Finite Geometries and their Applications

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My aim is to continue an established program of research in discrete and applied mathematics, specializing in finite geometries and designs with applications to communication theory. Modern technology deals with the storage and transmission of digital information.The groundwork was laid in the nineteen forties by Claude E Shannon, a mathematician and engineer. In particular he showed how to digitize [band-limited] analog signals in a practical way. Many practical problems in the communications area remain open. They give rise to basic mathematical problems which still remain unsolved despite the efforts of notable mathematicians and Fields medalists. I continue to work on these problems, such as the following: 1.the existence problem for finite projective planes and block designs. 2.the longest linear MDS codes problem I have contributed some results [e.g. for the order 10 case for planes] towards a partial solution of 1 and I have co-authored a major result for problem 2. For 1, the main tool is a collection of results in blocking sets which I have developed. The work in 2, published in the Inventiones, was inspired by Segre's brilliant use of the Hasse-Weil theorem for curves. The main practical examples of these MDS codes are the widely-used and ubiquitous Reed-Solomon codes which first came to prominence for error control with the invention of the compact disc player. The broad agenda of my ongoing research program continues to be my work in finite geometries and designs and related areas of discrete mathematics such as graph theory, but always with an eye to applications.I have written two patent applications in the communications area in the past and am working on two others.I am eager to vigorously pursue this ongoing research programme.