Graphs and Polynomials

许多现实世界中数学应用的基础是图，它是由对象(顶点)组成的模型，以及指示对象之间关系的有序或无序对(边)。计算机和社交网络、存储设施和调度都会产生图，对于这些图，突出问题(无论是由连通性还是资源分配组成)可以用数学术语重新表述为底层图的性质。*对于许多这样的问题，模型包括关联函数，这些函数后来被证明是最简单的类型，即多项式。网络可靠性衡量网络的稳健性，假设顶点总是在工作，而边以固定的概率独立运行。色多项式计算正确给图的顶点上色的方法的数量，以便由边连接的顶点(表示某种形式的不相容)以不同的颜色上色。此外，有时研究与图的性质(如独立或团)有关的数字序列的最好方法是形成所谓的生成多项式并研究后者的数学性质。*在我的研究计划中，将研究所有这类图多项式的代数性质和解析性质，以便更深入地了解这些图性质的重要应用和理论基础。方法和工具将从各种数学领域(如实数和复数分析、代数和概率)发展而来，这些方法和工具也可以应用于组合结构构成基础的其他环境。多项式的零点将发挥重要作用，因为它们的位置可以产生关于函数的近似及其系数的形状(例如序列是否为单峰)的许多有用信息。经典的结果和新的技术(包括一元多项式和多元多项式，如Gauss，Schur，Hermite，Beraha，Kahane，Weiss，Chudnovsky和Seymour，Borcea和Branden的结果)将在研究中发挥重要作用。由于跨学科的数学方法论和计算理论之间的相互作用，预计将出现新的定界和逼近图形多项式的方法，其强度比以前已知的方法更强。此外，邻域复形的同调将为我们提供有关图的3-色性这一难题的信息。*这项研究不仅对研究未解决的图问题的理论家和对Potts模型中局部相互作用和全局行为之间的相互作用感兴趣的物理学家有用，而且对那些正在实现图着色算法并正在设计最优(或接近最优)网络的应用环境(调度和交通网络)中的人也是有用的。

Underlying many real-world applications of mathematics are graphs, which are models that consists of objects (vertices), and ordered or unordered pairs (edges) that indicate relationships between the objects. Computer and social networks, storage facilities, and scheduling all yield graphs for which the salient problems (whether they consist of connectivity or resource allocation) can be reformulated in mathematical terms, as properties of the underlying graphs.******For a number of these problems, the models include associated functions, which turn out to be of the simplest kind, namely polynomials. Network reliability measures the robustness of a network, under the assumption that vertices are always working but the edges operate independently with a fixed probability. Chromatic polynomials count the number of ways to properly colour the vertices of a graph so that vertices joined by an edge (indicating some form of incompatibility) are coloured differently. Moreover, sometimes the best way to study sequences of numbers that relate to a property of graphs (such as being independent or being a clique) is to form what is called a generating polynomial and to study mathematical properties of the latter. ******In my research program, the algebraic and analytic properties of all such graph polynomials will be investigated, in order to get a deeper understanding of both the important applications at hand, and the theoretical underpinnings of the graph properties in question. Methods and tools will be developed from a variety of areas of mathematics (such as real and complex analysis, algebra and probability) that can also be applied in other settings where combinatorial structures form the basis. Zeros of polynomials will play a prominent role, as their location can produce much useful information about approximations of the functions and the shape of their coefficients (such as whether the sequence is unimodal). Classical results and new techniques (both for univariate and multivariate polynomials, such as those of Gauss, Schur, Hermite, Beraha, Kahane, Weiss, Chudnovsky and Seymour, Borcea and Branden) will play an important role in the research. It is anticipated that novel methods for bounding and approximating the graph polynomials will arise, stronger than previously known approaches, from the interplay between the interdisciplinary mathematical methodology and computational theory. As well, homology of neighbourhood complexes will inform us on the difficult problem of 3-colourability of graphs.******The research will be useful not only to theoreticians who work on outstanding graph problems and physicists who are interested in the interplay between local interactions and global behavior in Potts models, but also to those in applied settings (scheduling and transportation networks) who are implementing algorithms for graph colourings and are designing optimal (or near optimal) networks.