C. Anantharaman-Delaroche, H. Araki, A. Connes, J. Cuntz, E.G. Effros, U. Haagerup, V.F.R. Jones , M.A. Rieffel and D.V. Voiculescu

Multiplication of numbers does not depend on order: 3 x 2 = 6 and 2 x 3 = 6. Multiplication of numbers is commutative. On the other hand, multiplying a function f(x) by x and then taking the derivative is not the same as first taking the derivative and then multiplying by x. Actually: (xf)' - xf' = f. Other examples where commutativity fails are provided by matrix multiplication.

With the advent of quantum mechanics the usual way of thinking of physical observables as numerical functions became inadequate. In quantum mechanics physical quantities are represented by operators on a Hilbert space i.e. by infinite matrices. The fact that matrices need not commute, exactly models Heisenberg's uncertainty principle, the impossibility of sharp simultaneous measurement of position and momentum of a particle. Operator algebras appeared as an outgrowth of quantum physics. But gradually it was realized that they provide the natural framework for generalizing geometry, topology and measure theory in a fundamental noncommutative way of wide applicability.

The commutative algebra of continuous numerical functions on a compact space can be realized faithfully as multiplication operators on a Hilbert space of square-integrable functions. The quantum generalization is given by self-adjoint non-commutative algebras of operators on Hilbert spaces.

Figure 1 represents a flow with an irrational slope on a 2-dimensional torus meeting a closed transversal. Much of this situation is encoded in a noncommutative torus, and 'vector-bundles' on it. Yang-Mills theory on these 'vector-bundles' is of current interest in string theory in physics.

The knot in figure 2 is one of a pair having the same Jones polynomial. Equality of the polynomials follows from an argument borrowed from statistical mechanics. The connection between low-dimensional topology and statistical mechanics grew out of subfactors. Factors are operator algebras with trivial center and play the role of scalars in the noncommutative setting. Thus subfactors are 'noncommutative' Galois theory. But the Galois group must be replaced by a tower of algebras that have a topological and a statistical mechanical interpretation. Thus the connection was made between these two apparently quite unrelated fields.

The noncommutative mathematics of operator algebras has grown in many directions and has made unexpected connections with other parts of mathematics and physics. This can also be seen from the titles of the 8 areas into which the program is divided: noncommutative geometry, simple C*-algebras, noncommutative dynamical systems, subfactors, quantization, quantum field theory, free probability theory, noncommutative Banach spaces.

Since the 1984-85 MSRI program in Operator Algebras, developments have continued at a rapid pace, and interactions with other fields such as elementary particle physics and quantum groups continue to grow. These topics will be emphasized: Noncommutative geometry, Simple C* algebras, Noncommutative dynamical systems, Subfactors, Quantization, Algebraic quantum field theory, Free probability theory, and Operator spaces.