Topics in random matrix theory and spectral theory of operators on Riemannian manifolds.

Large systems consisting of many elementary components obeying simple binary interaction rule are very effectiverepresentatives of complex systems. As the number of constituents increases collective behaviour involving largenumber of the elementaries emerges. It turns out that eigenvalues of large random matrices --- square array of numbers drawn with theaid of the throw of a die (which accounts for the randomness) and itscontinuous generalization involving differentiation---also behave in a similar manner. In these projects the behaviour of thethe eigenvalues in both instances will give value insights into the model complex systems.

Both pure and applied mathematicians in the widest sense of the term. This would include analysts whose interests include the asymptotic analysis of non-linear equations, those who work in the area of numerical linear algebra, statisticians and probabilists working in the area of multivariate statistics in particular multivariate analysis of variance. This would also include those working in an area call Geometric Analysis where questions about the topology of manifolds, in particular Riemanian manifolds is recast as a problem in partial differential equations, in particular the heat equation. The outcome of our research especially in the methodology will also be of interest to those electrical enegineers working in the area of wireless communications, since the proposed research which deals with the general question of linear statistics, where the Shannon capacity and a related quantity which characterise the error in a Multi-Input-Multi-Output system is a possible example. Therefore the outcome will be one of considerable commercial interest to telecommunication/mobile phone service providers in building capacity and other related matter such as generating wider coverage. In the longer term, better, more efficient and cheaper means of communication will be beneficial to the wider public. In addition to establishing new links with mathematicians from the Asia Pacific to develope futher colloborations Asia and Europe, better and cheaper mobile phone calls has the obvious advantage that easy communication with the rapidly expanding economies in the Far East would forster, generate and enhanced existing economic contacts. So in this way the possible technological enhancement enjoyed by beneficiaries of the new methodology developed in the proposed research would have potential impact on the quality of life of mobile phone users either for social or commercial interactions. The output of our research will be dissminated widely intially amongst the academic institutions; Universities and Research Institutes through seminars and giving talks at conferences. As Profs. Basor, Cheng, Wong and the PI and Co-I tend to have quite a few invitations and have extensive connections in the academic communities. In the second phase the PI will seek out his colleagues in the area of wireless communications in Imperial College and organize workshops to propagate the outcome of our research in order to generate further collaboration in this rapidly moving field.