Alain Connes

College de France

Primary Section: 11, Mathematics
Membership Type: International Member (elected 1997)

Research Interests

I have recently worked in the areas of theoretical physics, number theory, topology and noncommutative geometry. In theoretical physics, I collaborated with T. Damour and P. Fayet on gravitational monopoles in connection with tests of the equivalence principle, I perfected my work on the geometric meaning of the classical Lagrangian of the Standard Model of particle physics, and collaborated with M. Douglas and A. Schwarz to show that the noncommutative torus (a basic example of noncommutative space) appears in the classification of BPS states of 11 dimensional supergravity. In number theory, I 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 my previous work on the classification of factors. In topology, I showed (with D. Sullivan and N. Teleman) that the quantized calculus provides local formulas for topological Pontrjagin classes, and collaborated with M. Gromov and H. Moscovici on the Novikov conjecture. I wrote a 700 page book on noncommutative geometry and developed E-theory with Nigel Higson, who used it with G. Kasparov to solve, for a large class of groups the conjecture that P. Baum and I had proposed. Finally, I solved with H. Moscovici ( by developing the theory of cyclic cohomology for Hopf algebras) the transversal index problem for foliations.

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