Mathematical Sciences: Cohomology of Sheaves, and Applications

9401460麦康奈尔这个项目一开始就涉及一个新的计算机代数系统Sheafhom的升级，用于研究拓扑空间上的派生的轮子类别。在Sheafhom中，基本项是有限维向量空间、余链复合体、正则胞胞复合体、层和这些对象之间的态射。有核、像、上同调、张量和楔积等函数。该系统适用于重要的滑轮复合体，如相交的上同调滑轮和其他倒置滑轮。其次，该项目涉及将Sheafhom应用于凸多面体和环状簇的交同调。麦克马伦猜想刻画了单凸多面体的f向量。斯坦利对麦克马伦猜想(单向)的证明使用了与单纯多面体相关的环簇的上同调。关键事实是模为Lefschetz类的上同调是一个生成于二次的分次环。对于任意有理凸多面体，人们可以问它的模为Lefschetz类的环面簇的交上同调是否是环，以及它是否生成于二次。该项目将以这些问题的初步结果为基础，Sheafhom将在解决这些问题方面发挥作用。计算机代数系统是用于代数计算的程序。任何程序都可以加法，但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。像Maple或数学这样优秀的通用系统是可用的，但当然它们并不具备每个数学家所需要的所有功能。近年来，几个学科已经建立了自己的专用系统--代数学家的Cayley/Magma，数论学者的Pari，以及其他十几个学科。研究人员打算开发一个用于代数拓扑学的计算机代数系统Sheafhom。其次，他打算将Sheafhom应用于研究凸多面体。凸多面体是人们熟悉的实体，如立方体、金字塔或百面钻石。凸多面体是第四维或更高维中的同一种物体。自1980年以来，代数几何已成为研究多面体的主要工具。这很令人惊讶，因为代数几何包含了一些已知的最抽象的数学，而多面体就像水晶一样，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，这将促进我们对凸多面体的理解。***

9401460 McConnell This project involves at the outset the upgrading of a new computer algebra system, Sheafhom, for research in the derived category of sheaves on a topological space. In Sheafhom the basic items are finite-dimensional vector spaces, cochain complexes, regular cell complexes, sheaves, and morphisms among these objects. There are functions like kernel, image, cohomology, and tensor and wedge products. The system works with important complexes of sheaves, such as the intersection cohomology sheaves and other perverse sheaves. Second, the project involves bringing Sheafhom to bear on convex polytopes and the intersection homology of toric varieties. McMullen's conjecture characterized the f-vectors of simplicial convex polytopes. Stanley's proof of (one direction of) McMullen's conjecture used the cohomology of the toric variety associated to a simplicial polytope. Key facts were that the cohomology, modulo the Lefschetz class, is a graded ring generated in degree two. For an arbitrary rational convex polytope, one can ask whether the intersection cohomology of its toric variety, modulo the Lefschetz class, is a ring, and whether it is generated in degree two. The project will build on preliminary results on these questions, and Sheafhom will play a part in approaching them. A computer algebra system is a program for algebraic calculation. Any program can add numbers, but a computer algebra system combines whole formulas: the input (2x + 7) + (4x - 3) automatically becomes 6x + 4. 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 every function that every mathematician needs. In recent years several disciplines have acquired their own special-purpose systems--Cayley/Magma for algebraists, Pari for number theorists, and a dozen others. The investigator intends to develop Sheafhom, a computer algebra system for algebra ic topology. Second, he intends to apply Sheafhom to study convex polytopes. Convex polyhedra are familiar solid bodies like cubes, pyramids, or hundred-faced diamonds. A convex polytope is the same kind of body in the fourth dimension or higher. Since 1980, algebraic geometry has become a major tool for studying polytopes. This is surprising, since 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 that will advance our understanding of convex polytopes. ***