Graphs and Patterns in Mathematics and Theoretical Physics

摘要奖：DMS-0107455首席研究员：Mikhail Lyubich 2001年在纽约州立大学石溪分校举行的数学和理论物理图形与模式国际会议将汇集数学和理论物理中以图形为核心的几个分支的研究人员，希望鼓励这些领域之间富有成效的互动。这次会议是为了庆祝丹尼斯·沙利文的60岁生日，并讨论了沙利文最近感兴趣的一些话题。要强调的主题包括(1)图和代数，(2)图和离散黎曼几何和离散规范理论，(3)图和动力系统中的分叉模式，(4)图和量子场论和拓扑学。关于会议主题的迷你课程将面向研究生和初级研究人员。在这次会议的标题中使用的图形是由一些“顶点”组成的数据结构，通常画成点，由“边”连接，通常画成从一个顶点开始到另一个顶点结束的曲线。这种图是一种常见的组合结构，在交通运输和调度问题(例如，考虑道路或电话线的网络)中每天都会遇到，但在许多不明显的方面很有用。最近图形在几何和物理理论中的一些应用源于理查德·费曼的著名想法，即用图形来描述量子理论的部分内容，这些图形代表了一个粒子在时空中的可能寿命。费曼关于量子理论量可以代表全部可能历史的总和的概念，从一个大型的、连续的计算简化为由不同的可能的费曼图索引的总和。以类似的方式，现代几何学家和物理学家正在探索通过简化为处理一族图形来处理特定空间上所有可能的几何结构的非常大的集合的方法。

AbstractAward: DMS-0107455Principal Investigator: Mikhail LyubichThe International Conference on Graphs and Patterns inMathematics and Theoretical Physics at SUNY Stone Brook in 2001will bring together researchers in several branches ofmathematics and theoretical physics in which graphs play acentral role, hoping to encourage fruitful interactions betweenthese fields. This conference is a celebration of DennisSullivan's sixtieth birthday and addresses some of the topics ofrecent interest to Sullivan. Themes to be emphasized include (1)graphs and algebra, (2) graphs and discrete Riemannian geometryand discrete gauge theory, (3) graphs and bifurcation patterns indynamical systems, (4) graphs and quantum field theory andtopology. Minicourses on conference topics will be pointedtowards graduate students and junior researchers.A graph in the sense used in the title of this conference is adata structure consisting of some number of "vertices," usuallydrawn as dots, joined by "edges," usually drawn as curvesbeginning at one vertex and ending at another. This kind ofgraph is a common combinatorial structure, encountered every dayin problems of transportation and scheduling (consider a networkof roads or telephone lines, for example), but useful in manyless obvious ways. Some of the recent uses of graphs in geometryand physical theory grow out of Richard Feynman's celebrated ideaof describing parts of quantum theory with graphs that representpossible lifetimes for a particle in space-time. Feynman'snotion that a quantum theoretic quantity may represent a sum overall possible histories is reduced from a large, continuouscalculation to a summation indexed by the different possibleFeynman graphs. In a similar manner, modern geometers andphysicists are exploring ways to work with the very largecollections of all possible geometric structures on a particularspace by reducing to manipulations of a family of graphs.