Mathematical Sciences: Topology, Arithmetic Groups and Toric Varieties

9704535 McConnell项目由三部分组成。首先，设G = SL(n，R)，设是G的算术子群，设X是G的对称空间，其中X/是一个可以定义自同构形式的集合；它们为朗兰兹猜想的部分提供了一种拓扑方法。对于X/Gamma上同调上的每个Hecke算子T， MacPherson和McConnell定义了一个细胞复合体W(T)，它允许人们通过细胞技术找到算子，即仅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正在计算SL(4)中某些Gamma在5度的X/Gamma的上同调。目标是确定cuspyidal类和Hecke对它们的作用。第三部分涉及麦康奈尔开发的一套计算机程序Sheafhom。Sheafhom提供链络合物，光谱序列，束和其他对象的模型。迄今为止，最大的应用是它的算法，用于寻找任何反常的环变体的交同源性（IH）。一个目标是计算环型的交积，并将其应用于凸多面体。Sheafhom也被用于Ash-McConnell的工作。计算机代数系统是用代数而不是数字进行计算的程序。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，都是可用的，但它们当然不具备每个数学家所需的一切。近年来，一些学科被赋予了自己的专用系统——代数学家的Cayley/Magma，数论学家的Pari等等。McConnell编写了Sheafhom，一个用于代数拓扑的计算机代数系统。这个程序大约有1万行，是用最活泼（也是最高效）的编程语言之一Lisp编写的。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、金字塔或百面钻石。凸多面体是四维或更高维度的同一类物体。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，因为代数几何包含了一些已知的最抽象的数学，而多面体，如晶体，是非常具体的物体。Sheafhom将使代数拓扑和几何中的一些困难的计算成为可能；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有更松散的关系。这些都与朗兰兹猜想有关，朗兰兹猜想是一组非常深刻的思想，将数论与数学的其他几何部分联系起来。＊＊＊

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. ***