Graphs and Patterns in Mathematics and Theoretical Physics

AbstractAward：DMS-0107455首席研究员：米哈伊尔·柳比奇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和模式国际会议将汇集数学和理论物理几个分支的研究人员，在这些分支中，图形发挥着中心作用，希望鼓励这些领域之间富有成效的相互作用。 这次会议是庆祝丹尼斯沙利文的六十岁生日，并解决了一些最近感兴趣的沙利文的主题。 重点学科包括：（1）图与代数，（2）图与离散黎曼几何与离散规范理论，（3）图与动力系统中的分叉模式，（4）图与量子场论与拓扑学。 关于会议主题的迷你课程将面向研究生和初级研究人员。在这个会议的标题中使用的图形是由一些“顶点”组成的结构，通常画成点，由“边”连接，通常画成曲线，从一个顶点开始，到另一个顶点结束。 这种图是一种常见的组合结构，在运输和调度问题中每天都会遇到（例如，考虑道路网络或电话线），但在许多不太明显的方面很有用。 最近图形在几何和物理理论中的一些应用源于理查德·费曼的著名思想，即用代表时空中粒子可能寿命的图形来描述量子理论的一部分。 Feynman的量子理论量可以表示所有可能的历史的总和的概念，从一个大的，连续的计算减少到由不同的可能Feynman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如何通过简化到一族图的操作来处理特定空间上所有可能的几何结构的非常大的集合。

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.