I am working with Bernard Deconinck on the theory and computation of Abelian functions. My primary goal is to provide the infrastructure to make Abelian functions as computationally accessible as trigonometric and hyperbolic functions thus allowing experimental advances in non-linear waves, combinatorial optimization, complex analysis, number theory, and algebra. Since all Abelian functions can be written in terms of homogenous rational functions of the Riemann theta function, Riemann theta functions are one focus of my research. I also examine algebraic-geometric methods for constructing periodic solutions to integrable nonlinear partial differential equations.
In particular I hope to accomplish the following in my Ph.D research:
implement the computation of the Riemann theta function and its derivatives as well as the computation of period matrices of complex plane algebraic curves in Python,
design and implement algorithms for computing Fay’s prime form, integrating differentials of the second and third kind on Riemann surfaces, and solving integrable equations such as the Korteweg-de Vries, Kadomtsev-Petviashvili, and nonlinear Schrodinger equations;
produce a constructive solution to the Schottky problem: given a Riemann matrix can we construct an algebraic curve which generates it?
This work is a continuation of the work by a previous student of Bernard’s, Matt Patterson.