![\"Quillen\"](\"http:\/\/static.guim.co.uk\/sys-images\/Guardian\/Pix\/pictures\/2011\/6\/23\/1308850106362\/Daniel-Quillen-007.jpg\")
<\/p>\nNotices 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.<\/p>\n
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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.<\/p>\n
Superconnections<\/strong><\/p>\n 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<\/a> 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<\/a> and here<\/a>). Superconnections in many respects seems the more natural thing to consider in the context of supermanifolds than the classical vector bundle connections. <\/p>\n<\/p>\n The original algebraic formulation of superconnections as differential operators on the algebra of differential forms with values in endomorphisms of a \\(Z_{2}\\)-graded vector bundle is due to Quillen. He introduced the notion as a way to encode the difference of the chern characters<\/a> of two vector bundles, largely motivated by topological K-theory<\/a>. <\/p>\n 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. <\/p>\n Link<\/strong><\/p>\n AMS Notices<\/a> (opens PDF)<\/p>\n References<\/strong><\/p>\n [1] Daniel Quillen, Superconnections and the Chern character, Topology<\/em>, 24<\/strong>(1):89\u201395, 1985<\/p>\n [2] Florin Dumitrescu, Superconnections and Parallel Transport, arXiv:0711.2766v2 [math.DG], 2007.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading The late Daniel Quillen<\/span>