Motives and Chern-Simons classes

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Every area of mathematics and its applications gains froma geometric point of view. Sometimes one can solve a problemjust using topology, which describes the rough shapeof an object. But topology is not always enough to tell uswhether a given equation can be solved. We are led to set upmore subtle notions such as motives whichinclude more information about which algebraic equationscan be solved. (The name motive comes from the ideaof a motif in music, a short melody which is developedin various ways.) But mathematicians face massive challengesin trying to understand motives. This problem includestwo of the Clay Institute's famousmillion-dollar problems, the Hodge conjectureand the Birch-Swinnerton-Dyer conjecture.The proposed research seeks to study the motivesof a special class of examples based on fibre bundles.Fibre bundles are spaces such as the Mobius stripwhich are made by twisting simpler spaces together.In this situation, we see the prospectof complete understanding of motives.