。第二个项目（与 D. Chan 合作）是之前关于复杂射影代数曲面上最大阶滑轮的工作的延续。主要目标是对此类最大阶滑轮的各种类别进行分类。这项工作遵循非交换代数几何中最近发展的思想。第三个项目（与 A. Ram 联合）是研究是否可以沿着单级 Heckalgebras 的思路为有理 Cherednik 代数建立 Springer 对应关系。另一个目标是研究这些代数是否与 N. Wallach 处理 Springer 对应关系的方法有关。最后一个项目（与 M. Banagl 联合）是研究是否存在与与有理数定义的半简单代数群相关的局部对称空间的简化 Borel-Serre 紧化相交链滑轮兼容的自对偶滑轮。在过去的几十年里，数学和理论物理各个分支之间的相互作用已被证明特别丰富了所有领域。这些联系中的一个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。 代数几何是研究作为几何对象的多变量多项式方程的解的数学领域。 它不仅在数学中得到应用，而且在计算机科学、编码理论、机器人技术和物理学中的弦理论等领域都有应用。该提案的前两个项目使用该主题的工具来研究广泛的数学家和物理学家感兴趣的“代数”。将整个代数类描述为几何对象是有用的，该项目的目标就是获得这样的描述。 这项研究的第三个方面是称为表征理论的主题的一部分。这里的主要研究对象是“表示”，即对自然界中的对称性信息进行编码的过程。该提案的最后一部分处于三个主题的交叉点：数论（研究数系统）、表示论和拓扑。在拓扑空间中进行研究以确定哪些属性在弹性变形（例如扭曲和拉伸）下不会改变。 本研究中要研究的空间编码数论信息。目标是研究这些空间是否存在本身满足某种对称性的不变量。