A. Muhammed Uludağ

Arithmetic and Geometry Around Hypergeometric Functions

This volume comprises lecture notes, survey and research articles originating from the CIMPA Summer School Arithmetic and Geometry around Hypergeometric Functions held at Galatasaray University, Istanbul during June 13-25, 2005. A wide range of topics related to hypergeometric functions is covered, thus giving a broad perspective of the state of the art in the field.

Fundamental groups of some quadric-line arrangements

Abstract In this paper we obtain presentations of fundamental groups of the complements of three quadric-line arrangements in P 2 . The first arrangement is a smooth quadric Q with n tangent lines to Q , and the second one is a quadric Q with n lines passing through a point p ∉ Q . The last arrangement consists of a quadric Q with n lines passing through a point p ∈ Q .

Orbifolds and Their Uniformization

This is an introduction to complex orbifolds with an emphasis on orbifolds in dimension 2 and covering relations between them.

Hypergeometric Galois Actions

This is an expository article about groups generated by two isometries of the complex hyperbolic plane.The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension of the period defined using Chens iterated integrals and captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. As an application, we give an algorithm in proving nontriviality for a class of homologically trivial algebraic cycles obtained from special compact Riemann surfaces. The moduli space of compact Riemann surfaces is the space of all biholomorphism classes of compact Riemann surfaces. The harmonic volume can be regarded as an analytic section of a local system on the moduli space. It enables a quantitative study of the local structure of the moduli space. We explain basic concepts related to the harmonic volume and its applications of the moduli space.We give an overview of the proof for Mirzakhanis volume recursion for the Weil-Petersson volumes of the moduli spaces of genus

Fundamental groups of a class of rational cuspidal plane curves

g

Covering relations between ball-quotient orbifolds

hyperbolic surfaces with

Binary quadratic forms as dessins

n

GALOIS COVERINGS OF THE PLANE BY K3 SURFACES

labeled geodesic boundary components, and her application of this recursion to Wittens conjecture and the study of simple geodesic length spectrum growth rates.We outline a project to study the Galois action on a class of modular graphs (special type of dessins) which arise as the dual graphs of the sphere triangulations of non-negative curvature, classified by Thurston. Because of their connections to hypergeometric functions, there is a hope that these graphs will render themselves to explicit calculation for a study of Galois action on them, unlike the case of a general dessin.

Jimm, a Fundamental Involution

We compute the presentations of fundamental groups of the complements of a class of rational cuspidal projective plane curves classified by Flenner, Zaidenberg, Fenske and Saito. We use the Zariski-Van Kampen algorithm and exploit the Cremona transformations used in the construction of these curves. We simplify and study these group presentations so obtained and determine if they are abelian, finite or big, i.e. if they contain free non-abelian subgroups. We also study the quotients of these groups to some extend.