Blog post: ajb: The late Daniel Quillen

Notices of the AMS (November) contains a 15 page obituary to Daniel Quillen (1940-2011), written by some rather large names in mathematics, including the late Loday.

Quillen is most famous for his contributions to algebraic K-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978.

Superconnections

My first exposure to the ideas of Quillen was via his superconnection [1]. The notion of a superconnection can be thought of a a generalisation of a vector bundle connection in which we replace the connection one-form with an arbitrary, but Grassman odd (pesudo)differential form. (I'll be slack on details, but you can read more here and here). Superconnections in many respects seems the more natural thing to consider in the context of supermanifolds than the classical vector bundle connections.

The original algebraic formulation of superconnections as differential operators on the algebra of differential forms with values in endomorphisms of a $latex Z_{2}$-graded vector bundle is due to Quillen. He introduced the notion as a way to encode the difference of the chern characters of two vector bundles, largely motivated by topological K-theory.

The geometric understanding came much later in the work of Florin Dumitrescu [2]. The relation between parallel transport along superpaths and superconnections on a vector bundle over a manifold are made explicit in that work.

Link

AMS Notices (opens PDF)

References

[1] Daniel Quillen, Superconnections and the Chern character, Topology, 24(1):89–95, 1985

[2] Florin Dumitrescu, Superconnections and Parallel Transport, arXiv:0711.2766v2 [math.DG], 2007.

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