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The motivic fundamental group |
Background:
topological, algebro-geometric and motivic π_{1} The
topological fundamental group π_{1} was developed in the
late nineteenth century as a tool for describing covering spaces in topology
and multi-valued solutions of differential equations. Its link to arithmetic
comes from the analogy between fundamental groups classifying covering spaces
and Galois groups classifying field extensions. In the 1960’s Grothendieck used finite étale
maps to define an algebro-geometric fundamental group
which was the pro-finite completion of the topological one. The
term motivic fundamental group refers to two related
ideas which have grown from Grothendieck’s program.
On the one hand, it is natural to think of the ℓ-adic
representations of the algebraic π_{1}, the flat vector
bundles corresponding to representations of the topological π_{1}, and the p-adic
representations of the crystalline π_{1} as being different
realizations of one underlying motivic object. On the
other hand, the focus in recent years has been on developing a powerful
abstract theory of motives. Using a construction of Beilinson,
Deligne and Goncharov could
in a precise sense endow the “unipotent part” of the
group ring of π_{1} with the structure of a motive. When the
space is the Riemann sphere minus three points, the periods of this motive are
the “multiple zeta numbers”. Recently, Faltings and Hadian have successfully applied the abstract theory of
motives to the fundamental group and have obtained strong new results in diophantine geometry. Format In
the workshop there will be 17 hours of lectures by invited speakers, three or four
lectures each day. There will be fairly long lunch breaks which can be used for
informal discussions. The lectures will focus on three basic programs: the work
of Deligne and Goncharov
(extending earlier work of Ihara, Drinfeld, and others)
on the unipotent motivic π1,
the program of Faltings, Hadian,
Kim to prove finiteness results in Diophantine geometry using the motivic fundamental group, and the “section conjecture” of Grothendieck relating the existence of a rational point to
a splitting of the arithmetic fundamental group. Aim It
is hoped that participants learn how to use the motivic
fundamental group (beyond knowing about its classical, etale,
deRham, and crystalline realizations) to attack
classical questions in diophantine geometry,
arithmetic algebraic geometry, and even physics. [Back] |