Combinatorics on Partially Ordered Sets

9970957研究者计划继续她在部分有序集合上的组合学研究，在与代数、拓扑和几何关系的方向上进一步扩展。提出的工作来自于研究者对她在非交叉分区格和代数组合方面的工作之间的关系的日益认识。利用结构组合学和代数组合学的技术，她考虑了两个主要的研究方向：部分有序集上的组合统计的行列式和部分有序集的拓扑方面。这个项目的动机是令人信服的证据，时间是有利的，以寻求一个适当的统一的一些重要的独立问题。研究者在不同的数学领域——低维拓扑、几何、数学物理、组合学——的研究人员所获得的许多结果、证明和猜想中发现了惊人的相似性。这些问题构成了大量深刻而多样的问题，表明这些相似之处不仅仅是巧合。因此，总体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.