Combinatorics entering the third millennium

In the later part of the century, attitudes changed. When the 1998 film Good Will Hunting featured a famous mathematician at the Massachusetts Institute of Technology who had won a Fields Medal for combinatorics, many found this somewhat unbelievable.2 However, life followed art in this case when, later in the same year, Fields Medals were awarded to Richard Borcherds and Tim Gowers for work much of which was in combinatorics. In this chapter, I have attempted to tease apart some of the interrelated reasons for this change, and perhaps to throw some light on present trends and future directions. I have divided the causes into four groups: the influence of the computer; the growing sophistication of combinatorics; its strengthening links with the rest of mathematics; and wider changes in society. I have told the story mostly through examples. If the views and examples are somewhat personal, perhaps this is better than having no views at all.