Topological and Algebraic Combinatorics

主要研究员：Braun，Benjamin J.建议编号：DMS-0758321机构：肯塔基大学研究基金会标题：拓扑和代数组合PI研究拓扑和代数组合中的问题。Ehrhart多项式理论由于与交换代数、代数几何、组合数学、离散几何和凸几何以及数论的联系而受到数学界的广泛关注。PI研究Ehrhart多项式的根和系数，特别关注自反多面体。PI还研究关于图和偏序集同态复形的问题，继续代数拓扑学在组合学中的应用。PI研究了可能的同伦测试图，并研究了偏序集同态复形和偏序集序维度之间的关系。最后，在与Richard Ehrenborg的合作中，PI研究了由非凸多边形的三角剖分产生的联合面体的边界复形的单纯形子复形。历史上，数学一直由离散结构和连续结构之间的相互作用驱动。由组合学、代数和拓扑学相互作用而产生的当代问题延续了这一传统。对多面体的研究始于古代，起源于欧几里得的立体几何。埃尔哈特理论是从组合和算法的角度研究多面体的一种当代方法，它从一个具有整数顶点的多面体产生一个多项式，该多项式计算该多面体的积分膨胀中的格点。众所周知，这些多项式的根和系数携带着一些组合和几何数据，但关于这条线到底能走多远，还有许多悬而未决的问题。对图形的研究始于公元1700年的S和欧拉的调查，并已在大多数科学和工程领域找到了现代应用。通过转移到连续世界来研究图的色数，导致研究具有由所考虑的图的对称性产生的对称性的拓扑空间。到目前为止，这方面的调查取得了成功，仍有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.