This project concerns the ring of invariants of a set of generic nxn matrices over a field K, and the trace ring, which is the noncommutative ring generated by the ring of invariants and the generic matrices. The fundamental problem is to find presentations for these algebras, which would solve various explicit combinatorial problems. These combinatorial problems have interest in their own right, and currently seem more likely to be solved than the general problem of finding presentations. Other questions to be considered are whether the quotient field of C(n,r) is purely transcendental over K and what is the structure of the variety of representations of a group G in the trace ring. The research supported concerns matrix theory, ring theory and the calculation of invariants. In particular, this involves an old problem of determining what mathematical functions are preserved under various transformations. This is of interest in many areas of science as well as in mathematics.
