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Fundamental groups, multiple polylogarithms, and Diophantine geometry
At least since the sixties, a good deal of serious work in arithmetic geometry has involved ideas and techniques from homology theory. One might cite the resolution of the Weil conjectures concerning equations over finite fields, Faltings' finiteness theorem for rational points on hyperbolic curves, and Wiles' proof of Fermat's last theorem among many other celebrated results. More recently, several people (including Grothendieck) have emphasized the importance of adapting non-linear invariants such as homotopy groups into an arithmetic setting. Rather surprisingly, it is the motivic fundamental group (as defined and studied by Deligne) that appears to have the most direct influence on questions of a Diophantine nature. In this lecture, we will explain the general notions necessary to make this statement precise.
Thursday, October 21, 2004, 245 Altgeld Hall, 4 p.m.
Mathematics Colloquia homepage