Combinatorial Models for Manifolds and Vector Bundles

这个项目主要研究流形和向量丛的两种组合模型：Gelfand和MacPherson的组合微分流形和拟阵流形理论以及Mnev和Mani的局部多边形流形理论。主要的焦点是丛理论和每个模型中的空间分类。在这两个模型中，集合论的视角都指向拓扑学和组合学之间深刻而意想不到的新关系：本研究的主要目的是扩展这些关系。该项目的目标既包括在拓扑学中的组合应用(例如，拓扑不变量的组合公式)，也包括在组合学中的拓扑学结果(特别是在定向拟阵和多面体的偏序集的拓扑上)。Gelfand和MacPherson理论的分类空间源于某些定向拟阵的偏序集，并且与真实的Grassman空间有密切的联系。这项研究的目的之一是更深入地了解组合Grassman的拓扑，并将这些知识应用于实向量丛和各种组合问题。Mnev和Mani的理论似乎也引起了拓扑学和组合学中感兴趣的分类空间的出现，将PL流形和凸多面体联系在一起。作者还将研究定向拟阵中的一种新的“非欧几里德”病理及其对组合微分流形的影响。这些拓扑学组合模型的大部分力量来自于将拓扑学概念转化为组合类比的优雅。例如，人们可以通过简单地将拓扑式的每个元素转化为其组合等价物来推导出拓扑不变式的组合公式。从组合学到拓扑学的转换已被证明同样有用：这些模型提供了一个新的框架，在其中观察各种组合主题。这两个模型都揭示了对各种组合问题的矢丛观点，这些问题最初似乎与实矢丛没有联系，导致了意想不到的拓扑证明。这个项目将建立在这一领域的最新成果的基础上，该领域的深度在于将数学的组合方面(植根于计算事物)与拓扑侧(植根于可塑性和空间连续性)相结合。

9803615Anderson This project centers on two combinatorial models for manifolds andvector bundles: Gelfand and MacPherson's theory of combinatorial differentialmanifolds and matroid bundles and Mnev and Mani's theory of locally polytopalmanifolds. The primary focus is on the bundle theory and classifying spaceswithin each model. In both of these models, a bundle-theoretic perspectiveleads to deep and unexpected new relations between topology and combinatorics:the primary objective of this research is to expand on these relations. Theproject's goals include both combinatorial applications in topology (forinstance, combinatorial formulas for topological invariants) and topologicalresults in combinatorics (particularly on the topology of posets of orientedmatroids and polytopes). The classifying spaces for Gelfand and MacPherson'stheory derive from certain partially ordered sets of oriented matroids, andhave close ties to the real Grassmannian. Among the objectives of thisresearch are a closer knowledge of the topology of the combinatorialGrassmannian and applications of this knowledge both to real vector bundles and to various combinatorial problems. Mnev and Mani's theory also appearsto give rise to a classifying space of interest in both topology andcombinatorics, relating PL manifolds and convex polytopes. The author willalso study a new type of ``non-Euclidean'' pathology in oriented matroids andits implications for combinatorial differential manifolds. Much of the power of these combinatorial models for topology derivesfrom the elegance with which topological concepts translate into combinatorial analogs. One can, for instance, derive combinatorial formulas for topologicalinvariants by simply translating each element of a topological formula intoits combinatorial equivalent. The translation from combinatorics to topologyhas proven equally useful: these models have provided a new framework in whichto view various combinatorial topics. Both models have revealed avector-bundle perspective on various combinatorial problems that originallyappeared to have no connection to real vector bundles, leading to unexpected topological proofs. This project will build on recent results in this areawhose depth lies in uniting the combinatorial side of mathematics, rooted incounting things, to the topological side, rooted in plasticity and spatialcontinuity.***