A paper of 6-12 pages related to algebraic combinatorics is due on the last class. The paper should be printed in some flavor of TeX (e.g., LaTeX). It is strongly recommended that all figures be drawn with some graphics package (such as xfig) that can be incorporated into LaTeX, the reason being that knowing how to do this will prove invaluable for your future career.

The paper need not have any original mathematics, but it should consist of more than paraphrasing from a single source. There should be a bibliography with at least two references. You can choose any topic related to algebraic combinatorics, but your topic needs to be approved in advance. In particular, the paper must involve some algebra (linear algebra, groups, rings, fields, …). Therefore please let me know (in person or by email) your proposed paper topic. Include a couple of sentences about what material you plan to cover.

Some suggestions for paper topics, along with ideas for sources:

Combinatorial Commutative Algebra

Stanley, Richard P. Combinatorics and Commutative Algebra. 2nd ed. Boston, MA: Birkhäuser, 1996. ISBN: 0817638369 and 3764338369.

Miller, Ezra, and Bernd Sturmfels. Combinatorial Commutative Algebra. New York, NY: Springer, c2005. ISBN: 0387223568.

My book Combinatorics and Commutative Algebra has many possible topics, as does Combinatorial Commutative Algebra by Miller and Sturmfels.

Topological Combinatorics (For those Who Know Some Algebraic Topology)

Buy at MIT Press Graham, R. L., M. Grötschel, and L. Lovász, eds. Handbook of Combinatorics. Vol. 2. Cambridge, MA: MIT Press, 1995. ISBN: 0262071711.

The article “Topological methods” by Anders Björner in Handbook of Combinatorics, vol. 2, pp. 1819-1872, is a good place to get started.

Linear Algebra and Combinatorics

Buy at MIT Press Graham, R. L., M. Grötschel, and L. Lovász, eds. Handbook of Combinatorics. Vol. 2. Cambridge, MA: MIT Press, 1995. ISBN: 0262071711.

See the articles “Tools from linear algebra” by C. D. Godsil, pp. 1707-1748, and “Tools from higher algebra” by Noga Alon, pp. 1749-1783, in Handbook of Combinatorics, vol. 2.

Graph Eigenvalues

Cvetković, Dragoš M. Spectra of Graphs: Theory and Application. 3rd ed. Edited by Dragoš M. Cvetković, Michael Doob, and Horst Sachs. New York, NY: Academic Press, 1979. ISBN: 0121951502.

Biggs, Norman, ed. Algebraic Graph Theory. 2nd ed. Cambridge, UK: Cambridge University Press, 1993. ISBN: 0521458978.

A huge topic. The book Spectra of Graphs, 3rd ed., by Cvetkovic, Doob, and Sachs will get you into the literature. The book Algebraic Graph Theory by Biggs also has a lot of information.


Stein, Sherman K. Algebra and Tiling: Homomorphisms in the Service of Geometry. Edited by Sherman K. Stein and Sándor Szabó. Washington, DC: Mathematical Association of America, 1994. ISBN: 0883850281.

See Algebra and Tiling by S. K. Stein and S. Szabó. I hope to cover some of these topics in class.

Combinatorics and Symmetry

Buy at MIT Press Graham, R. L., M. Grötschel, and L. Lovász, eds. Handbook of Combinatorics. Vol. 1. Cambridge, MA: MIT Press, 1995. ISBN: 0262071703.

This includes such topics as permutation groups, association schemes, block designs, and codes. See Chapters 12-16 in Handbook of Combinatorics, vol. 1.

Course Meeting Times

2 sessions / week, 1.5 hours / session

Course Description

This course introduces techniques for studying intersection theory on moduli spaces such as homogeneous varieties, the Deligne-Mumford moduli space of stable curves and the Kontsevich moduli spaces of stable maps. The course emphasizes how one can deduce global geometric properties of moduli spaces and the objects they parameterize using intersection theory.

The topics include:

  1. Littlewood-Richardson rules for Grassmannians,
  2. Basic results about the divisor theory and cohomology of Mg due to Harer, Zagier, Arbarello and Cornalba,
  3. A study of Brill-Noether theory and the theory of limit linear series in order to prove that Mg is of general type if g is greater than or equal to 24,
  4. Recent developments due to Farkas, Gibney, Keel, Khosla and Morrison about the ample and effective cones of Mg,
  5. The Gromov-Witten invariants of simple homogeneous varieties, and
  6. The ample and effective cones of Kontsevich moduli spaces.


Algebraic Geometry (18.725). This is a first year graduate class in algebraic geometry at the level of the second and third chapters of R. Hartshorne (Algebraic geometry. New York, NY: Springer-Verlag, 1977). Familiarity with algebraic topology helpful.

Required Text

There is no required text for this course. However, there are many recommended readings.


Since there are no problem sets or exams for this course, the grade is based primarily on participation in the class.

Course Info

As Taught In
Spring 2006
Learning Resource Types
Lecture Notes