A reader pointed out this great video lecture by Jim Simons as well as this (very academic) study by Jim Simons- some might enjoy the study so we also posted an abstract and link to it, but make sure either way to check out the video below.

Abstract

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring ˆK , elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f. Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection.

Jim Simons – Mathematics, Common Sense, and Good Luck: My Life and Careers

Jim Simons is an American hedge fund manager, mathematician, and philanthropist. In 1982, Jim Simons founded Renaissance Technologies, a private investment firm based in New York with over \$15 billion under management. Simons retired at the end of 2009, as CEO, of what is one of the world’s most successful hedge funds. Jim Simons’ net worth is estimated to be \$10.6 billion.

Don’t expect to glean any market tips or trading secrets from James Simons, who steadfastly refuses to disclose the method behind his remarkable record in investing. Instead, listen to this mathematician, hedge fund manager and philanthropist sum up a remarkably varied and rich career, and offer some “guiding principles” distilled along the way.

Jim Simons: Structured Vector Bundles Define Differential K-Theory – Introduction

This paper grew out of the effort to construct a simple geometric model for differential K-theory, the fibre product of usual K-theory with closed differential forms, [4],[5],[6]. The model which finally emerged also fulfilled our long standing wish for a simple and straightforward codification of complex vector bundles with connection.

Considering pairs of connections whose Chern-Simons difference form is exact defines an equivalence relation in the space of all connections on a given bundle. We call a pair, V = (V, {?}), consisting of a vector bundle together with a particular such equivalence class, a structured bundle. As is true for vector bundles, structured bundles have additive inverses up to trivial structured bundles: given V there is a W such that their direct sum is equivalent to a bundle with trivial holonomy (Theorem 1.15).

By defining Struct to be the commutative semi-ring of isomorphism classes of structured bundles, and using the standard Grothedieck device to turn Struct into a commutative ring, we obtain ˆK , a functor from smooth compact manifolds with corners into commutative rings. As in ordinary K, every element of ˆKmay be written as V ?[n], where V is a structured bundle and [n] is the trivial structured bundle of dim n. ˆK achieves the above desired codification of connections and serves as the sought after geometric model of differential K-theory.

Defining four natural transformations into and out of ˆK we develop in the first four sections the diagram with exact diagonals and boundaries, where the sequence along the upper boundary may be identified (via ch ? C) with the Bockstein sequence for complex K-theory (the long exact sequence associated to the short exact sequence of coefficients 0 ? Z ? C ? C/Z ? 0), and that along the lower boundary comes from de Rham theory. Here, ?BGL means all closed forms cohomologous to Chern characters of complex vector bundles, and ?GL means all closed forms cohomologous to pull-backs by maps into GL = union of the GL(n,C) of the transgression of the Chern character form.  is the map which simply drops the connection, and ch is the Chern-Weil map applied to the Chern character polynomial. The fibre product statement above is related to the commutative square on the right half of the diagram.

The work’s main technical innovation is embodied in Proposition 2.6, where it is shown that all odd forms modulo ?GL arise as the Chern-Simons difference forms between the trivial connection and arbitrary connections on trivial bundles. A corollary, as implied by the diagram above, is that every element of ?BGL arises after stabilizing as the Chern character form of some connection in any bundle whose Chern character is the given cohomology class. In particular, if a bundle has zero characteristic classes over C, then there is a connection on that bundle, stabilized by adding in a trivial bundle, with vanishing Chern-Weil forms.

Full study here.