Collaborative Research: The Role of Curvature in Combinatorics

近年来，来自几何的思想推动了组合学中一些最令人兴奋的发展，如Gromov双曲群和CAT(0)空间、组合Morse理论、组合Ricci曲率、组合微分流形和矩阵束。几何中最重要的统一概念是曲率。现在，通过这些不同的几何和组合理论，曲率正在成为组合学中一个强大的工具和基本的统一概念。这个重点研究小组将探索一些特定的组合曲率的概念，推动当前的组合工作，以及曲率作为组合学本身连贯几何视觉的基础的作用。一个多世纪以来，曲率的概念一直是几何和物理中统一的大概念之一。例如，我们对曲率的现代理解的鼻祖高斯表明，欧几里德几何与其他几何的区别在于，它是一个零曲率空间的几何。作为一项应用，他表明，正是地球表面的曲率使得（在一张平坦的纸上）绘制地球表面的地图不可能准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度的光滑空间，爱因斯坦观察到黎曼几何正是描述他的广义相对论（宇宙的曲率是引力的结果）的正确设置。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。粗略地说，组合学是研究可以用有限数量的信息来描述的对象。这正是计算机所能做的数学。这种类型的数学似乎与高斯、黎曼和无数其他人的几何研究相去甚远。然而，有越来越多的组合现象可以被看作是光滑空间曲率事实的有限类似物。这一建议的目标是将曲率作为一个组合概念得到一致的理解。此外，汇集了来自不同数学学科的研究人员，我们希望以曲率作为统一的主题，弥合几何、组合学、代数和拓扑之间的鸿沟。

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.