Title: Zeros of Harmonic Polynomials and Related Applications
Speaker: Azizah Alrajhi
Date/Time: Friday, October 22nd, 2:00 pm
Abstract: In this thesis, we study topics related to harmonic functions, where we are interested in the maximum number of solutions of a harmonic polynomial equation and how it is related to gravitational lensing. In Chapter 2, we studied the conditions that we have to have on the real or complex coe cients of a polynomial p to get the maximum number of distinct solutions for the equation $p(z) - \bar{z}^2 = 0$, where deg p = 2. In Chapter 3, we discuss the lens equation when the lens is an ellipse, a limacon, or a Neumann Oval. Also, we discuss a counterexample for a conjecture by C. Beneteau and N. Hudson in [1]. We also in particular discuss estimates related to the maximum number of solutions for the lens equations for the Neumann Oval.
[1] C. Beneteau and N. Hudson. A survey on the maximal number of solutions of equations related to gravitational lensing. Complex Analysis and Dynamical System, Trends Math. Cham: Birkhauser/Springer, 23-38.