Mathematics around the Heisenberg Group

Prof. Roger Howe
Professor Emeritus
Yale University

In the mid 1920s, Werner Heisenberg formulated the CCR – canonical commutation relations – describing the relationship between the operations of measuring position and of measuring momentum of a particle in quantum mechanics. These have been fundamental to the later, dramatically successful development of subatomic physics. Shortly after Heisenberg’s work, Hermann Weyl pointed out that the CCR defined the relations of a Lie algebra, whose associated group is a two-step nilpotent group with one dimensional center. Today, this group (and its increasingly large set of cousins) is known as the “Heisenberg group”. Over the remainder of the 20th century, appreciation grew of the fundamental role of the Heisenberg group in disparate mathematical topics, including harmonic analysis, partial differential equations, invariant theory and representation theory, in both finite and infinite dimensions. This talk will review and attempt to summarize some of the manifold connections between these topics that are mediated by the Heisenberg group