Non-Commutative Geometry and Mathematical Physics

Abstract

However, the flow of knowledge in the reverse direction, from mathematics to physics, has been more limited. There has been prejudice among physicists that modern mathematics is unnecessary for the future evolution of physics. In the early 1960's, Res Jost introduced series of lectures with the observation that for many physicists, an appropriate knowledge of mathematics comprised, a familiarity with the Greek and Latin alphabets. This reference to the study of perturbation theory reflected the enormous distance between the mathematics and physics communities, just 30 years ago. To put quantum field theory on logical level, mathematics has been developed over these 30 years in the areas of partial differential equations, in probability theory, in analysis, and in geometry of infinite dimensional manifolds. In other words, one developed tools to study functions on the infinite dimensional configuration space or phase space of the quantum fields. This work was carried out by the traditional community of mathematical physicists, but it was more or less ignored by the majority of theoretical physicists. Eventually, theoretical physicists themselves began to take an interest in abstract mathematics. Beginning with the mathematical study of gauge theories in the late 1970's, and carried on in widespead way due to the popularity of string theory among theoretical physicists in the mid 1980's, the mathematical methods used by physicists became more sophisticated. Through string theory, links were made between problems worked on by physicists and frontier problems in algebraic geometry, number theory, and representation