Problems in Complex Analysis

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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.