Groups and Physics---Dogmatic Opinions of a Senior Citizen, Volume 44, Number 1

T he editors of the Notices have invited me to ruminate publicly on two topics which have engaged my interest for sixty years. In so doing, I hope to convey some of the pleasure and excitement which I experienced while revisiting the relations between groups and physics. Both are huge subjects and their interactions, multifaceted. In 1939–40 when I was a graduate student at Princeton, the original Fine Hall was one of the world’s most revered shrines for mathematical devotees. We were told that this charming building had been modelled on a medieval manor house. Therefore, on either side of the main entrance a niche and pedestal had been provided to accommodate the obligatory statues of protective saints. Perhaps because of the abhorrence of idolatry in the Presbyterian tradition from which the university sprang, these niches were empty. Perhaps the university authorities were waiting for the Church to name patron saints for mathematics! History records that one day as he approached the entrance, Lefschetz paused, turned to his companion and said, “Regardez, Claude, une place pour Saint Elie et l’autre pour Saint Hermann.” He was gently poking fun at Claude Chevalley. We all knew that in Chevalley’s pantheon, Elie Cartan and Hermann Weyl were the greatest mathematicians of the twentieth century, perhaps of any century, surpassed, if at all, only by Gauss. I have no reason to disagree with Chevalley’s judgment. Both Cartan and Weyl were prodigious mathematicians. Further, it was these two giants who revealed to a largely unappreciative world the power, depth, and beauty of the ideas of Sophus Lie, Wilhelm Killing, and Isai Schur about continuous groups and their representations and taught us how to apply them to physics and to diverse areas of mathematics.