Math 8680: Topics in Combinatorics
Graphs on surfaces

Fall 2014

Location, time: Monday-Wednesday-Friday 9:05 - 9:55pm in Vincent Hall 2 
Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
Office hours: Tuesdays 9:05-9:55am. 
Study group(!): Tuesday 1:25-2:15 in Vincent Hall 203a.
Check out their schedule.
We will assume knowledge of basic abstract algebra, and complex analysis (e.g. analytic functions, Cauchy integral formula). Although we will review them, it would be help to have also seen a little topology of covering spaces and fundamental groups, along with some basic representation theory of finite groups,
Course content:
We plan to cover as much as we can of the beautifully written book:
S.K. Lando and A.V. Zvonkin, Graphs on Surfaces and Their Applications.
People with our library subscription can get a PDF or $25 softcover edition.
Some of the topics we should discuss are
  • Riemann surfaces as ramified coverings of the sphere
  • Hurwitz numbers, permutation factorizations
  • Maps, hypermaps, Goulden and Jackson's formula counting cacti
  • Belyi functions and dessins d'enfants
  • Harer and Zagier's formula counting 1-face planar maps on surfaces by genus, and various proofs, including
    • Wick's formula and integrals over the space of Hermitian matrices
    • Lass's proof via the BEST theorem.
Here are a few survey articles that give some overview on these topics:
Grading: Grad students who are registered for the class and want to get an A should attend regularly, and give a talk once during the semester on a paper listed below, or some other paper that you can convince the instructor is related:
Papers for student talks:
Back to Reiner's Homepage.