University of David
In the 1980s he made at the very least three outstanding contributions which will shape mathematics for a long time: the invention of the “determinant line” of an elliptic partial differential operator as a tool in index theory, the concept of a “superconnection” in differential geometry and analysis, and the Loday-Quillen theorem relating cyclic homology to algebraic K-theory. The first of these came from thinking about the relation of index theory to anomalies in quantum field theory. Determinant lines were a familiar idea in algebraic geometry, and defining regularized determinants by means of zeta functions was standard in quantum field theory and had been studied bymathematicians such as Ray and Singer. Nevertheless, the simple idea that any Fredholm operator has a determinant line in which its determinant lies and that the role of the zeta function is to “trivialize” the determinant line (i.e., identify it with the complex numbers) brought a new perspective to the subject. “Superconnections” came from thinking about the index theoremfor families of elliptic operators and also about Witten’s ideas on supersymmetry in quantum theory. When one has a bundle whose fibers are compact Riemannian manifolds, there is a virtual vector bundle on the base which is the fiberwise index of the Dirac operators on each fiber. The index theorem for families gives a formula for the Chern character of this virtual vector bundle. Quillen’s idea was to combine the formula expressing the index of a single Dirac operator D as the supertrace of the heat kernel expD2 with the identical-looking definition of the Chern character form of a connection in a finite-dimensional vector bundle as the fiberwise supertrace of expD2, where now D denotes the covariant derivative of the connection, whose curvature D2 is a matrix-valued 2-form. He aimed to prove the index theorem for families by applying this to the infinite-dimensional vector bundle formed by the spinor fields along the fibers, defining a superconnection D, with expD2 of trace class, by adding the fiberwise Dirac operator to the natural horizontal transport of spinor fields. Superconnections are now very widely used but, after the first short paper in which he gave the definition and announced his project, Quillen himself did not return to the index theorem for families, as Bismut published a proof of it the following year along Quillen’s lines. Only two of his subsequent papers involved superconnections. One of them (joint with his student Mathai) was extremely influential, though it dealt only with finite-dimensional bundles. It gave a beautiful account of the Thom class of a vector bundle in the language of supersymmetric quantum theory and has provided a basic tool in geometrical treatments of supersymmetric gauge theories. The last phase of Quillen’s work was mostly concerned with cyclic homology. He was attracted to this from several directions. On one side, cyclic cocycles had been invented as a tool in index theory, and the Connes “S-operator” is undoubtedly but mysteriously connected with Bott periodicity, whose role in general algebraic K-theory Quillen had constantly tried to understand. More straightforwardly, cyclic homology is the natural home of the Chern character for the algebraic K-theory of a general ring. Yet again, it seemed that cyclic theory ought somehow to fit into the framework of homotopical algebra of Quillen’s first book. Connes was a virtuoso in developing cyclic cohomology by means of explicit cochain formulae, but to someone of Quillen’s background it was axiomatic that these formulae should not be the basis of the theory. In trying to find the “right” account of the subject, he employed a variety of techniques, pursuing especially the algebraic behavior of the differential forms on Grassmannians when pulled back by the Bottmap. One notable success has already been mentioned, his proof of a conjecture of Loday which, roughly, asserts that cyclic homology is to the Lie algebras of the general linear groups exactly what algebraic K-theory is to the general linear groups themselves.3 In a paper written in 1989, dedicated to Grothendieck on his sixtieth birthday, he succeeded in giving a conceptual definition of cyclic homology but still wrote that “a true Grothendieck understanding of cyclic homology remains a goal for the future.” He continued to make important contributions to the subject throughout the 1990s, mostly jointly with Cuntz, but I amfar fromexpert on this phase of his work, and refer the reader to Cuntz’s account. Nevertheless, on the whole I think he felt that, in T. S. Eliot’s words, the end of all his exploring of Connes’s work had been to arrive at where he started and know the place for the first time.