**ODTÜ-B****Ý****LKENT
ALGEBRAIC GEOMETRY SEM****I****NAR**

**“The Universal Arithmetic Curve”**

by

### Muhammed UludaG

### (GALATASARAY University)

**Abstract:
**I will discuss the limit space F of the category of
coverings C of the "modular interval" as a deformation retract of the
universal arithmetic curve, which is by (my) definition nothing but the
punctured solenoid S of Penner. The space F has the advantage of being compact,
unlike S. A subcategory of C can be interpreted as ribbon graphs, supplied with
an extra structure that provides the appropriate morphisms for the category C.
After a brief discussion of the mapping class groupoid of F, and the action of
the Absolute Galois Group on F, I will turn into a certain
"hypergeometric" galois-invariant subsystem (not a subcategory) of
genus-0 coverings in C. One may define, albeit via an artificial construction,
the "hypergeometric solenoid" as the limit of the natural completion
of this subsystem to a subcategory. Each covering in the hypergeometric system corresponds
to a non-negatively curved triangulation of a punctured sphere with flat
(euclidean) triangles. The hypergeometric system is related to plane
crystallography. Along the way, I will also discuss some other natural
solenoids, defined as limits of certain galois-invariant genus-0 subcategories
of non-galois coverings in C. The talk is intended to be informal, relaxed and
audience friendly.

** **

**Date:** 15 May 2009 Friday

**Time:** 15:40

**Place:** Bilkent, Mathematics Seminar
Room SA-141

** **

**Tea and cookies will be served before the
talk.**