Analysis and CR Geometry in Several Complex Variables

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A basic object of study in every calculus class is the derivative: it is a quantity that describes the rate of change of a function. For example, the derivative of position is velocity, and in economics, the code word for derivative is marginal. The derivative is a foundational concept in mathematics and the basis for much of the application of mathematics to physics and the other sciences, economics, and statistics. In applications, equations often contain functions and their derivatives. Such equations are called differential equations, and they describe the world: Newton's Laws of Motion, Maxwell's equations for electromagnetic radiation, Schrodinger's equations in quantum mechanics are all examples of differential equations. The derivative has a straight forward generalization to complex valued functions, and it is called the complex derivative in this situation. In order for the complex derivative of a complex valued function to exist, however, the function must satisfy a particular differential equation. In other words, complex differentiable functions have extra structure, and this extra structure provides far-reaching applications to a wide variety of subjects, including every areas of mathematics, engineering, physics, chemistry, and economics. This proposal investigates complex differentiable functions of several variables. Adding in additional variables creates significant new challenges, but the applications to mathematics and science are numerous and varied, and, as the PI believes, profound.In several complex variables, understanding the dbar-Neumann and Kohn Laplacians is a major driver of research questions. The PI and his collaborators will examine questions in the both the pseudoconvex and nonpseudoconvex categories. Together with Phillip Harrington of the University of Arkansas, the PI will prove existence and regularity results for the dbar-Neumann and complex Green operators on nonpseudoconvex and/or unbounded domains. These questions are beyond the scope and tools in the bounded pseudoconvex case for technical reasons, and the PI will develop new approaches to tackle these questions. In the pseudoconvex case, solvability is well-understood, and the questions instead focus on the relationship between the type of the boundary and the regularity of solutions. In the finite type case, the PI and Albert Boggess of the University of Arizona have an ongoing collaboration to establish pointwise estimates for the complex heat kernel on quadric submanifolds. There are few examples which are accessible to direct computation, and quadric submanifolds provide a large such class, including examples of higher codimension, all of which have a formulas that can be exploited. Quadric submanifolds have a very regular curvature structure, and the PI is also interested in the regularity of solutions to the (tangential) Cauchy-Riemann equations in the exponentially flat case. He has an ongoing collaboration with Khanh Tran of the National University of Singapore to establish L^p and Holder bounds for the Bergman and Szego kernels as well as the dbar-Neumann and complex Green operators.