Group Theory

Abstract

The tale of mathematics and physics has been one of love and hate, of harmony and discord, and of friendship and animosity. From their simultaneous inception in the shape of calculus in the seventeenth century, through an intense and interactive development in the eighteenth and most of the nineteenth century, to an estrangement in the latter part of the nineteenth and the beginning of the twentieth century, mathematics and physics have experienced the best of times and the worst of times. Sometimes, as in the case of calculus, nature dictates a mathematical dialect in which the narrative of physics is to be spoken. Other times, man, building upon that dialect, develops a sophisticated language in which—as in the case of Lagrangian and Hamiltonian interpretation of dynamics—the narrative of physics is set in the most beautiful poetry. But the happiest courtship, and the most exhilarating relationship, takes place when a discovery in physics leads to a development in mathematics that in turn feeds back into a better understanding of physics, leading to new ideas or a new interpretation of existing ideas. Such a state of affairs began in the 1930s with the advent of quantum mechanics, and, after a lull of about 30 years, revived in the late 1960s. We are fortunate to be witnesses to one of the most productive collaborations between the physics and mathematics communities in the history of both.