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

在过去的六年里，我与几位博士生以及比利时、法国、德国、意大利和墨西哥的同事合作进行了研究，研究朝着三个方向发展：（1）发展理解小波变换和Wigner函数的统一群论框架，（2）在矩阵域上构造矢量相干态，研究Berezin量化的后续问题;（3）非欧曲面上的类小波变换和时频变换的构造。这项工作是数学兴趣的理论平方可积群表示和矩阵模型，以及应用问题的信号分析非平坦的几何形状。在今后五年中，我打算继续在这三个方向上努力。维格纳函数的工作将继续在一般对称空间，作为早期结果的推广。这，应该打开了定义信号变换的一些有趣的信号几何形状的可能性。在数学上，它将导致对维格纳函数的更深入的理解，如在相空间上定义的那样。在矩阵域上构造矢量相干态是我们提出的一个新思想，它扩展了标准的矢量相干态理论。许多半单李群的齐性空间自然是矩阵域。我们在那里介绍的矢量相干态导致了用矩阵变量表示的解析函数空间。目前，我正在研究Berezin型量子化，使用这种相干态及其相关的再生核。一个相关的问题是研究这种量子化理论的经典极限，特别是因为它们似乎触及了非对易几何的问题。另一个有趣的问题，我们打算深入探讨的是构造矢量相干态的超对称哈密顿的可能性。小波和时间-频率变换的工作将建立在不同小组最近取得的一些成果之上。我们提出了一个组理论和几何模型开发这样的变换。这项工作已经引起了研究界的广泛关注。

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.