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PROPOSED RESEARCH PROJECTS


The following projects or project areas are proposed for the consideration of prospective PhD students.
 
 

Computational algebraic topology

(Dr Graham Ellis)
  • Development of methods/algorithms for making explicit topological calculations. A  downloadable preprint gives a flavour of the mathematics involved.
  • Applications of algebraic topology in Concurrency Theory (or the theory of parallel computation). A recent article gives a flavour of the mathematics involved.

Weights calculations for finite dimensional and affine Lie algebras:

(Dr John Burns)

Lie groups and Lie algebras (their tangent spaces at the identity) are central to much of mathematics and physics. For instance the Lie group SU(2) can be regarded as the set of states of an electron at rest at the origin. Lie groups and their Lie algebras are studied concretely by their action on a vector space, a so called representation, and (for compact groups) the (irreducible) represenrations are determined by the the eigenvalues of their maximal abelian subgroups, the so called weights of their representations.

In [1] new formulae involving sums of weights were proven . This project will use these formulae to:

1. Improve some known algorithms for computing weights and develop new ones.

Most methods for obtaining the weights of a Lie algebra representation involve a trawl of all “ height levels “ less than a maximal height that can be computed from the highest weight. Thus the multiplicity of certain linear combinations of roots are calculated and that very small number of combinations with positive multiplicities are weights of the representation. Most of the cost of the algorithm therefore derives from computing a zero multiplicity most of the time. The approach of this project will be to keep track of sums of certain weights sums ( “fundamental sums”) as we trawl and when the coefficients of the coordinates of the sums hit various bounds (determined by the theorems in [1]), the problem can be simplified. This part of the project will involve a considerable computing dimension. In addition to improvements and new methods in the Lie algebra case, some of the results will extend to affine Lie algebras in the case of integrable and restricted representations. This is an area of interest to many branches of mathematics at the moment.

2.Investigate the topological and geometric content of ( “fundamental sums”).

In [1] the applications of some “fundamental sums” . to the topology of compact symmetric spaces and generalised flag manifolds were studied . Similar investigations will be carried out for other “fundamental sums” as they are identified.

[1] Weight sum formulae in Lie algebra representations (J.Burns and M. Clancy), Journal of Algebra 257 (2002).



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