Nil algebras, algebraic algebras and algebras with finite Gelfand-Kirillov dimension

Noncommutative rings are the fertile meeting grounds for many areas of mathematics and physics, as noted by T.Y.Lam in his book 'A first course in noncommutative rings'. The project Investigated the structure of noncommutative rings, especially nil rings, Golod-Shafarevich algebras and rings appearing in noncommutative algebraic geometry. Nil rings are rings in which every element to some power is zero. They are related to torsion groups and were used for solving the General Burnside Problem in Group theory. Nil rings are algebraic and Jacobson radical. The Jacobson radical plays a crucial role in the study of the general structure of rings. Key results related to nil rings obtained under the grant: 1. Solution of the Makar-Limanov conjecture by constructing nil algebras which, after extending the base field, contain noncommutative free subalgebras in two generators. 2. Constructing finitely generated, infinite dimensional nil algebras with the smallest known growth; these algebras have approximately n² elements of degree n, linearly independent over the base field, for almost all n. 3. Providing the first example of Jacobson radical not nil algebras with the smallest possible growth, that is the Gelfand-Kirillov dimension two (it is known that the Jacobson radical algebras with the Gelfand-Kirillov dimension less than two are nil).
Another idea investigated using the grant was Golod-Shafarevich algebras, answering open questions by Field's medallist Efim Zelmanov. An algebra is called Golod-Shafarevich if the number of generating relations of each degree is not large, and they were used to solve the Tower of fields conjecture in Number Theory, the General Burnside Probelm in Group theory and the Kurosh Problem in noncommutative algebra. It is known that Golod-Shafarevich algebras have exponential growth; I have shown that there are Golod-Shafarevich algebras whose infinite dimensional homomorphic images have exponential growth. Later,
together with L. Bartholdi I showed that if the number of defining relations of each degree is polynomially bounded then such Golod-Shafarevich algebras have homomorphic images of polynomial growth. This shows that there are algebras with polynomial growth which satisfy the prescribed relations, under mild assumptions about the number of defining relations of each degree.
Algebras related to noncommutative projective geometry were also investigated. In particular, the structure of domains with quadratic growth which have a non trivial derivation was completely determined, showing that the Artin proposed classification of domains holds for rings with derivations.

In addition, in a collaborative effort I used algebraic methods in the area of numerical linear algebra as follows. In many practical applications, e.g. arising in solving differential equations numerically, we need to solve a linear system of equations Ax=b where A is a nonsingular matrix with a special block structure. Very often the block matrices A(i,j) are sparse and many of them are zero. Numerical algorithms ought to exploit the structure of the matrix A. The numerical properties of solutions of a nonsingular system of linear equations Ax=b , with A partitioned into blocks, were studied using a classical iterative refinement (IR) algorithm and a k-fold iterative refinement (RIR) algorithm, using only single precision. We proved that RIR has superior numerical quality to IR.