A central simple algebra is an algebra over a field whose centrer is exactly that field, its dimension over its center is finite and it has no nontrivial two-sided ideals. The easiest example is the matrix algebra over a field. Division algebras (of finite dimension over their center) are also central simple algebras. The first example of a noncommutative algebra of this type is Hamilton's quaternion algebra which has proved to be useful in Physics. Nowadays, central simple algebras are also used in error correcting codes.
I am interested in the classification of central simple algebras. In particular, I am interested in solving the word problem for the Brauer group of a field - the group that encodes most of the relevant information about central simple algberas over that field.
I have also made contributions to the study of computational aspects of these algebras, in particular solving polynomial equations over division algebras and finding the left eigenvalues of matrices over division algebras.
My list of publication can be found in my cv.