"Quantization, coherent states and complex orthogonal polynomials: applications to physics and signal analysis"

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My research over the past six years has been a continuation as well as extension of an ongoing programme, involving several graduate students and the following coworkers: J.-P. Antoine (Louvain-la-Neuve), F. Bagarello (Palermo), M. Engli·s (Prague), J.-P. Gazeau (Paris), The work has been an outgrowth of our decades long research on coherent states and square-integrable group representations and their use in quantization, wavelet analysis and signal processing. Most of the earlier results of the work have been published in two monographs while some of the more recent work has been described in two review articles, one of them being in the Encyclopedia of Mathematical Physics. This work has had applications to signal analysis and image processing, in quantum optics and atomic physics and on the mathematical side, in the field of harmonic analysis and time-frequency and wavelet analysis. My work over the next few years is expected to continue along the above lines, as well as moving into a number of new directions. One of these new areas of work will be in a generalization of coherent states on Hilbert spaces to more general Hilbert modules. This, apart from its intrinsic mathematical value, in the study of related subnormal operators, would also find applications to problems in non-commuting quantum mechanics and more generally to the theory of non-commutative spaces. We also expect to study the problem of entanglement in quantum computing within this setting. A second new direction of work will be in the area of orthogonal polynomials related to coherent states and a third to the connection between Baysian duality in classical statistics and coherent states. The well known problem of the electron in a constant magnetic field has a particularly interesting algebraic structure. We have studied this in some detail in the past couple of years and are now in the process of generalizing the system to include pseudo-bosons. My research touches on some of the most actively worked upon areas of mathematical physics.