Problems in Complex Analysis

首席研究员计划研究复分析中的各种问题。长期目标是理解柯西-黎曼方程和相关主题。问题要么直接涉及证明估计，提高对这些方程的基本关键概念的理解，要么直接涉及应用估计。(i)我们将与Gautam Bharali和Berit Stensønes共同研究有限型伪凸域。我们未来三年的目标是提高我们对边界几何形状的理解。这是发展柯西-黎曼方程结果的必要前提。Bharali，Stensones和PI最近通过更好地理解高阶正切的复杂曲线而取得了进展。(ii)我们将与T。Dinh，F. Rong，Nessim Sibony和Erlend F.复射影空间中黎曼曲面的叠层。我们未来三年的目标是提高我们对更高维度叠层的理解。作者最近发现了一些黎曼曲面叠层的例子。(iii)黎曼面对叶理的研究引出了关于水流的基本问题。在二维空间中，正闭合流可以近似为黎曼曲面积分流。在更高维度中，对电流进行类似的几何解释将是有用的。我们建议与科曼一起研究这个问题。第一种情况是三维电流。(iv)PI还建议与Lina Lee和Yuan Zhang合作。我们正在研究涉及不变度量和无限型伪凸域的各种问题。(v)PI还计划继续与Loredana Lanzani合作研究复杂的div-curl定理，并与Klas Diederich合作研究D 'Angelo域的Cauchy Riemann方程的保持器估计和其他问题。我们计划与处于不同职业阶段的数学家合作。一些高级数学家Diederich，Løw，Sibony和Stensønes。我们相信，我们四个人一起对几个复变量和复动力学领域有了很好的概述。这有利于我们的联合项目，也有利于我们与年轻数学家的合作。我们计划与四个职业中期的数学家，Bharali，Coman，Dinh和Lanzani以及4个博士后，Lee，Rong，Zhang和Wold合作。我们还与研究生Crystal Zeager和Taeyong Ahn合作。其中5人为女性。

The principal investigator plans to work on various problems in complex analysis. The long term goal is to understand the Cauchy-Riemann equation and related topics. Problems either are directly concerned with proving estimates, improving understanding of key concepts basic for these equations or applying estimates. (i) We will work jointly with Gautam Bharali and Berit Stensønes on finite type pseudoconvex domains. Our objectives for the next three years is to improve our understanding of the boundary geometry. This is a necessary prerequisite to develop results on the Cauchy-Riemann equations. Bharali, Stensones and the PI have recently made progress by developing a better understanding of complex curves tangent to high order. (ii) We will work jointly with T. Dinh, F. Rong, Nessim Sibony and Erlend F. Wold on laminations by Riemann surfaces in complex projective space. Our objective for the next three years is to improve our understanding of laminations in higher dimension. The authors have recently found some examples of Riemann surface laminations.(iii) The study of foliations by Riemann surfaces leads to basic questions about currents. In two dimensions, positive closed currents can be approximated by currents of integration of Riemann surfaces. In higher dimension, it would be useful to have a similar geometric interpretation of currents. We propose to work on this topic with Coman. The first case is that of currents in three dimensions. (iv) The PI also proposes joint collaborations with Lina Lee and Yuan Zhang. We are working on various problems involving invariant metrics and infinite type pseudoconvex domains. (v) The PI also plans to continue the project with Loredana Lanzani on a complex div-curl theorem and with Klas Diederich on Holder estimates for the Cauchy Riemann equation of D'Angelo domains and other questions.We plan to work with mathematicians at different stages of their carreers. Some senior mathematicians are Diederich, Løw, Sibony and Stensønes. We believe that together the four of us have a good overview of the fields of several complex variables and complex dynamics. This benefits our joint projects but also our work with younger mathematicians. We plan to work with four midcareer mathematicians, Bharali, Coman, Dinh and Lanzani and also with 4 postdocs, Lee, Rong, Zhang and Wold. Also we work with graduate students, Crystal Zeager and Taeyong Ahn. Of these, 5 are females.