Combinatorics on Partially Ordered Sets

9970957调查员计划继续她的研究在组合学的偏序集，进一步扩大它的方向与代数，拓扑和几何的关系。 拟议中的工作出现从调查员的认识增加之间的关系，她的工作格的非交叉分区和方面的代数组合。 利用结构和代数组合学的技术，她认为调查的两条主线：从组合统计的偏序集和拓扑方面的偏序sets.This项目的动机是令人信服的证据表明，时间是有利的寻求一个适当的统一的一些重要的独立问题。 调查人员遇到了惊人的相似之处之间的一些结果，证明，并获得了各种研究人员在不同领域的数学工作-低维拓扑，几何，数学物理，组合数学。 这些构成了一个深刻而多样的问题的临界质量，表明相似之处不仅仅是巧合。 因此，首要目标是在理解这些问题背后的组合学及其相似性的原因方面取得进展。

9970957The investigator plans to continue her research in combinatorics on partially ordered sets, expanding it further in the direction of relations with algebra, topology, and geometry. The proposed work emerges from the investigator's increased awareness of relations between her work on the lattice of noncrossing partitions and aspects of algebraic combinatorics. Using techniques of structural and algebraic combinatorics, she considers two main lines of investigation: determinants from combinatorial statistics on partially ordered sets and topological aspects of partially ordered sets.This project is motivated by compelling evidence that the time is propitious for seeking a suitable unification of a number of significant independent problems. The investigator encountered striking similarities among a number of results, proofs, and conjectures obtained by a variety of researchers working in different areas of mathematics -- low-dimensional topology, geometry, mathematical physics, combinatorics. These constitute a critical mass of deep and diverse problems, indicating that the similarities are not mere coincidences. Thus, and overarching goal is to make progress toward understanding the combinatorics underlying these problems and the reasons for their similarity.