Algebraic and topological combinatorics of posets

研究者将开发和应用组合方法来研究组合学和相关领域中出现的简单复合物和细胞复合物的拓扑结构。一个主要的焦点是处理比组合学中许多流行方法所设计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。研究人员将继续致力于开发具有少量关键单元的离散莫尔斯函数的构造技术，重点是部分有序集的有序复形和相关的自由分辨率（包括单项式理想，以及单项式或环上的剩余域的求解）。至少对于有序复合体来说，要有实质性的改进，可能需要更好地理解控制关键细胞之间梯度路径的非常丰富的结构。在相关工作中，她还计划研究庞加莱级数在单项式环和环上的自由分辨力，例如试图更好地理解对于环，庞加莱级数是否为有理。她还打算研究（非几何）格中的模块元素，并尝试将字典学的贝壳性推广到复合体的骨架，这再次受到构建小自由分辨率和边界图色数（通过更好地理解特征多项式）的潜在应用的激励。拓扑组合,特别是连接下界,过去用来确定并验证复杂性理论下界在某些类型的算法的运行时间,推断结果在交换代数多项式之间的关系通过自由理论的决议,也给下界的数量颜色需要一个图的顶点以这样一种方式,没有两个顶点共享优势是相同的颜色。研究者有兴趣进一步开发用于研究拓扑结构的组合技术（例如最近引入的摩尔斯理论的离散版本），让潜在的应用引导方向。莫尔斯理论是一种经典理论，它通过从下到上逐级观察物体，在结构发生根本变化的时刻记录基本数据来分析物体的拓扑结构；最近，罗宾·福尔曼（Robin Forman）引入了这个看似内在连续概念的离散版本。研究人员的主要重点之一是开发实用的机制，使这个理论上非常强大的工具方便地用于实际（在许多情况下非常复杂）的例子。

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.