Math 7590: Fall 2009: Dessins

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

Additional Course Information

  • Course Materials
  • 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
    Office:         Lockett 258:   578.1656
    Office Hours:   TTh 10:30am or by appointment
    Web Page:	URL:
                    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:

  • (Barycentric subdivision=triangulations) of graphs embedded on surfaces
  • Finite index subgroups of SL_2(Z): integral invertible 2by2 matrices
  • Algebraic curves defined over a field of algebraic numbers: Belyi curves
  • with applications to:
  • Combinatorial extension of the Whitney-Tutte polynomial by Bollobas & Riordan
  • Jones polynomial of knots and links
  • Generalization of spanning tree technology to quasi-trees (one-faced embeddings)
  • Grothendieck's Lego-Teichmuller space
  • 	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