Collaborative Research: The Role of Curvature in Combinatorics

近年来，来自几何的思想推动了组合数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、组合微分流形和拟阵丛。几何中的中心统一概念是曲率概念。现在，通过这些不同的几何和组合理论，曲率也成为组合数学中的一个强有力的工具和基本的统一概念。这个重点研究小组将探索组合曲率驱动当前组合工作的一些具体概念，以及曲率作为组合数学本身连贯几何愿景的基础的作用。一个多世纪以来，曲率概念一直是几何学和物理学中重要的统一概念之一。例如，我们现代理解曲率的创始人高斯表明，欧几里得几何与其他几何不同，因为它是零曲率空间的几何。作为一个应用，他证明了正是地球表面的曲率使得(在一张扁平的纸上)不可能绘制出准确地描绘所有长度和角度的地球表面地图。里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恰恰是描述他的广义相对论(其中宇宙的曲率是引力的结果)的正确背景。部分由于爱因斯坦的工作，上个世纪见证了对光滑空间曲率的深入研究。组合学，大致定义为研究可以用有限数量的信息来描述的对象。这正是计算机可以完成的数学运算。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，最好地被视为关于光滑空间的曲率的事实的有限类似。这一提议的目的是将曲率作为一个组合概念达成一致的理解。此外，将来自不同数学学科的研究人员聚集在一起，我们希望以曲率为统一主题，弥合几何、组合学、代数和拓扑学之间的鸿沟。

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.