Algebraic and topological combinatorics of posets

研究人员将开发和应用组合方法来研究组合学及相关领域中出现的单纯复合体和细胞复合体的拓扑结构。一个主要的焦点是处理比组合数学中的许多流行方法被设计来处理的更一般的拓扑类型的复合体，例如在证明连通性下界的新技术中。研究人员将继续她正在进行的工作，以开发构造具有较少临界单元的离散Morse函数的技术，重点是偏序集的序复和相关的自由分解(既用于单项理想，也用于解决单项式或环面上的剩余场)。实质性的改进，至少对于有序复合体来说，可能需要更好地理解支配关键细胞之间的梯度路径的非常丰富的结构。在相关工作中，她还计划研究Poincare级数在单项环和环面环上的自由分解，例如，试图更好地理解这样的Poincare级数将是有理的。她还打算研究(非几何)格中的模元，并试图将词典可剥离性推广到复杂结构的骨架上，这再次受到了构造小自由分辨率和有界图色数(通过更好地理解特征多项式)的潜在应用的启发。过去，拓扑组合学，特别是连通性下界，已经被用来确定和验证某些类型算法的运行时间的复杂性理论下界，通过自由分解理论推导出关于多项式之间关系的交换代数中的结果，并且还给出了为图的顶点着色所需的颜色数量的下界，使得共享一条边的两个顶点都不是相同的颜色。研究人员有兴趣进一步开发组合技术(例如最近推出的离散版本的莫尔斯理论)来研究拓扑结构，让潜在的应用为方向指引方向。莫尔斯理论是一个经典的理论，它通过从下到上逐步观察对象来分析对象的拓扑结构，在结构发生根本变化的时刻跟踪基本数据；最近，罗宾·福尔曼引入了这个看似内在连续的概念的离散版本。研究人员的主要关注点之一是开发实用的机器，使这个理论上非常强大的工具便于在真实(以及在许多情况下非常复杂)的例子中使用。

The investigator will develop and apply combinatorial methods for studying the topological structure of simplicial complexes and cell complexes arising in combinatorics and related fields. A major focus is to deal with complexes of more general topological type than many of the prevalent methods within combinatorics were designed to handle, e.g. in new techniques for proving connectivity lower bounds. The investigator will continue her ongoing effort to develop techniques for constructing discrete Morse functions with few critical cells, with emphasis on order complexes of partially ordered sets and on related free resolutions (both for monomial ideals, and also for resolving a residue field over a monomial or toric ring). Substantial improvement, at least for order complexes, will likely require better understanding of the very rich structure governing gradient paths between critical cells. In related work, she also plans to study Poincare' series for free resolutions over monomial and toric rings, for instance trying to better understand for toric rings which such Poincare' series will be rational. She also intends to study modular elements in (non-geometric) lattices and to try to generalize lexicographic shellability to skeleta of complexes, motivated again by potential applications to constructing small free resolutions and also to bounding graph chromatic number (via better understanding of characteristic polynomial). Topological combinatorics, and in particular connectivity lower bounds, have in the past been used to determine and verify complexity theory lower bounds on the running time for certain types of algorithms, to deduce results in commutative algebra about relations among polynomials via the theory of free resolutions, and also to give lower bounds on the number of colors needed to color the vertices of a graph in such a way that no two vertices sharing an edge are the same color. The investigator is interested in further developing combinatorial techniques (such as a recently introduced discrete version of Morse theory) for studying topological structure, letting potential applications guide the way. Morse theory is a classical theory which analyzes the topological structure of an object by viewing the object progressively from bottom to top, keeping track of essential data at those moments in time where fundamental changes in structure take place; recently Robin Forman introduced a discrete version of this seemingly inherently continuous notion. One of the investigator's major focuses is to develop to practical machinery for making this theoretically very powerful tool convenient to use on real (and in many cases very complex) examples.