Date of Award

5-11-2025

Date Published

June 2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Dan Coman

Second Advisor

Loredana Lanzani

Keywords

Complex Analysis;Harmonic Analysis;Numerical Applications;Pluripotential Theory;Several Complex Variables

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

There are two research themes within complex analysis presented. The first is developing generalizations of the Unified Transform Method as a novel way to compute fluid flows and numerically solve boundary value problems for holomorphic functions and solutions to the Laplacian and complex Helmholtz equation. The second is finding a formula for the pluricomplex Green function with two poles of equal weights for the bidisk. The Unified Transform Method (also sometimes known as the Fokas Method) is a technique for numerically solving boundary value problems for linear and integrable nonlinear PDEs. Crowdy reformulated the Unified Transform Method in the complex analytic setting in the plane. In section 3.2, we extend Crowdy's notion of transform pair for the Laplacian and complex Helmholtz equation to any bounded convex domain in the plane. In section 3.3, we use the Szegő kernel to develop a new transform pair for analytic functions. Applications include mixed boundary value problems and fluid flow computations. The work on the Unified Transform Method is joint work with L. Lanzani, E. Luca, and S. Llewellyn Smith. Pluricomplex Green functions are the fundamental solutions to the complex Monge Ampère operator. In 2017, Kosiński, Thomas, and Zwonek proved the equality between the pluricomplex Green function and the Lempert function for the bidisk with two poles of weight one. We find a formula for the pluricomplex Green function for the bidisk with two poles of weight one. We divide the bidisk into two open regions, where the formula is found explicitly on the first region, and the other region is the union of a family of hypersurfaces. On each hypersurface, the formula is explicit up to a unimodular constant that is the root of a sixth degree polynomial.

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