Computation of capacity and automorphisms of the spectral ball

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This research program is made of two distinct subprograms: the computation of capacity and the classification of the automorphisms of the spectral ball. CAPACITY is a central notion in potential theory with many important applications in complex analysis, approximation theory and number theory. Computing the capacity of a general compact set is a very hard problem. However, the recent algorithm developed by Ransford and myself offers a practical way of computing the logarithmic capacity of a large class of compact subsets of the plane. One of the main objective of this research program is to exploit, extend and improve this new method. Here is a short list of projects. 1- To extend the algorithm to other capacities, for example, Riesz capacities, weighted capacities (in R^n). 2- To relax the hypotheses under which the algorithm converges, especially HCP. 3- To enhance the performance of the algorithm by exploiting the special form of the linear programs involved. 4- To test the strong isoperimetric inequality for counterexamples in the case of disconnected sets. 5- To improve the precision on the capacity of special sets, for example the Cantor set. 6- To develop a software for automatic computation of capacity. The classification of the AUTOMORPHISMS OF THE SPECTRAL UNIT BALL of M_n is still unfinished, 17 years after the Ransford-White conjecture was stated. Although recent progress have been made, the conjecture is still open even in the case n=2. We suggest an experimental approach to some of the underlying questions. Moreover, we would like to exploit the reformulation of the problem in the theory of cohomology. The problem of understanding the automorphisms of the spectral ball arises from the spectral Nevanlinna-Pick problem. This one in turn comes from a question of robust stability in the field of control theory.