Course Description

This class will be an introduction to mulitiplication sets, vector space realizations, and applications to problems in combinatorics, including some very interesting applications. The first two weeks (taught by An Huang) will be on multiplication sets, and introducing combinatorial problems, while the last two weeks (taught by Erik Carlsson) will deal with vector space realizations and applications to those problems.

Here is a tentative syllabus:

  • First week: Concept of mulitiplication set. Multiplication sets as functions on another set. Examples: Permutation sets, Similarity sets, Quotient sets.
  • Second week: Some combinatorics problems, the card game SET as a mulitplication set. Counting multiplication elements that multiply to one. Counting Permutation invariant polynomials.
  • Third week: Vector space realizations of mulitplication sets. Minimal realizations. Trace functions. Realization of permutation sets. Block diagrams. Equality of vector space realizations.
  • Fourth week: Applications, group functions and the mulitplication to one problem. Trace functions and permutation invariant polynomials. Group functions and the SET question.