Geometry - Groups and Lattices

几何 - 群和格

• 批准号: 9701444
• 负责人: John Conway
• 金额: $ 26.1万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-15 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9701444&HistoricalAwards=false
• 关键词: Geometry; Groups; Lattices

Conway 9701444 This award funds the research of Professor John Conway on knots, groups, and quadratic forms. (1) Knots are of increasing interest to a wide variety of scientists, and are intimately connected with 3-manifolds. Conway has two students who are studying them computationally in terms of his new naming system. One of the students is trying to prove a classification proof for simplex knots using Thurstonıs geometrical techniques, and the other plans to apply similar ideas to the computational study of 3-mainfolds. (2) Groups, and the Monster Group , in particular, form the second topic of study. Conway plans to develop practical algorithms for computing in the Monster and other large groups using a new and simplified construction for it. Conwayıs student Chris Simons is studying the Monster using a different technique (hyperbolic reflections), and they plan to marry the techniques. (3) Lattices and Quadratic Forms are closely related. Conwayıs student W. Schneeberger is working on a proof of their conjecture that an integer positive-definite quadratic form that represents the numbers from 1 to 290 will represent all numbers. Conway intends to continue the work (mostly geometrical and algebraic) that he and Sloane have been doing over the last decade. The proposed work on (1) and (3) should produce tables of information of great value to many workers in those fields. The work on (2) is more speculative but more fundamental. This project involves research in Algebra, Number Theory, and Knot Theory. Algebra can be though of as the study of symmetry in the abstract. As such, Algebra has direct applications to areas of physics and chemistry. In particular, the modern theory of gauge fields in physics uses algebra extensively. Number Theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. Within the last half century, it has become indispensable tools in diverse applications in areas such as data transmission, data processing, and communication systems. Knot Theory deals with the mathematical theory of knots. This branch of mathematics has applications in biology and chemistry. In particular, biologists and mathematicians have joined forces to understand what type of knotting allows DNA to replicate so easily.

康威 9701444 该奖项资助约翰·康威教授对纽结、群和二次形式的研究 (1) 纽结越来越引起众多科学家的兴趣，并且与 3 流形密切相关。康威有两个学生。一名学生正在尝试使用瑟斯顿的几何技术来证明单纯形结的分类证明，而另一名学生则计划将类似的想法应用到他的新命名系统中。 3-mainfolds（2）群，特别是 Monster Group 的计算研究，构成了第二个研究主题，计划使用新的简化的构造算法开发用于 Monster 和其他大型群计算的实用方法。康威的学生克里斯·西蒙斯正在使用一种不同的技术（双曲线反射）研究怪物，他们计划将这些技术结合起来（3）。康威的学生 W. Schneeberger 密切相关。正在证明他们的猜想：表示 1 到 290 之间的整数正定二次形式将表示所有数字。康威打算继续他和斯隆一直在做的工作（主要是几何和代数）。关于（1）和（3）的拟议工作应该会产生对这些领域的许多工作者具有重要价值的信息表。关于（2）的工作更具推测性，但更基础。代数、数论和结论可以被认为是抽象的对称性研究，因此，代数可以直接应用于物理和化学领域，特别是现代物理学中的规范场理论。数论的历史根源在于对整数的研究，解决诸如整数能否整除之类的问题。它是数学最古老的分支之一，几个世纪以来一直被人们追求纯粹的美学。后半段内的原因。世纪以来，它已成为数据传输、数据处理和通信系统等领域中不可或缺的工具。结理论涉及结的数学理论，特别是生物学和化学领域。数学家们联手研究什么类型的打结可以让 DNA 如此轻松地复制。