The series of lectures will cover several aspects of matrix theory over fields and rings and applications of the results to general questions in algebra. Open problems will be identified and discussed.

The topics covered will include:

- Factorisation problems over fields and rings, especially over the ring of integers. The question of existence of factorisations and the number of factors required will be examined;

- Representation theory and matrix semigroups. The possibility of finding efficient multiplication algorithms using these tools will also be discussed;

- Nonnegative and Patterned Matrices, including spectral and combinatorial questions and completion problems;

- Boolean matrices and low rank factorisation problems. In particular, some results on the problem of factoring nonnegative matrices of low rank as a product of nonnegative rectangular matrices of sizes reflecting the rank will be presented;

- Similarity problems for matrices over rings, especially over the ring of integers;

- Applications of matrix theory to other parts of Mathematics, and to areas such as communications and information technology.