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September 07, 2010
Tuesday, September 07
Algebra Seminar
Time: 14:30
Speaker: Sheldon Joyner (Western)
Title: "On a certain PSL(2,Z)1-cocycle"
Room: MC 107

Abstract: Classically, if some manifold M is equipped with an action of a subgroup G of PSL(2,Z) under which a certain space F of functions on M transforms via a 1-cocycle, the latter is referred to as an automorphy factor, and the functions F are said to be G-modular. In this talk, I will produce an injective 1-cocycle of PSL(2,Z) into a certain group of formal power series which extends the well-known identification of the fundamental group of P1{0,1,} with associated Chen series. This cocycle may be regarded as a quasi-automorphy factor for sections of the universal prounipotent bundle with connection on PSL(2,Z) - in particular for the polylogarithm generating series Li(z). I will go on to show that the quasi-modularity of Li(z) may be used to give a family of proofs of the analytic continuation and functional equation for the Riemann zeta function. Moreover, under this cocycle, the involutive generator of PSL(2,Z) maps to the Drinfeld associator, while the infinite cyclic generator maps to an Rmatrix, in Drinfeld's formal model of the quasi-triangular quasi-Hopf algebras, thereby producing a representation of PSL(2,Z) into tensor products of certain qtqH algebras. Underlying the whole story is a path space realization of PSL(2,Z) using Deligne's idea of tangential basepoint.