Preface

Summary

This little book is based on the Earle Raymond Hedrick Lectures that I gave at the Joint Mathematics Meetings of the American Mathematical Society and the Mathematical Association of America in Orono, Maine on August 7–9, 1991. I have been interested in quadratic forms for many years, but keep on discovering new and simple ways to understand them. The “topograph” of the First Lecture makes the entire theory of binary quadratic forms so easy that we no longer need to think or prove theorems about these forms—just look! In some sense the experts already knew something like this picture—but why did they use it only in the analytic theory, rather than right from the start?

Mark Kac's famous problem “Can one hear the shape of a drum?” when applied to n-dimensional toroidal “drums” leads to the question of which properties of quadratic forms are determined by their representation numbers. What, in other words, do we know about a lattice when we are told exactly how many vectors it has of every possible length?

Since sight and hearing were now involved, I took as the theme of the lectures the idea that one should try to appreciate quadratic forms with all one's senses, and so arose the title “THE SENSUAL FORM” for my Hedrick Lectures, and also the topics for the first two of them.

I could not settle on a single topic for the third of these lectures, even when I came to give it.