Combinatorics on Partially Ordered Sets

这位研究人员计划继续她在偏序集上的组合学研究，将其进一步扩展到与代数、拓扑和几何的关系的方向。这项拟议的工作源于研究人员对她在不相交划分格上的工作与代数组合学方面之间的关系的日益认识。利用结构和代数组合数学的方法，她考虑了两条主线：偏序集合上的组合统计的行列式和偏序集合的拓扑方面。这个项目的动机是令人信服的证据，即时间有利于寻求一些重要的独立问题的适当统一。研究人员发现，在低维拓扑学、几何学、数学物理学、组合学等不同数学领域的不同研究人员获得的许多结果、证明和猜想之间存在惊人的相似之处。这些构成了一系列深刻而多样的问题，表明这些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和它们相似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.