Math 7590: Fall 2009: Dessins
# Math 7590: Graphs on Surfaces (a.k.a. Dessins D'enfant

### Additional Course Information

### Basic Course Structure

Math 7590: Seminar in Topology: Graphs on Surfaces
Time/Place: 1:30 p.m. - 3:00 pm TTh --- 119 Lockett Hall.
Instructor: Neal Stoltzfus
Email: stoltz@math.lsu.edu
Office: Lockett 258: 578.1656
Office Hours: TTh 10:30am or by appointment
Web Page: URL: http://www.math.lsu.edu/~stoltz/Courses/F09/7590/
This site will contain this document on course information,
and links to additional web resources.
Prerequisite: Point-set topology, fundamental groups and modules over rings, basic Galois theory.
Textbook: Lando-Zvonkin, Graphs on Surfaces and Their Application

Textbook Website:
Lando-Zvonkin Textbook

Author Abstract: Graphs drawn on two-dimensional surfaces have always attracted researchers by
their beauty and by the variety of difficult questions to which they give rise. The theory of such
embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely
unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models,
and has become a kind of a focus of a vast field of research. The book provides an accessible
introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois
group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral
method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of
Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation
theory. The presentation is concrete throughout, with numerous figures, examples (including computer
calculations) and exercises, and should appeal to both graduate students and researchers.

Instructor Choice: The study of fixed embeddings of graphs in surfaces is known by many
names: dessin d'enfant, rotation systems, combinatorial maps, ribbon graphs, inter alia.
The course will develop the equivalences between the following objects:

with applications to:
We will cover Chapters: 1,2,3(parts),4 and 6, if time permits
Portfolio & Project: The portfolio for the course primarily consists of a project which develops a
topic related to graphs embedded on surfaces. An oral presentation of your
project will be made during the last weeks of class. A written report will be submitted
by the Final Exam date. More portfolio Information .
Grades: Grading will be weighted as follows:
Project: 50%
Research Paper Reviews: 20%
Other "portfolio" items: 30%

Last update: 29 August, 2009