Math 7590 Information

Additional Course Materials: Math 7590: Dessins d'Enfant

Polynomial Invariants of Combinatorial Maps

Quasi-Tree Expansions

  • Fundamental Expansion
  • Sokal Survey: maultivariate Tutte
  • Bollobas-Riordan-Whitney-Tutte Polynomials: BRWT

  • Application to Knots: DFKLS
  • Chmutov/Pak
  • Partial Duality & Signed BRWT
  • Knot Theory and Combinatorial Basics & Reference Guides

  • Kauffman AMM article
  • Chmutov: Tutte Reference Guide
  • Chmutov: Matroids Reference Guide
  • Chmutov: Links and the Jones polynomial Reference Guide
  • Chmutov: Ribbon graphs and the Bollobaiís-Riordan polynomial
  • Combinatorial Knot Theory
  • The last item is a first draft LaTeX version of a Book by Lou Kauffman -- This book is an introduction to knot theory and to Witten's approach to knot theory via functional integral.

    Riemann Surfaces & Maps

    Galois Invariants

  • Galois Invariants (and Braids): Ellenberg
  • Enumeration

  • Akhmedov: Gluing Surfaces
  • Search MathSciNet for Liskovets: Plane Trees
  • Belyi Functions

  • L. Schneps: Dessins d'enfants on the Riemann Sphere, in The Grothendieck Theory of Dessins d'Enfants, LMS Lecture Notes 200, Cambridge U. Press, 1994.
  • Galois Groups & Covers: Excellent, elementary notes on rudiments of Dessins by David Meredith
  • Mathematica Notebook on Dessin construction
  • Maps

  • Jones-Singerman: Maps Hypermaps and Triangle Groups: Article in Leila Schneps: Grothendiecks Theory of Dessins: London LNIM 200
  • Ribbon Graphs: see Sergei Chmutov's papers on the archive
  • Riemann-Hurwitz

    If you have taken algebraic topology, here is a set of exercises on the Riemann-Hurwitz formula.

    Riemann Surfaces

    For an introduction to the field of meromorphic functions on a Riemann surfaces, linear fractional transformations and the upper half plane two dimensional geometry:
  • Jones-Singerman: Complex Functions: An Algebraic and Geometric Viewpoint
  • Kirwan, Frances: Complex Algebraic Curves
  • Fulton, W.: Algebraic Curves: (A ring theoretic approach)
  • Harvey Cohn: Conformal Mappings on Riemann Surfaces (Classical approach)
  • Surface Classification & Triangulation

  • The Classification Theorem for Compact Surfaces : Book by Jean Gallier. Includes triangulability proof
  • Conway's ZIP proof
  • A simple triangulation method for smooth manifolds
  • Permutations & Coverings

    Magma Code

    I computed an example of a cartographic group of a 3-constellation in magma. However, you are strongly encouraged to support sage different approach to mathematics software at Sage Home , the free python-based follow-up to magma.

    Mathematica Notebook

    Here is a Mathematica Notebook on using Mathematica and its Combinatorica package to work with permutations and draw graphs.

    TWF 252: Burnside & Frobenius

  • This Week Math Physics: Section 7
  • This is a discussion of the distinctions between group actions and representations in the study of properties of finite groups.

    Classification of Coverings and Relationship with Group Actions

    Prof. Moeller has made available an excellent set of notes on this relationship.

    Algorithms for Permutations

  • Combinatorica Package in Mathematica
  • magma on chaos.math.lsu.edu (Online Reference Manual magma reference)
  • sage on chaos.math.lsu.edu and from William Stein
  • Sources

  • Projects

    Problems

    Comments --> Last update: 1 September , 2009