The Milliman Lectures

Alain Connes and Luis Caffarelli

Our 2002-2003 Milliman Lecturer was Janos Kollár from Princeton University. Professor Kollár is one of the leading algebraic geometers of his generation, and the titles and content of his three lectures reflected his research activities: (1) What is the biggest multiplicity of a root of a degree d polynomial? (2) What are the simplest algebraic varieties? (3) Rationally connected varieties over finite fields. An article in our Autumn 2002 newsletter discussed Professor Kollár and his research - see this page (PDF file) for more information.

Our 2003-2004 Milliman Lecturer, Professor Alain Connes, is scheduled to visit our department, and deliver three lectures, during the week of May 31-June 4, 2004.

Professor Connes, currently a professor at the College de France and at the Institut des Hautes Études Scientifiques in Paris, has been one of the world's most influential mathematicians since he received his Ph.D. from École Normale Supérieure in 1973. His thesis ``A classification of factors of type III'' was a major, stunning breakthrough in the classification of operator algebras which play a central part in the modern formulation of quantum mechanics. For example, Heisenberg's Uncertainty Principle is expressed as the equation PQ - QP = h/(2 pi i), where P and Q are the momentum and position operators.

Connes has since received almost all the awards and honors that are open to mathematicians. He was awarded the Fields Medal in 1982 for his contributions ``to the theory of operator algebras, particularly the general classification and structure theorem of factors of type III, classification of automorphisms of the hyperfinite factor, classification of injective factors, and applications of the theory of C<sup>*</sup>-algebras to foliations and differential geometry in general.'' The Fields Medal is the highest honor in mathematics, awarded every four years to a mathematician under the age of forty. In 2001, the Royal Swedish Academy of Sciences awarded Connes the Crafoord Prize ``for penetrating work on the theory of operator algebras and for having been a founder of non-commutative geometry.'' This prize, worth half a million dollars, pays tribute to fields not covered by the Nobel Prizes.

He was awarded the Prix Aimée Berthé in 1975, the Prix Pecot-Vimont in 1976, the Gold Medal of the Centre National de la Recherche Scientifique in 1977, the Prix Ampere from the Académie des Sciences in Paris in 1980 and in 1981 the Prix de l'Electricité de France. When he was elected to the French Académie des Sciences in 1982 he was one of only thirteen mathematicians in the Académie. He has been elected a foreign member of the Danish Academy of Sciences (1980), the Norwegian Academy of Sciences (1993), the Canadian Academy of Sciences (1995), and the American National Academy of Science (1997).

At around the times Connes received the Fields Medal, his interests turned to differential geometry. He soon realized that operator algebras, particularly non-commutative ones, offered deep new insight into geometry. His primary occupation over the past two decades has been the creation, development, and application of non-commutative geometry. Major ingredients in the theory are differential geometry, cyclic cohomology (a concept he introduced), operator algebras, and K-theory. The naive idea is this. Given a geometric object, say a topological space X, one may consider continuous functions from X to the real or complex numbers. Such functions may be added and multiplied, giving the collection of all such functions the structure of a ring. The multiplication in this ring is commutative, meaning that if f and g are two such functions, then fg=gf. In ``good'' situations it is possible to recover the original space X from the ring. Thus, the algebraic object contains all the geometric information and one may pass back and forth between the algebraic and geometric worlds. Each of the two worlds, geometric and algebraic, has its respective advantages and limitations, but now one has a dictionary that allows one to apply algebraic ideas when geometric ideas alone are not sufficient, and conversely. This is an enormously powerful idea and is a theme that pervades mathematics.

Connes' idea is simply that when one has a non-commutative ring, meaning that a product fg need not be the same as a product gf, one should view this as if it were the ring of functions on some imaginary geometric object, a non-commutative space. At first this seems bizarre, but when one encounters ``pathological'' geometric objects, the dictionary referred to earlier breaks down. However, Connes' fundamental insight was that if one allows non-commutative rings, then by associating to the pathological space an appropriate non-commutative ring, the dictionary can now be extended to new geometric situations, albeit at the expense of having to accept that fg may not equal gf. To make this an effective tool one must translate into algebraic terms all the standard geometric ideas and methods, and then use these algebraic tools in a situation where the geometric tools are not directly applicable.

The development of this idea over the past 25 years has led, not only to solutions to outstanding problems in mathematics, but perhaps more excitingly, to new conceptions of space-time based on quantum field theory. Amazingly, these ideas of ``non-commutative space'' seem to be exactly what is called for in the development of string theory. Connes has applied these ideas to gravitational monopoles and the renormalization problem in quantum field theory, and has shown that the noncommutative torus, a basic example of noncommutative space, appears in the classification of BPS states of 11-dimensional supergravity.

Connes' book Noncommutative Geometry is a dazzling tour-de-force in which he lays out his vision, and illustrates it with a host of applications to operator algebra problems, theoretical physics, particle physics, and differential geometry. Connes has also used these ideas as a possible approach to math's most famous unsolved problem, the Riemann hypothesis (Science, 26 May 2000, p. 1328). He found a spectral interpretation of the zeros of the Riemann zeta function and a geometric interpretation of the explicit formulas of number theory as a trace formula on a natural noncommutative space related to adeles and to his previous work on the classification of factors. One reviewer of Connes' book said his work produced a ``feeling of intense jubilation.''

The 2004-2005 Milliman Lecturer will be Professor Luis Caffarelli from the University of Texas at Austin.

Paul Smith