Collaborative Research: The Role of Curvature in Combinatorics

DMS-0101506乔纳森McCammond在最近几年的想法，从几何已经推动了一些最令人兴奋的发展，组合，如格罗莫夫双曲群和CAT（0）空间，组合莫尔斯理论，组合里奇曲率，组合微分流形和拟阵丛。几何学中的中心统一概念是曲率。现在，通过这些不同的几何和组合理论，曲率正在成为一个强大的工具和基本的统一概念，在组合学以及。 这个重点研究小组将探讨一些具体的概念combinatorialcurvature驱动当前的组合工作，也是曲率的作用，作为一个连贯的几何视觉组合本身的基础。曲率的概念一直是一个伟大的统一概念，在几何和物理超过一个世纪。 例如，高斯，我们的现代理解曲率的创始人，表明欧几里德几何是区别于其他几何作为几何空间的曲率为零。作为一个应用，他表明，正是地球表面的曲率使得不可能绘制地球表面的地图（在一张平的纸上）精确地描绘出所有的长度和角度。黎曼将高斯的工作推广到更高维的光滑空间，爱因斯坦观察到黎曼几何正是描述他的广义相对论的正确背景爱因斯坦观察到黎曼几何正是描述他的广义相对论的正确背景（宇宙的曲率是引力的结果）。部分由于爱因斯坦的工作，上个世纪人们对光滑空间的曲率进行了深入的研究。粗略地定义，组合数学是研究可以用有限的信息量来描述的对象。 这正是计算机可以做的数学。 这种类型的数学似乎远离几何调查高斯，黎曼和无数其他人。 然而，有越来越多的组合现象，可以最好地被视为是有限的类似物的事实曲率的光滑空间。 这个建议的目的是要达到一致的理解曲率作为一个组合的概念。此外，汇集来自各种数学学科的研究人员，我们希望弥合几何，组合学，代数和拓扑之间的鸿沟，使用曲率作为统一的主题。

DMS-0101506Jonathan McCammondIn recent years ideas from geometry have driven some of the most exciting developments in combinatorics such as Gromov hyperbolic groups and CAT(0) spaces, combinatorial Morse theory, combinatorial Ricci curvature, combinatorial differential manifolds and matroid bundles. The central unifying notion in geometry is that of curvature. Now, through these diversegeometric and combinatorial theories, curvature is emerging as a powerful tool and fundamental unifying concept in combinatorics as well. This Focused Research Group will explore some of the specific notions of combinatorialcurvature driving current combinatorial work, and also the role of curvature as the basis for a coherent geometric vision of combinatorics itself.The notion of curvature has been one of the grand unifying concepts in geometry and physics for well over a century. For example, Gauss, the originator of our modern understanding of curvature, showed that Euclidean geometry was distinguished from other geometries as being the geometry of a space with zero curvature. As an application he showed that it isprecisely the curvature of the surface of the Earth which makes it impossible to draw a map of the Earth's surface (on a flat piece of paper) that accurately portrays all lengths and angles.Riemann generalized Gauss's work to smooth spaces of higher dimensions, and Einstein observed that Riemannian geometrywas precisely the right setting in which to describe his theory of general relativity (in which the curvature of the universe is the result of gravitational forces). Partly as a result of Einstein's work, the last century saw an intensive investigation into the curvature of smooth spaces.Combinatorics, roughly defined, is the study of objects which can be described by a finite amount of information. This is precisely the mathematics that computers can do. This type of mathematics seems far removed from the geometric investigations of Gauss, Riemann and countless others. However, there is a growing collection of combinatorial phenomena which can best be viewed as being finite analogues of facts about the curvature of smooth spaces. The goal of this proposal is to come to acoherent understanding of curvature as a combinatorial notion. In addition, bringing together researchers from a variety of mathematical disciplines, we wish to bridge the chasms between geometry, combinatorics, algebra and topology, using curvature as the unifying theme.