Mathematical Sciences: Topology, Arithmetic Groups and Toric Varieties

数学科学：拓扑、算术群和环面簇

• 批准号: 9704535
• 负责人: Weiping Li
• 金额: $ 7.8万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-15 至 2001-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704535&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Arithmetic; Groups; Mathematical; Sciences; Topology; Arithmetic; Groups

9704535 McConnell The project has three parts. First, let G = SL(n,R), let Gamma be an arithmetic subgroup of G, and let X be the symmetric space for G. The spaces X/Gamma are one setting where automorphic forms can be defined; they provide a topological approach to parts of the Langlands conjectures. For each Hecke operator T on the cohomology of X/Gamma, MacPherson and McConnell have defined a cell complex W(T) which allows one to find the operator by cellular techniques, i.e., using only a finite amount of combinatorial data. They will investigate W(T) and extend the definition to more G. In the second part, Ash and McConnell are computing the cohomology of X/Gamma in degree five for certain Gamma for SL(4). The goal is to determine the cuspidal classes and the Hecke action on them. The third part concerns Sheafhom, a suite of computer programs McConnell has developed. Sheafhom provides models of chain complexes, spectral sequences, sheaves, and other objects. The largest application to date is its algorithm for finding the intersection homology (IH) of toric varieties in any perversity. One goal is to compute the intersection product on IH of toric varieties, with applications to convex polytopes. Sheafhom has also been used in the Ash-McConnell work. A computer algebra system is a program for calculation with algebra, as opposed to numbers. Any program can find 2 + 2, but a computer algebra system combines whole formulas: the input (2x + 7) + (4x - 3) becomes 6x + 4 automatically. This capacity is more abstract, hence more flexible. Excellent general-purpose systems, like Maple or Mathematica, are available, but of course they don't have everything that every mathematician needs. In recent years, several disciplines have been given their own special-purpose systems-- Cayley/Magma for algebraists, Pari for number theorists, and a dozen others. McConnell has written Sheafhom, a computer algebra system for algebraic topology. The program, some 10,000 lines long, is written in Lisp, one of the liveliest (and most efficient) programming languages. The goal is to apply Sheafhom to study convex polytopes. Convex polyhedra are solid bodies with flat faces, like cubes, pyramids, or hundred-faced diamonds. A convex polytope is the same kind of body in the fourth or higher dimension. Since 1980, algebraic geometry has become a major tool for studying polytopes. This is surprising, because algebraic geometry includes some of the most abstract mathematics known, while polytopes, like crystals, are very concrete objects. Sheafhom will make possible some difficult computations in algebraic topology and geometry; these will advance our understanding of convex polytopes. McConnell's project actually has two other parts that are more loosely related to Sheafhom. These are connected with the Langlands conjectures, a very deep set of ideas that relates number theory to other, more geometric parts of mathematics. ***

9704535 McConnell该项目有三个部分。 首先，令G = SL（N，R），让Gamma为G的算术子组，让X为G。空格X/Gamma是可以定义自动形式的设置；他们为兰兰兹猜想的一部分提供了拓扑方法。 对于每个Hecke操作员，X/Gamma的共同体学，MacPherson和McConnell都定义了一个细胞复合物W（t），该元件允许通过细胞技术找到操作员，即仅使用有限量的组合数据。 他们将研究W（t）并将定义扩展到更多G。在第二部分中，Ash和McConnell正在计算SL（4）的某些Gamma中X/Gamma的X/Gamma的共同体学。 目的是确定尖锐的类别和对它们的Hecke行动。 第三部分涉及Sheafhom，麦康奈尔已经开发了一系列计算机程序。 Sheafhom提供了链复合物，光谱序列，滑轮和其他物体的模型。 迄今为止，最大的应用是其在任何变态中查找感谢您的曲折品种的相交同源性（IH）的算法。 一个目标是计算在复曲面品种的IH上的相交产品，并应用于凸多属的应用。 Sheafhom也已用于Ash-McConnell工作。 计算机代数系统是用于用代数计算的程序，而不是数字。 任何程序都可以找到2 + 2，但是计算机代数系统结合了整个公式：输入（2x + 7） +（4x -3）自动变为6x + 4。 这种能力更抽象，因此更灵活。 出色的通用系统，例如Maple或Mathematica，但是当然，它们没有每个数学家需要的一切。 近年来，已经为几个学科提供了自己的特殊用途系统 - 代数主义者的Cayley/Magma，数字理论家的Pari和其他十二个学科。 麦康奈尔（McConnell）写了Sheafhom，这是一种用于代数拓扑的计算机代数系统。 该程序长约10,000行，用LISP编写，是最活泼（也是最有效）的编程语言之一。 目的是应用sheafhom来研究凸多属。 凸多面体是固体，具有平坦的脸，例如立方体，金字塔或一百个面孔的钻石。 在第四或更高维度中，凸多属是相同的物体。 自1980年以来，代数几何形状已成为研究多面体的主要工具。 这是令人惊讶的，因为代数几何形状包括已知的一些最抽象的数学，而晶体（如晶体）是非常具体的对象。 Sheafhom将使代数拓扑和几何形状在代数拓扑和几何形状中进行一些困难的计算；这些将促进我们对凸多属的理解。 McConnell的项目实际上还有其他两个与Sheafhom更松散的部分。 这些与Langlands的猜想有关，这是一个非常深刻的思想，将数字理论与数学的其他更多几何部分相关联。 ***