FORMANEK 96-10118 This project has two parts, both concerned with nxn matrices over a field. The first part is the classification of the irreducible n-dimensional representations of certain infinite groups, such as braid groups, mapping class groups, and Torelli groups. While there is presently no possibility of a complete classification in all dimensions, a classification in low dimensions should be possible. Note that many representations of braid groups have been constructed by both mathematicians and physicists, especially in connection with the Yang-Baxter equation. The second part is the study of the invariants and identities of nxn matrices. This is research in ring theory and group theory, two branches of algebra. The problems and methods to attack them are combinatorial, and deal with particular rings and groups, rather than general theory. As noted above, some of these questions are also of interest to physicists.
