Geometry and Topology in Mainstream Culture

主流文化中的几何和拓扑

• 批准号: 0822022
• 负责人: Jeffrey Weeks
• 金额: $ 14.78万
• 依托单位: Weeks Jeffrey R
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-10-01 至 2011-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0822022&HistoricalAwards=false
• 关键词: Geometry; Topology; Mainstream; Culture

Geometric topology was once a fringe topic, well studied in graduate schools but almost unknown in mainstream culture. Happily, in recent years public interest has increased. The key to effectively presenting geometric topology to a broad audience is to keep in mind that explanations using words alone are sure to fail. Explanations using pictures will convey a vague sense of the intended meaning. But to deeply understand the topology of a multi-connected space, the learner must experience the space for him or herself. One of the P.I.'s missions in life --- and the subject of the present project --- is to provide computer software that enables children and other non-specialists to experience 2-manifolds, 3-manifolds, and other structures directly. For example, the P.I.'s existing Torus Games have already proved effective with students from fifth grade on up, leading them to the mind-stretching concept of multiconnected spaces (specifically, the flat 2-torus and the flat Klein bottle). The present project aims to maintain this kid-friendly approach while extending the software's mathematical scope to include spherical and hyperbolic 2-manifolds as well as 3-manifolds.While mathematicians' and physicists' understanding of space has made tremendous progress, the general public's understanding still lags behind. Part of the reason is that, while the modern concept of space is not inherently difficult to understand, it is extremely difficult to explain in words. Explanations with pictures work a little better, but in practice they too usually prove inadequate. The only truly effective way for a student (or nonspecialist adult) to fully grasp a new concept of space is for the student to experience the new space for him or herself. The P.I.'s existing Torus Games software lets students develop a gut-level understanding of spaces that are finite yet have no boundaries; this software is widely used in middle and high schools and is effective with students from fifth grade on up. The present project aims to maintain this kid-friendly approach while extending the software's mathematical scope to include curved spaces as well as flat ones, and 3-dimensional spaces as well as "toy" 2-dimensional ones. Cosmologists are currently investigating the hypothesis that the real universe might be finite, but no matter whether that hypothesis is ultimately accepted or rejected, spaces of all sorts (flat and curved, finite and infinite, 2-dimensional and 3-dimensional) play a pervasive role throughout physics and mathematics, and merit a broader understanding in our culture as a whole.

几何拓扑学曾经是一个边缘话题，在研究生院中得到了很好的研究，但在主流文化中几乎不为人知。令人高兴的是，近年来公众的兴趣有所增加。要有效地向广大观众展示几何拓扑，关键是要记住，仅用文字解释肯定会失败。 使用图片的解释将传达一种模糊的意图。但要深入理解多连通空间的拓扑结构，学习者必须亲自体验空间。其中一个私家侦探的使命在生活中-和本项目的主题-是提供计算机软件，使儿童和其他非专业人员的经验，2流形，3流形，和其他结构直接。例如，P.I.现有的Torus游戏已经被证明对五年级以上的学生有效，引导他们认识到多连通空间的思维拉伸概念（特别是平坦的2-torus和平坦的Klein瓶）。本项目的目标是保持这种儿童友好的方法，同时将软件的数学范围扩展到包括球面和双曲2-流形以及3-流形。虽然数学家和物理学家对空间的理解已经取得了巨大的进步，但普通公众的理解仍然落后。 部分原因是，虽然现代空间概念本身并不难理解，但用语言解释却极其困难。用图片来解释会更好一些，但在实践中，它们通常也是不够的。 对于学生（或非专业的成年人）来说，完全掌握一个新的空间概念的唯一真正有效的方法是让学生亲自体验新的空间。 私家侦探现有的Torus Games软件可以让学生对有限但没有边界的空间进行深入的理解;该软件广泛用于初中和高中，对五年级以上的学生有效。 本项目旨在保持这种儿童友好的方法，同时扩展软件的数学范围，包括弯曲空间以及平面空间，三维空间以及“玩具”二维空间。 宇宙学家目前正在研究一个假设，即真实的宇宙可能是有限的，但无论这个假设最终被接受还是拒绝，各种空间（平坦和弯曲，有限和无限，二维和三维）在整个物理学和数学中发挥着普遍的作用，并且值得在我们的文化中得到更广泛的理解。