Project: Research

- Smoktunowicz, Agata (Principal Investigator)

Status | Finished |
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Effective start/end date | 1/08/06 → 31/07/11 |

Total award | £494,573.00 |
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Funding organisation | EPSRC |

Funder project reference | EP/D071674/1 |

Period | 1/08/06 → 31/07/11 |
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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.

## Primitive algebraic algebras of polynomially bounded growth

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

## Jacobson radical non-nil algebras of Gelfand-Kirillov dimension 2

Research output: Contribution to journal › Article › peer-review

## Iterative refinement techniques for solving block linear systems of equations

Research output: Contribution to journal › Article › peer-review

## Graded algebras associated to algebraic algebras need not be algebraic

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

## EXTENDED CENTRES OF FINITELY GENERATED PRIME ALGEBRAS

Research output: Contribution to journal › Article › peer-review

## GK-DIMENSION OF ALGEBRAS WITH MANY GENERIC RELATIONS

Research output: Contribution to journal › Article › peer-review

## Makar-Limanov's conjecture on free subalgebras

Research output: Contribution to journal › Article › peer-review

## The prime spectrum of algebras of quadratic growth

Research output: Contribution to journal › Article › peer-review

## The Jacobson radical of rings with nilpotent homogeneous elements

Research output: Contribution to journal › Article › peer-review

## An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension

Research output: Contribution to journal › Article › peer-review

## IMAGES OF GOLOD-SHAFAREVICH ALGEBRAS WITH SMALL GROWTH

Research output: Contribution to journal › Article › peer-review

## IMAGES OF GOLOD-SHAFAREVICH ALGEBRAS WITH SMALL GROWTH

Research output: Contribution to journal › Article › peer-review