Courses
Jyväskylä Summer School
In July 2006 there was a summer school in mathematics in Jyväskylä, Finland.
Lecture notes taken by Tuomas Puurtinen:
All lecture notes in one document: Lectures.
Hints to some exercises in Prof Wolfart's lectures: Hints.
Prof. Jones's lecture notes: Jones
lecture.
Solutions to Prof. Jones's exercises: Solutions.
Some useful reading material concerning Dessins d'Enfants and Belyi Functions:
Dessins d'Enfants:
bipartite maps and Galois groups Gareth Jones, Southampton, U.
K.
Regular
Cyclic Coverings of the Platonic Maps Gareth A. Jones and David
B. Surowski
ABC
for polynomials, Dessins d'Enfants and Uniformization  A survey
Jürgen Wolfart
MA1: Dessins d'Enfants: Function Theory and Algebra
of Belyi Functions on Riemann Surfaces, 4 ECTS
Lecturer: Prof. Jürgen Wolfart (Johann Wolfgang GoetheUniversität,
Frankfurt, Germany)
Dates: 24th July  4th August 2006
Smooth complex projective algebraic curves are compact Riemann
surfaces, and conversely every compact Riemann surface can be seen as
such an algebraic curve. In the last 25 years, we have been able to say
much more about this classical correspondence: the curves can be
defined by equations with coefficients in number fields if and only if
certain functions with very special ramification properties exist, the
"Belyi functions". There are some beautiful reformulations of
this fact
using special triangulations of Riemann surfaces ("dessins d'enfants")
and concerning their uniformisation, i.e. their representation as
quotients of the hyperbolic plane by special kinds of Fuchsian groups.
Moreover, dessins and Belyi functions encode deeper properties of the
algebraic curves/Riemann surfaces, such as symmetries and their
behaviour under algebraic conjugations, in other words under the action
of the absolute Galois group.
The course will introduce this very fascinating topic, present many
examples and lead to interesting open questions. It is intimately
related to the other course MA2 given by Gareth Jones.
Students attending the course should have a solid basic knowledge in
Function Theory, Group Theory, Topology and Algebra. Highly recommended
is some knowledge about Riemann Surfaces, e.g. along
 Chapters 4 and 5 in "Complex Functions" by Jones and Singerman
(Cambridge UP 1986) or
 Chapter I and II in Miranda's "Algebraic Curves and Riemann Surfaces"
(AMS 1991).
MA2: Dessins d'Enfants: Combinatorics and Group Theory
of Belyi Functions on Riemann Surfaces, 4 ECTS
Lecturer: Prof. Gareth Jones (University of Southampton,
UK)
Dates: 24th July  4th August 2006
Compact Riemann surfaces are equivalent to smooth complex projective
algebraic curves, in the sense that they can be defined by equations involving
polynomials with complex coefficients. Belyi's Theorem states that these
coefficients can be chosen from an algebraic number field if and only
if the surface admits a Belyi function, a meromorphic function with very
special branching properties. Dessins d'enfants (children's drawings)
are maps on surfaces, in which an embedded graph provides a combinatorial
"picture" of a Belyi function, and one can reformulate Belyi's
Theorem to state that a compact Riemann surface is defined over an algebraic
number field if and only if it corresponds to a dessin. This important
result links together three different theories: that of Riemann surfaces,
which involves analysis and geometry (usually hyperbolic), that of algebraic
number fields, which involves algebra and number theory, and that of maps
on surfaces, which involves combinatorics and topology. Many of the connections
between these different theories are provided by certain groups associated
with a dessin: its automorphism group, which describes its symmetry properties,
its monodromy group, which describes the branching of the associated Belyi
function, and a subgroup of a triangle group, which shows how the surface
and the dessin are obtained from a tessellation of a simply connected
Riemann surface, such as the hyperbolic plane. Understanding this situation
involves the study of finite groups, arising as automorphism and monodromy
groups of dessins, and Fuchsian groups, acting as groups of isometries
of the hyperbolic plane.
Another important group here is the absolute Galois group, the automorphism
group of the field of algebraic numbers. This infinite group can be regarded
as a limit of the family of (finite) Galois groups of the algebraic number
fields. It has a natural action on dessins, by acting on the coefficients
of the polynomials and rational functions which define the corresponding
surfaces and Belyi functions. In 1984 Grothendieck observed that this
action is faithful, allowing us to "see" the whole of the Galois
theory of algebraic number fields by studying the action of the absolute
Galois group on dessins.
This course, which is closely related to the other course MA2 given by
Juergen Wolfart, will concentrate mainly on the grouptheoretic and combinatorial
aspects of this subject, covering a number of important theorems, instructive
examples, and open questions.
Students attending the course should have a solid background in elementary
Group Theory, Topology and Algebra, including some basic Galois Theory.
Almost any undergraduate algebra textbook, e.g. "Abstract Algebra"
by J. J.
Rotman (Pearson Prentice Hall), will cover the basic algebra of groups
and fields required.
"Complex Function Theory" by G. A. Jones and D. Singerman (Cambridge
University Press) covers the background on Riemann surfaces, while J.
J.
Rotman "Galois Theory" (Springer) covers the Galois Theory required.
It would be useful, but not essential, to look at some survey articles
about maps and hypermaps, e.g. "Maps, Hypermaps and Triangle Groups",
pp.115145 in "The Grothendieck Theory of Dessins d'Enfants"
(ed. L. Schneps, London Math. Soc. Lecture Note Series 200, Cambridge,
1994),
and/or two manuscripts accessible here: G. A. Jones: Fuchsian Groups,
and J. Wolfart: abc for polynomials, dessins d'enfants, and uniformization
 a survey.
