Mathematical Sciences: Topology, Arithmetic Groups and Toric Varieties

小行星9704535 该项目有三个部分。 首先，设G = SL（n，R），Gamma是G的一个算术子群，X是G的对称空间. 空间X/Gamma是可以定义自守形式的一种设置;它们为朗兰兹拓扑的一部分提供了一种拓扑方法。 对于X/Gamma上同调的每个Hecke算子T，MacPherson和McConnell定义了一个胞腔复形W（T），它允许人们通过胞腔技术找到算子，即，只使用有限数量的组合数据。 他们将研究W（T），并将定义扩展到更多的G。 在第二部分中，Ash和McConnell计算了SL（4）中某些Gamma的X/Gamma的五次上同调。 我们的目标是确定尖点类和它们上的Hecke作用。 第三部分涉及Sheafhom，这是McConnell开发的一套计算机程序。 Sheafhom提供了链复合体、谱序列、层和其他对象的模型。 迄今为止最大的应用是它的算法，用于寻找任何反常的复曲面品种的交叉同源性（IH）。 一个目标是计算的交集产品的IH环面品种，与应用凸多面体。 Sheafhom也被用于Ash-McConnell的工作。 计算机代数系统是一个用代数计算的程序，而不是数字。 任何程序都可以找到2 + 2，但计算机代数系统会将整个公式组合起来：输入的（2x + 7）+（4x - 3）自动变成6x + 4。 这种能力更抽象，因此更灵活。 优秀的通用系统，如Maple或Mathematica，是可用的，但当然它们没有每个数学家需要的一切。 近年来，几个学科都有了自己的专用系统--代数学家的Cayley/Magma，数论学家的Pari，以及其他十几个学科。 麦康奈尔写了Sheafhom，一个代数拓扑的计算机代数系统。 这个程序大约有10,000行，是用Lisp编写的，Lisp是最简单（也是最有效）的编程语言之一。 我们的目标是应用Sheafhom来研究凸多面体。 凸多面体是具有平面的实体，如立方体、金字塔或百面金刚石。 凸多面体是第四维或更高维中的同类物体。 自1980年以来，代数几何已成为研究多面体的主要工具。 这是令人惊讶的，因为代数几何包括一些已知的最抽象的数学，而多面体，如晶体，是非常具体的对象。 Sheafhom将使代数拓扑和几何中的一些困难计算成为可能;这些将促进我们对凸多面体的理解。 麦康奈尔的项目实际上还有另外两个部分与Sheafhom的关系更为松散。 这些都与朗兰兹定理有关，朗兰兹定理是一套非常深刻的思想，它将数论与数学中其他更几何的部分联系起来。 ***

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. ***