Interactions between Algebra, Algebraic Geometry and Topology

在这个提议中，研究者和合作者研究代数、代数几何和拓扑学中的问题。其中三个项目研究在代数的各个分支中自然产生的非交换代数家族。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。研究这类代数的方法是用超曲面上某些自反层的模空间来研究这类问题。另一个目标是调查这些结果的算术应用。第二个项目（与D. Chan）是以前关于复射影代数曲面上的极大阶层的工作的继续。主要目标是分类的各种类的最大阶的这种层。这项工作遵循最近发展的思想在noncommutative algebraicgeometry。第三个项目（与A. Ram）的目的是研究有理Cherednik代数是否可以沿着单阶化Heck代数的思路建立Springer对应。另一个目标是研究这些代数是否能与N相关。瓦拉赫对斯普林格信件的处理方法。最后一个项目（与M. Banagl）的目的是研究与有理数上定义的半单代数群相关的局部对称空间的约化Borel-Serre紧化是否存在与交链层相容的自对偶层。在过去的几十年里，数学的各个分支和理论物理之间的相互作用已经被证明是特别丰富的所有领域。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。 代数几何是数学领域，研究解决方案的多变量多项式方程作为几何对象。 它不仅在数学中得到了应用，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几个受其影响的领域也得到了应用。该计划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”。将这类代数的整个族描述为一个几何对象是有用的，本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表征理论的一部分。这里的主要研究对象是“表征”，即对自然界中对称性的信息进行编码的过程。这一建议的最后一部分站在三个学科的交叉点：数论（研究数字系统），表示论和拓扑学。在拓扑学中，研究空间是为了确定哪些性质在弹性变形（如扭曲和拉伸）下不会改变. 本研究所研究的空间编码数论信息。目标是研究这些空间是否存在自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.