Coxeter groups are of central importance in several areas of algebra, geometry, and
combinatorics. This clear and rigorous exposition focuses on the combinatorial aspects of …

Isaac Newton encrypted his discoveries in analysis in the form of an anagram, which
deciphers to the sentence``It is worthwhile to solve differential equations''. Accordingly, one …

This book began as a set of lecture notes for a postgraduate course at the University of
Sydney in 1982. The aim was to start with vector spaces, introduce sesquilinear forms, and …

If a point is picked at random inside a regular simplex, octahedron, 600-cell, or other
polytope, what is its average squared distance from the centroid? In n-dimensional space …

A unitary reflection is a linear transformation of a complex vector space that fixes each point
in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is …

Moonshine forms a way of explaining the mysterious connection between the monster finite
group and modular functions from classical number theory. The theory has evolved to …

These lecture notes for the IAS/Park City Graduate Summer School in Geometric
Combinatorics (July 2004) provide an overview of root systems, generalized associahedra …

The first half of this article is expository; it contains a brief survey of the famous ADE
classification, and how it applies to six kinds of objects, some old and some relatively new …

This essay tells of a newly discovered connection among root systems, graphs, and
matroids. Root systems are sets of vectors which satisfy certain requirements of symmetry …

In 1986 Cappelli, Itzykson and Zuber classified all modular invariant partition functions for
the conformal field theories associated to the affine A1 algebra; they found they fall into an …