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来自比利时、法国、德国、意大利和墨西哥的同事合作进行了研究，研究方向有三个：(1)建立理解小波变换和Wigner函数的统一群理论框架；(2)在矩阵域上构造矢量相干态；研究随后的Berezin量化问题；(3)在非欧几里德曲面上构造类小波变换和时频变换。工作是在数学兴趣的平方可积群表示和矩阵模型的理论，以及在非平坦几何信号分析的应用问题。在接下来的五年里，我打算继续在这三个方向上努力。关于维格纳函数的工作将在一般对称空间上继续，作为先前结果的扩展。这将为定义一些有趣的信号几何形状的信号变换提供可能。在数学上，它将导致对相空间上定义的维格纳函数的更深入理解。在矩阵域上构造矢量相干态是我们对矢量相干态标准理论的扩展，提出了一个新的思想。许多半单李群的齐次空间是自然的矩阵域。我们在这里引入的向量相干态引出了解析函数的空间，用矩阵变量表示。目前，我正在研究berezin型量化，使用这些相干态及其相关的再现核。一个相关的问题是研究这些量子化理论的经典极限，特别是因为它们似乎涉及到非交换几何的问题。我们打算深入探讨的另一个有趣的问题是为超对称哈密顿量构造矢量相干态的可能性。关于小波和时频变换的工作将建立在不同小组最近取得的一些结果的基础上。我们提出了一种发展这种变换的群理论和几何模型。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.