函数空间、几何和Mahler测度

项目名称： 函数空间、几何和Mahler测度

项目编号： No.11471113

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 黄寒松

作者单位： 华东理工大学

项目金额： 60万元

中文摘要： 本项目拟研究乘法算子构成的算子代数性质，及其符号的函数性质之间的联系；同时考虑当符号为多项式时，其定义的Mahler测度，从数论和几何、代数角度分析其性质。乘法算子的约化子空间问题一直是算子论专家和函数论专家关注的一个重要课题，我们已经在这方面作了很多工作；这些工作启发了算子理论和函数论之间深入的联系，以及一些新问题的考虑，如Chang-Marshall定理的应用。Mahler测度和和超越数论、动力系统有着密切的联系。从Mahler测度的一些特殊值可以看到它和代数曲线及几何之间的联系；也有大量有意义的问题值得考虑，寻求其和函数论、几何之间的联系。

中文关键词： von；Neumann代数；函数空间；Mahler测度；多项式

英文摘要： This project will focus on the study of the properties of operator algebra induced by multiplication operators defined on function spaces, and that of their symbols. In particular, when the symbols are polynomials, their Mahler measures are of special interest, in view of number theory, geometry and algebra. The reducing subspace problem of multiplication operator has been the focus of operator-theorists and function theorists, for which we have done a lot of deep jobs, that inspire some deep connection between operator theory and function theory, and stimulates some new questions, such as the application of Chang-Marshall's theorem. It is known that Mahler measure has close connection with transcendental number theory, dynamical system, and also with algebraic curve. Also, there are quantity of problems that are valuable, which may indicates some connections with other aspects of mathematics.

英文关键词： von Neumann algebra;function spaces;Mahler's measure;polynomial