Abstract Effros/Popa/Takesaki The three PI's and their students will continue their investigations in operator algebra theory. Edward Effros is embarking on a major program of understanding non-commutative functional analysis in collaboration with Zhong-Jin Ruan, using the theory of "quantized" Banach spaces. Sorin Popa will be continuing his investigation of subfactor theory in both its analytic and combinatorial aspects. Masamichi Takesaki will be taking advantage of the enormous progress that he and his collaborators have made in finding the intrinsic underlying structure of von Neumann algebras to make further progress. This work has enabled him to formulate a paradigm for classification problems in mathematics that bypasses the non-classifiability theory of Mackey. It is only now becoming clear that the abstruse notions of quantum physics pondered by Bohr, Einstein, and Heisenberg in the early part of this century, will have profound effect on modern technology. In particular, we will be able to exploit the classifically inconceivable predictions of quantum mechanics to build new kinds of computers, measure physical quantities with unprecedented accuracy, and make new materials with remarkable properties. These possibilities were to some degree foreseen by John von Neumann, who invented the correct mathematical language for quantum mechanics. He was the first mathematician to realize that since mathematics is based on fundamentally mistaken ideas of geometry and matter arising from classical physics, it was incumbent on mathematicians to develop new forms of "quantized mathematics". For that purpose he introduced the area of operator algebras, which now constitutes one of the most exciting branches of modern mathematics. This subject has important applications to geometry, algebra, probability theory, and mathematical physics. The participants in this grant are pursuing some of the most exciting directions in this field. These include the exploration of quantized symmetry (Popa: subfactor theory), the quantized version of infinite dimensional analysis (Effros: operator spaces), and understanding the classification problems of operator algebra and its deep implications for such problems throughout mathematics (Takesaki: von Neumann algebras).
