Geometry - Groups and Lattices

该奖项资助约翰·康威教授对结、群和二次型的研究。结与3流形密切相关，引起了越来越多的科学家的兴趣。康威有两个学生正在用他的新命名系统对它们进行计算研究。其中一名学生正在尝试使用Thurstonıs几何技术证明单纯形结的分类证明，另一名学生计划将类似的想法应用于3-主褶皱的计算研究。(2)群体，尤其是怪物群体，形成了第二个研究主题。康威计划使用一种新的简化结构，为“怪物”和其他大型群体开发实用的计算算法。Conwayıs学生克里斯·西蒙斯正在用另一种技术（双曲反射）研究怪物，他们计划将这两种技术结合起来。(3)格与二次型密切相关。Conwayıs学生W. Schneeberger正在证明他们的猜想，即表示1到290的整数正定二次型可以表示所有的数。康威打算继续他和斯隆过去十年来一直在做的工作（主要是几何和代数方面的工作）。关于(1)和(3)的建议工作应该产生对这些领域的许多工作者有很大价值的信息表。关于(2)的工作更具有推测性，但也更具有根本性。这个项目涉及代数、数论和结论的研究。代数可以被认为是对抽象对称的研究。因此，代数在物理和化学领域有直接的应用。特别是，现代物理规范场理论广泛地使用了代数。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。在过去的半个世纪里，它已经成为数据传输、数据处理和通信系统等各种应用领域不可或缺的工具。结理论研究结的数学理论。数学的这一分支在生物学和化学中都有应用。特别是，生物学家和数学家们已经联合起来了解是什么类型的结使DNA如此容易复制。

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.