Combinatorial Models for Manifolds and Vector Bundles

9803615安德森 这个项目的中心是流形和向量丛的两个组合模型：Gelfand和MacPherson的组合微分流形和拟阵丛理论和Mnev和Mani的局部多面体流形理论。 主要的重点是在捆绑理论和每个模型内的分类spaceswithin。 在这两个模型中，一个斗争理论的视角，导致深刻的和意想不到的拓扑学和组合学之间的新关系：本研究的主要目标是扩大这些关系。 该项目的目标包括拓扑学中的组合应用（例如，拓扑不变量的组合公式）和组合学中的拓扑结果（特别是有向拟阵和多面体的偏序集的拓扑）。 Gelfand和MacPherson理论的分类空间来源于定向拟阵的某些偏序集，与真实的格拉斯曼有着密切的联系。 本研究的目的是进一步了解组合格拉斯曼的拓扑结构，并将其应用于真实的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的兴趣，涉及PL流形和凸多面体的分类空间。 作者还将研究定向拟阵中的一种新的“非欧”病态及其对组合微分流形的影响。 这些拓扑学的组合模型的力量，很大程度上来自于将拓扑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公式的每个元素转换成其组合等价物来推导拓扑不变量的组合公式。从组合学到拓扑学的转换也被证明是同样有用的：这些模型提供了一个新的框架，在这个框架中可以查看各种组合主题。 这两个模型都揭示了矢量束的角度来看，各种组合问题，最初似乎没有连接到真实的矢量束，导致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之上，其深度在于将数学的组合方面（植根于计算事物）与拓扑方面（植根于可塑性和空间连续性）结合起来。

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.***