Pluripotential Theory, Hulls and Foliations

经典势理论建立于上世纪上半叶，它的发展主要（但不仅是）受到电物理学的兴趣的推动。多能理论要年轻得多，它的基本结果是在70 - 80年代贝德福德、泰勒、萨杜拉耶夫等人的基础著作中得到的。对这一领域的兴趣及其重要性是由复杂分析的需要和对其他领域（如动力系统）的应用决定的。虽然这一理论的许多基本结果已经被很好地理解，但我们将解决一些重要的开放性问题，涉及多极集的结构和多复格林函数的性质。项目的第二部分致力于研究在多项式和有理逼近的经典问题中自然出现的多项式和有理壳。尽管这些概念很重要，但多项式和有理壳的结构即使在Cn中的紧集合中也远未被理解，对于无界集合基本上根本不被理解。该项目的资金支持旨在促进国际合作、思想交流，特别是（也是最重要的）促进年轻数学家的发展。

Classical potential theory was built in the first part of the last century and its development was motivated mainly (but not only) by interests of electro-physics. Pluripotential theory is much younger and its basic results were obtained in 70’s - 80’s in the fundamental works of Bedford, Taylor, Sadullaev and others. Interest to this area and its importance are determined both by the needs of complex analysis and by applications to other fields, e.g. to dynamical systems. While many fundamental results of this theory are already well understood, we will address some important open problems concerning the structure of pluripolar sets and properties of the pluricomplex Green function. A second part of the project is devoted to the study of polynomial and rational hulls which appear naturally in the classical problem of polynomial and rational approximation. Despite the importance of these notions, the structure of polynomial and rational hulls is far from being understood even in the case of compact sets in Cn and is basically not understood at all for unbounded sets. Financial support for this project is aimed at fostering international collaboration, exchange of ideas, and especially (and foremost) at promoting young mathematicians.