Course Announcement

Combinatorics students will learn how combinatorics can be applied to problems of computer science, and computer science students will encounter many tools (especially orthogonal polynomials) that are standard in combinatorics, but have not yet made their way into theoretical computer science. Everyone involved should expect to encounter interesting research problems as well as learn techniques that they will find useful elsewhere in their research.

As such a course has not been taught before at MIT, we will produce lecture notes.

The following is a list of the topics that I presently plan to discuss. There are many relations among the topics listed, but I have forced them into a tree for ease of viewing.

• Eigenvalues of graphs
  • Second eigenvalue/mixing rate
    • Isoperimetric number, multicommodity flow, geometric embeddings and isoperimetry
    • Spectral partitioning, geometric embeddings and eigenvalues
    • Mixing rates of walks, estimation of volume of convex bodies
  • Isomorphism testing
    • Cospectral graphs
    • Relation of eigenvalues to automorphisms and testing isomorphism of graphs of bounded eigenvalue multiplicity
  • Pseudo-random dense graphs
    • Discrepancy bounds,
    • Paley graphs and Weil sums
    • A little Ramsey theory
  • Expander graphs
    • Random expanders
    • Explicit construction (probably Gabber-Galil)
    • Application to re-using random bits
  • Miscellaneous cute facts (if time allows such frivolity)
• Eigenvectors of graphs
  • Planar graphs
  • Regular polytopes
  • Cayley graphs
  • Strongly-regular graphs
    • Relation to spherical designs, extremal point sets, and density of sphere packings
    • Limits of regularity
  • Association schemes, Bose-Mesner algebra, and design of statistical experiments
• Error-correcting codes
  • Lower bounds
    • Naive volume bound
    • Elias's bound
    • Linear Programming bound
  • Constructions
    • Gilbert-Varshamov bound (random constructions)
    • Reed-Solomon Codes
    • Justesen Codes
    • Codes from expander graphs and superconcentrators
  • Weight distributions of codes
  • Constructing expanders from error-correcting codes
• A little extremal set theory
  • Erdos-Ko-Rado theorem
  • Ray-Chaudhuri--Wilson theorem
• Hashing
  (material might be pushed into next semester--advanced complexity theory)
  • Public coins versus private coins
  • Hash functions from error-correcting codes