Geometry - Groups and Lattices

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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.