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.

Conway 9701444该奖项为John Conway教授的研究提供了针对结，团体和二次形式的研究。 （1）对各种科学家的兴趣越来越多，并且与3个策略密切相关。康威（Conway）有两个学生，他们根据他的新命名系统在计算上研究他们。其中一名学生试图使用Thurstonıs的几何技术来证明单纯结的分类证明，而其他计划将相似的想法应用于3点毛的计算研究。 （2）组，尤其是怪物组构成了第二个研究主题。康威计划使用新的和简化的构造来开发怪物和其他大型群体中计算的实用算法。 Conwayıs学生Chris Simons正在使用不同的技术（双曲反射）研究怪物，他们计划与这些技术结合。 （3）晶格和二次形式密切相关。 Conwayıs的学生W. Schneeberger正在努力证明他们的概念，即代表1至290数字的整数积极二次形式将代表所有数字。 Conway打算继续他和Sloane在过去十年中一直在做的工作（主要是几何和代数）。关于（1）和（3）的拟议工作应为这些领域的许多工人提供巨大价值的信息表。 （2）上的工作更具投机性，但更基本。该项目涉及代数，数理论和结理论的研究。代数可以作为抽象的对称性研究。因此，代数直接应用于物理和化学领域。特别是，物理学领域的现代理论广泛使用代数。数字理论在整个数字的研究中具有历史根源，解决了诸如处理一个整数的问题的问题。它是数学最古老的分支之一，由于纯粹的审美原因而被追捕了许多世纪。在过去的半个世纪中，在数据传输，数据处理和通信系统等领域的潜水员应用程序中，它已成为必不可少的工具。结理论涉及结的数学理论。该数学分支在生物学和化学中有应用。特别是，生物学家和数学家联合起来，了解哪种类型的打结使DNA可以如此轻松地复制。