Research

My work is a synthesis of complex analysis and potential theory. Here is a synopsis of various projects I have been developing:

POLYNOMIAL APPROXIMATION

I have recently been investigating with collaborators the best uniform polynomial approximation to the "Checkmark Function" f(x;a)=|x-a| on the interval [-1,1]. Treating a as a parameter to vary in (-1,1), we examine the evolution of the minimax error E(a) for approximation with polynomials of given degree. We also track the positions of the Chebyshev alternation nodes and describe their phase transitions.

ELECTROSTATIC EQUILIBRIUM ON THE SPHERE

Place a unit amount of charge onto a sphere, which is free to redistribute into a configuration of lowest energy. Then, place point charges of like charge onto the sphere. How does the initial charge redistribute in the presence of these charges? In other words, what is the resulting equilibrium measure which will represent the charge distribution? Using logarithmic potential theory and stereographic projection, I am analyzing this problem with a fusion of complex analysis and potential theory, and it turns out that quadrature domains have an important role to play.

QUADRATURE DOMAINS IN SEVERAL COMPLEX DIMENSIONS

Quadrature domains are domains where integration on a chosen test class of functions coincides with a linear combination of point evaluations of the functions and their derivatives. For example, the simplest quadrature domain is the disc, with the test class of integrable harmonic functions: all these functions integrate to a multiple of the function value at the center by the harmonic mean value theorem! Quadrature domains in the plane have many striking properties and make an appearance in many physical and potential theory applications. Using as test class the space of square-integrable analytic functions (Bergman Space), I have published an account of some properties of quadrature domains in several complex variables. Among the fruit of this project has been a result along the lines of generalizing the Riemann Mapping Theorem: that every smooth bounded convex domain in several complex variables is biholomorphic to a quadrature domain.

BERGMAN PROJECTION ON ELLIPSOIDS:

In several complex variables, the orthogonal projection of L^2 onto the Bergman Space A^2 (so, the projection of square integrable functions to the subspace of analytic square integrable functions), is called the Bergman Projection. The behavior of this operation is of interest in the complex analysis, function theory, and geometry of several complex dimensions. I have shown that on ellipsoids in any number of complex variables, the Bergman Projection maps polynomials to polynomials without increasing degree, which is reminiscent of properties of the Dirichlet Problem in the plane. In the plane, there is also a connection to the so-called "Khavinson-Shapiro Conjecture," which posits that among bounded domains, only ellipses should have polynomial solutions to the Dirichlet Problem for the Laplacian when polynomial data is posed on the boundary.