Topological and Algebraic Combinatorics

摘要主要研究者：Braun，Benjamin J.提案编号：DMS -0758321机构：肯塔基州大学研究基金会标题：拓扑和代数组合学PI研究拓扑和代数组合学中的问题。 埃赫哈特多项式的理论是广泛的兴趣，数学界由于与交换代数，代数几何，组合数学，离散和凸几何，数论。 PI研究Ehrhart多项式的根和系数，特别关注自反多面体。 PI还研究了关于图和偏序集同态复体的问题，继续将代数拓扑应用于组合学。 PI研究可能的同伦测试图，并调查偏序集同态复形和偏序集序维数之间的联系。 最后，在与理查德·埃克林伯格的合作中，PI研究了由非凸多边形三角剖分产生的协多面体边界复形的单纯子复形。数学在历史上一直受到离散和连续结构之间相互作用的驱动。 当代的问题所产生的相互作用的组合学，代数和拓扑继续这一传统。 多面体的研究始于古代，起源于欧几里得的立体几何。 Ehrhart理论是从组合和代数的角度研究多面体的当代方法，从具有整数顶点的多面体产生多项式，该多项式计数该多面体的整数扩张中的格点。 已知这些多项式的根和系数带有一些组合和几何数据，但关于这条研究路线到底能走多远，还有许多悬而未决的问题。 对图的研究始于1700年的欧拉调查，并在现代科学和工程的大多数领域中找到了应用。 研究图的色数移动到连续世界导致调查的拓扑空间的对称性所产生的对称性的图正在考虑。 迄今为止，这方面的调查取得了成功，但仍存在许多悬而未决的问题。

ABSTRACTPrincipal Investigator: Braun, Benjamin J.Proposal Number: DMS - 0758321Institution: University of Kentucky Research FoundationTitle: Topological and Algebraic CombinatoricsThe PI investigates problems in topological and algebraic combinatorics. The theory of Ehrhart polynomials is of broad interest to the mathematics community due to connections with commutative algebra, algebraic geometry, combinatorics, discrete and convex geometry, and number theory. The PI studies roots and coefficients of Ehrhart polynomials, with particular focus on reflexive polytopes. The PI also investigates problems regarding graph and poset homomorphism complexes, continuing the application of algebraic topology to combinatorics. The PI studies possible homotopy test graphs and investigates connections between poset homomorphism complexes and poset order dimension. Finally, in joint work with Richard Ehrenborg, the PI studies simplicial subcomplexes of the boundary complexes of associahedra arising from triangulations of non-convex polygons.Mathematics has historically been driven by the interplay between discrete and continuous structures. Contemporary problems arising from the interaction of combinatorics, algebra, and topology continue this tradition. The study of polytopes began in antiquity, with roots in the solid geometry of Euclid. Ehrhart theory is a contemporary approach to studying polytopes from a combinatorial and algbraic perspective, producing from a polytope with integer vertices a polynomial counting lattice points in integral dilates of that polytope. The roots and coefficients of these polynomials are known to carry some combinatorial and geometric data, but there are many open questions about exactly how far this line of investigation can be taken. The study of graphs began in the 1700's with investigations by Euler, and has found modern day applications in most areas of science and engineering. Studying chromatic numbers of graphs by moving to the continuous world leads to investigations of topological spaces with symmetries arising from symmetries of the graphs under consideration. Investigations in this direction have been successful so far, and many open questions remain.