Groups, Lattices and Geometry

Princeton University, Princeton NJ

Investigators

Abstract

Abstract The main aim of the project is to describe, classify, enumerate and name the interesting discrete groups in low dimensional spaces, in as uniform a manner as possible. My "orbifold symbol" does this very effectively for those that act in 2-dimensional spherical, Euclidean or hyperbolic spaces, and as part of the project I am writing a book on this theory with Heidi Burgiel and Chaim Goodman-Strauss. This has been extended to the 3-dimensional spacegroups in a paper soon to be published by Conway, Delgado, Huson and Thurston. A book on quaternions and octonions that I am writing with Derek Smith will contain chapters on the subgroups of GO(4), and joint works with Marc LaFortune and Frank Swenton will be concerned with the classification and enumeration of knots and 3-manifolds via the associated discrete groups in hyperbolic 3-space. The symmetries that objects can have are of great interest to many different communities both inside mathematics (number theorists, geometers, group theorists, Lie algebraists and topologists) and out (physicists, crystallographers, and artists). This has had the unfortunate effect that often the same object is known under different names to different people. Moreover, the standard treatments are ignorant of the recent great developments in the mathematical theory. I aim to change all this by producing definitive classifications and systematic names for all the symmetry types of low-dimensional objects, and writing books and papers on the theory, addressed to wide audiences. I believe this will have a great educational effect, and as evidence can point to that of my previous books "Sphere Packings, Lattices and Groups" (with Neil Sloane) and "An Atlas of Finite Groups" (with several co-authors) that have become standard works by performing similar services for other structures.