NSF Award
1600371
This project combines two very powerful ideas of mathematics -- that of complex numbers and that of the calculus. Complex numbers include quantities such as the square root of negative one. The calculus studies how physical or geometric quantities vary in space or time. The combination of these ideas (called complex analysis) leads to far-reaching and beautiful results about smoothly varying complex quantities (called holomorphic functions). One may use these ideas to model various natural phenomena such as electrical attraction or the motion of liquids. These considerations also have surprising consequences in other parts of mathematics, such as the properties of prime numbers, the geometry of higher dimensional spaces, and the study of equations (called partial differential equations) used to describe many physical processes such as heat conduction and the propagation of waves. This research project studies the behavior of holomorphic functions as one approaches the boundary of the region in higher dimensional space where the function is defined. The investigator will also continue his successful mentoring of undergraduate student research in related topics.<br/><br/>The behavior of holomorphic functions of several variables is well-understood on strongly pseudoconvex domains. In this project, the goal is to study the extension of such results to new and more general types of domains. Among the questions under study are estimates for the solutions of the solutions of the inhomogeneous Cauchy-Riemann equations, understanding what happens when there are no such estimates, and the boundary behavior of holomorphic functions. The domains to be studied include piecewise smooth domains (in particular product domains), Levi-flat Stein domains in complex manifolds, and non-pseudoconvex domains, both with and without "holes." The research will involve both the investigation of general questions and the careful study of particular examples, which can exhibit unexpected phenomena associated to these domains. Tools employed in this project to study these problems include a priori estimates, integral formulas, and algebraic methods based on sheaf theory, as well as insights from other parts of mathematics such as partial differential equations, functional analysis, and differential geometry.
