Research Interests

I have always been fascinated by the rich interplay between the frontiers of mathematics and those of physics; these ideas have come to span almost every subfield in the two subjects. Sixty years ago, quantized field equations appeared a natural way to incorporate relativity into a quantum description of nature. This approach led to famous experimental checks, despite questionable renormalization calculations used to avoid divergences. Constructive quantum field theory (CQFT) eventually became the mathematical domain to ask whether quantum fields and the physical principles of renormalization have a logical foundation. My early work focused on establishing CQFT in space-times of two and three dimensions. This problem remains unresolved for the four-dimensional Yang-Mills equations: During the year 2000 understanding these equations was named one of seven mathematical millennium prize problems. I became intrigued with the invention of non-commutative geometry (NCG) in the 1980s, for that new mathematical structure allows one to incorporate quantization into the definition of space. Meanwhile, in physics, super-symmetry describes a relation between Bose particles and Fermi particles. My recent study of supersymmetric quantum fields led to new tools and examples for NCG and for CQFT.

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Primary Section

Section 11: Mathematics

Secondary Section

Section 13: Physics