Dear colleagues:

We announce the following seminar at IPMU
(the Institute for the Physics and Mathematics of the Universe).

Regards,

Satoshi Kondo (IPMU)
http://ipmu.jp/

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Speaker: Jiro Sekiguchi (Tokyo University of Agriculture and Technology)

Date: Dec. 1 (Tue) 2009, 13:15-14:45

Place:  Seminar Room at IPMU Prefab. B

Title: Systems of Uniformization Equations and Hyperelliptic Integrals

Abstract:
The main subject of this talk is systems of uniformization equations along
logarithmic free divisors which have solutions expressed in terms of
hyperelliptic integrals.
Such a logarithmic free divisor in C^3 is related with the
discriminant of a dihedral group
of order 2n. We construct fundamental solutions by Gaussian
hypergeometric functions
in addition to a solution expressed by a hyperelliptic integral, say
v. Let u_1, u_2 be fundamental
solutions of the hypergeometric equation implied by the system in question.
Then the image of the upper half plane by the map defined by the ratio
u_1/u_2 is a hyperbolic
triangle in the upper half plane with angles 0, ¥pi/2, ¥pi,/n. This is
a fundamental
domain of Hecke's triangle group. We now consider the case of
reflection groups of types
A_2, B_2, G_2. These are examples of dihedral groups. In these cases,
the inverse of the
map ¥phi=(u_1, u_2, v) of C^3 to C^3 is concretely constructed in
terms of elliptic functions
and Eisenstein series. The case A_2 was already obtained by K. Saito
who initiated and
developed the theory of logarithmic free divisors and formulated the
notion of the systems
of uniformization equations along logarithmic free divisors extending
the study of H. A.
Schwarz on hypergeometric differential equations. The result in this
talk is an extension
of his work to the case of the dihedral groups.
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You can check the location from
http://www.ipmu.jp/access/img/KashiwaCampusMap.png

The schedule of the seminar can be checked from
http://db.ipmu.jp/seminar/