This project will focus on the study of topics in complex analysis in one and several variables and in related geometric theories. The work is motivated by the observation that with respect to linear fractional transformations, certain structures in higher dimensions behave like their one-dimensional counterparts. The principal investigator will study ten interconnected problems that divide roughly into three areas: i) the study of estimates for the norm and spectrum of the Leray transform, a multivariate analogue of the Cauchy transform; ii) in higher dimensions, the local and global characterization of surfaces in terms of invariant curvatures and kernel symmetries; and iii) the study of the Szego projector and Bergman projector under proper holomorphic maps and on weighted spaces. In addition, the principal investigator will supervise undergraduate students on projects involving the study of problems that arise in the linear fractional geometry of the dual and double planes, and for the case of the complex plane, involving the continued study of an implementation of the Kerzman-Stein method as it pertains to the computation of canonical domain functions.
Several complex variables has a rich history of providing a context in which techniques from analysis and geometry support each other in important ways. This began with the pioneering work of Hartogs in the 1900's and it continued with the work of Oka in the 1950's. It was followed by work of Chern, Hormander, and Kohn, and more recently, for instance, by work of Catlin, D'Angelo, and Webster. The proposed work will follow in this tradition by drawing together elements of analysis and geometry in order to draw new connections in complex analysis between one and several variables. Essential to the project will be the involvement of undergraduate students who will participate directly in the research. Students will work in teams of two doing original research on geometric and computational problems that are motivated by the study of one complex variable. They will be expected to present their work externally and to publish their work in suitable journals. The end result is that there will be more students who are excited about mathematics, who have experienced the benefits of collaborative research, and who are better prepared for graduate work in mathematics. In this way the project also contributes to the greater mathematical and scientific infrastructure.
The award NSF Grant DMS-1002453 supported research in complex analysis at the interface of analysis and geometry. It also provided funding for nine undergraduates working on 5 summer REU projects in analysis and geometry. Complex analysis is a classical area of mathematics that has application to other areas of mathematics as well as the physical sciences. Techniques developed for its generalization to higher dimensions have found application to more areas of mathematics and in mathematical physics. The work supported by this grant uncovered analytic and geometric properties of basic structures in the subject. Outcomes include a transformation law for the Szego projector under proper holomorphic maps; a calculation of the spectrum of the Kerzman-Stein operator for a family of planar regions; a description of the spectrum of the Kerzman-Stein operator for regions with corners; and a local characterization of real hypersurfaces with constant invariant curvature. Results were published in peer-reviewed mathematics research journals. The grant also enabled the principal investigator to participate in regional conferences and workshops and to invite collaborators to his home institution. The 5 summer REU projects lasted 8-10 weeks and all resulted in publications in the College Math Journal, Pi Mu Epsilon Journal, or Rose-Hulman Undergraduate Math Journal. The students were majoring in mathematics, came from all grade levels, and had a desire to pursue an advanced degree in the sciences. In this way, the grant provided training for those among the next generation of scientists and engineers. At the end of the summer, students participated in MathFest where they interacted with like-minded students from across the country. The upper-level REU projects continued a program of proving theorems for analytic functions of a generalized complex variable. In this way, the projects reinforced learning that students gained through regular coursework.