Titlebook: Visions in Mathematics; GAFA 2000 Special Vo N. Alon,J. Bourgain,V. Milman Book 2010 Birkh?user Basel 2010 Finite.Mathematica.equation.func

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,γ-Functions of Representations and Lifting,rreducible representation π of .(.) the action of Φ. in π is well defined and thus it defines a number γ.(π). These gamma-functions and the associated .-functions were studied by J. Tate for . = 1 and by R. Go dement and H. Jacquet for arbitrary .

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Geometric and Unipotent Crystals,onal ..(g)-modules. It was shown in [Lu2] that Eashiwara’s crystal bases are the limits as . → 0 of Lusztig’s canonical bases. Later, in [K2] Kashiwara introduced a new combinatorial concept — . Kashiwara’s crystals generalize the crystal bases and provide a natural framework for their study.

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,Gau?—Manin Determinant Connections and Periods for Irregular Connections,r connection is calculated. The determinant of cohomology of the standard rank 2 Kloosterman sheaf is computed modulo 2 torsion. Periods associated to irregular connections are studied in the very basic exp(.) case, and analogies with the Gau?-Manin determinant are discussed.

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