UNIVERSITY OF MINNESOTA
SCHOOL OF MATHEMATICS

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 E-mail: reiner@math.umn.edu
Office hours:	Tuesdays 9:05-9:55am.
Study group(!):	Tuesday 1:25-2:15 in Vincent Hall 203a. Check out their schedule.
Prerequisites:	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: A. Zvonkin's Matrix Integrals and Map Enumeration: An Accessible Introduction G. Schaeffer's Lotharingien Seminaire talks on Trees, maps and Hurwitz numbers Goulden and Jackson's Transitive factorizations of permutations and geometry Herrlich and Schmithüsens Dessins d' enfants and Origami curves
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:	Abedessalam, Anderson and Miller, A tridiagonal approach to matrix integrals Ambjorn and Chekhov, The matrix model for dessins d'enfants Armstrong, Mingo, Speicher, and Wilson, The noncommutative cycle lemma Bernardi and Fusy, Unified bijections for planar hypermaps with general cycle-length constraints Arizmendi et al, Relations between cumulants in noncommutative probability Bernardi, An analogue of the Harer-Zagier formula for unicellular maps on general surfaces Biane and Dehornoy, Dual Garside structure of braids and free cumulants of products Bini, Goulden and Jackson, Transitive factorizations in the hyperoctahedral Group Bollobas and Riordan, A polynomial invariant of graphs on orientable surfaces Bose, Gundry and He, Gauge theories and dessins d' enfants: beyond the torus Carrell and Chapuy, Simple recurrence formulas to count maps on orientable surfaces Chapuy and Stump, Counting factorizations of Coxeter elements into products of reflections Chmutov and Vignes-Tourneret, Partial duality of hypermaps Ekedahl, Lando, Shapiro, and Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves Forrester and Warnaar, The importance of the Selberg integral Goulden and Jackson, The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group Goulden and Jackson, Transitive factorizations into transpositions and holomorphic mappings on the sphere Goulden and Jackson, A proof of the conjecture for the number of ramified coverings of the sphere by the torus Goulden and Jackson, Symmetric functions and Macdonald's result for top connexion coefficients in the symmetric group Goulden and Nica, A direct bijection for the Harer-Zagier formula Goulden and Yong, Tree-like Properties of Cycle Factorizations Goulden et al, Monotone Hurwitz numbers in genus zero Goupil, Reflection decompositions in the classical Weyl groups Guay-Paquet and Harnad, 2D Toda tau-functions as combinatorial generating functions Guay-Paquet and Harnad, Generating functions for weighted Hurwitz numbers Harnad and Orlov, Hypergeometric tau-functions, Hurwitz numbers and enumeration of paths Harer and Zagier, The Euler characteristic of the moduli space of curves Krattenthaler and Mueller, Decomposition numbers for finite Coxeter groups and generalized non-crossing partitions Lass, Demonstration combinatoire de la formule de Harer-Zagier McMullen, Moduli spaces in genus zero and inversion of power series Michel, "Case-free" derivation for Weyl groups of the number of reflection factorizations of a Coxeter element Mingo and Speicher, Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces Novak, Complete symmetric polynomials in Jucys-Murphy elements and the Weingarten function Speicher, Free probability theory and random matrices Turner, Riemann, Hurwitz, and branched covering spaces Zoladek, The topological proof of the Abel-Ruffini Theorem Zvonkin, Enumeration of weighted plane trees

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