Quantization, vector coherent states and wavelets on non-euclidean surfaces

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My research over the last six years, carried out in collaboration with several doctoral students and with co-workers in Belgium, France, Germany, Italy and Mexico has moved in three directions: (1) developing a unified group theoretical framework for understanding wavelet transforms and Wigner functions, (2) constructing vector coherent states over matrix domains, and studying the ensuing problem of Berezin quantization and (3) construction of wavelet-like and time-frequency transforms on non-Euclidean surfaces. The work is of mathematical interest in the theory of square-integrable group representations and matrix models, as well as to applied problems of signal analysis over non-flat geometries. In the next five years, it is my intention to continue working in all three directions. The work on Wigner functions will be continued over general symmetric spaces, as an extension of an earlier result. This, should open up the possibility of defining signal transforms for a number of interesting signal geometries. Mathematically, it would lead to a deeper understanding of the Wigner fuction, as defined on a phase space. The construction of vector coherent states over matrix domains was a new idea introduced by us, which extends the standard theory of vector coherent states. The homogeneous spaces of many semi-simple Lie groups are naturally matrix domains. The vector coherent states that we have introduced there lead to spaces of analytic functions, in terms of matrix variables. Currently, I am studying Berezin-type quantization, using such coherent states and their associated reproducing kernels. A related problem is to study the classical limits of such quantized theories, particularly since they seem to touch on questions of non-commutative geometry. Another interesting problem that we intend to explore in depth is the possibility of constructing vector coherent states for supersymmetric Hamiltonians. The work on wavelets and time-frequency transforms will build on some recent results obtained by different groups. We are proposing a group theoretical and a geometrical model for developing such transforms. The work has already attracted wide attention within the research community.