Mathematical Sciences: Several Complex Variables

David Barrett将研究与n维复杂空间和更一般流形中的光滑有界域的函数理论和几何性质有关的各种主题。一些研究课题涉及的问题在满足Levi形式的某些正性条件的域上得到了很好的理解，但在更一般的域上却知之甚少。关于标准函数空间中的近似理论、Bergman投影的正则性、生物全纯映射和固有映射的奇异性等问题都属于这一类。巴雷特还将研究与李维形式上满足很强条件的域和CR流形有关的问题。这些问题包括严格伪凸三维CR流形的全局嵌入问题和列维平面超曲面的全局性质问题。

David Barrett will work on various topics relating to function theoretic and geometric properties of smooth bounded domains lying in n-dimensional complex space and in more general manifolds. Several of the research topics concern questions which are fairly well understood on domains which satisfy certain positivity conditions on the Levi form but are poorly understood on more general domains. The questions raised about approximation theory in standard function spaces, regularity of the Bergman projection, and singularities of biholomorphic and proper maps fall into this category. Barrett will also investigate problems relating to domains and CR manifolds satisfying very strong conditions on the Levi form. These include questions about global embedding of strictly pseudoconvex three dimensional CR manifolds and about global properties of Levi-flat hypersurfaces.