A survey comparing the potential theoretic properties of lemniscates and quadrature domains, two classes of algebraic curves.
We discuss a nonlinear moving boundary problem where a domain grows by the normal derivative of its own harmonic Green's function.
A celebrated theorem of S. Richardson reduces this problem to a parallel investigation of quadrature domains,
special domains that generate an exterior gravity potential equivalent to that of finitely many point masses.
We review these connections and discuss exact solutions in the plane.
Then we consider the higher-dimensional case, where we pass from the real domain through the complex domain in order to apply hyperbolic methods
(characteristics and globalizing families) to study elliptic Cauchy problems.
Presentation of joint work with A. Eremenko answering a question of H.S. Shapiro and B. Gustafsson.
The mean-value property for harmonic functions states that the integral of any harmonic function over a disk
(resp. n-ball) equals a constant times point-evaluation at the center.
More generally, a so-called quadrature domain admits a formula for integration of any harmonic function in terms of a sum of weighted point-evaluations.
We state equivalent definitions that bring together perspectives from potential theory, holomorphic PDE, and function theory.
As an application, we describe a moving-boundary problem from fluid dynamics (Hele-Shaw flows) for which quadrature domains give exact solutions.
Our main goal is to address the lack of explicit examples in higher dimensions compared to the abundance of examples in the plane.
A mini-course in potential theory proving theorems in the real domain while passing through the complex domain (in several variables).
Discussion of the Khavinson-Shapiro conjecture and a connection to Bergman orthogonal polynomials.
Discussion of non-linear moving boundary problem and its infinite set of conservation laws.
Wilmshurst's conjecture bounds the number of zeros of a (complex-valued) harmonic polynomial. The proof of a known special case was extended to rational functions by D. Khavinson and G. Neumann. This turned out to resolve S. Rhie's conjecture in astronomy on the number of images produced by point-mass gravitational lenses. Other image-counting problems in gravitational lensing can also be reduced to problems in complex analysis. The number of images lensed by an elliptical galaxy leads to a transcendental harmonic map. We discussed a partial solution given by D. Khavinson and E. Lundberg,
followed by the complete solution given by W. Bergweiler and A. Eremenko.
We considered the Dirichlet problem with polynomial (or real-entire) data posed on (a component of)
the zero set of a polynomial. For any polynomial data posed on an ellipsoid the solution is again
a polynomial. We discussed some partial results toward proving the Khavinson-Shapiro conjecture
which states that this property characterizes ellipsoids.
We discussed the Laplacian growth (Hele-Shaw) problem in terms of singularities of the Schwarz potential.
Then we described how the singularities can be located using C^n techniques and gave new exact solutions
in the higher dimensional case and in the case where the medium is not homogeneous.
A philosophical promenade with a glimpse of some loose connections between math and the humanities leading to startling concequences.
(Somewhat of an attempt at the style of John Allen Paulos.)
We discussed the Laplacian growth (Hele-Shaw) problem in terms of singularities of the Schwarz potential.
Then we described how the singularities can be located using C^n techniques and gave new exact solutions.
A domain is called a "quadrature domain" if it admits a quadrature formula expressing integration of an analytic function as a finite sum of weighted point evaluations of the function and its derivatives.
In this talk, we discussed equivalent definitions including such notions as the Cauchy transform transform of a domain,
the Schwarz function of the boundary, and the conformal map from the disk.
Then we saw how quadrature domains arise in applications,
where it is often important to find an approximate quadrature formula for a given domain.
We discussed the Laplacian growth (Hele-Shaw) problem in terms of singularities of the Schwarz potential.
Then we described how the singularities can be located using C^n techniques and gave examples.
First, we made a loose analogy between Richardson's Theorem in Laplacian growth and the inverse scattering method for the KdV equation.
Then we discussed independence of order of work of sources and sinks and how this leads to a direct connection to integrable hierarchies in soliton theory.
We discussed directed graphs and permutation theory arising in dynamics of interval maps.
Then we posed a basic question in permutation theory motivated by considerations related to Sharkovsky's Theorem.
We discussed the propogation of singularities through C^n of solutions to the Laplace equation.
Then we showed how this gives an appealing geometric explanation of the source of singularities of the Schwarz function.
This presentation discussed joint work with D. Khavinson and subsequent work of W. Bergweiler and A. Eremenko on the problem of bounding the number of images lensed by an elliptical galaxy.
An expository talk about the harmonic analysis of a string plucked from the center versus near the edge.
A demonstration was given using a guitar and one volunteer who plucked the string in each location while I announced
where they had plucked it (without peeking).
We gave an elementary proof (using only vector calculus) that singularities of the Schwarz potential of a Laplacian growth
obey simple dynamics.
We discussed how so-called "lightning bolts" limit analytic continuation of harmonic functions.
Then we suggested that Clifford analysis might lead to a higher dimensional version.
This presentation discussed joint work with D. Khavinson studying gravitational lensing by elliptical galaxies and how complex dynamics can be used to bound the number of images that can be produced from a single source.
(jointly with L. Kuznia) This presentation discussed gravitational lensing by collinear point masses.
An expository talk based on Laplacian Growth.
A discussion of some examples where solutions to Dirichlet's problem develop singularities that can be located if the complexification of the boundary contains a special finite set.
Singularities of solutions to Dirichlet Problems are connected to well-posedness of boundary-value problems for wave equations. A polygon with vertices on the boundary and edges composed of characteristics establishes singularities. Finding such a polygon leads to studying self-maps of the circle.
An expository talk about Toral Automorphisms taken from Robert Devaney's book "Chaotic Dynamical Systems".
Discussion of a statistical converse to Sharkovsky's Theorem
Expository talk taken from George Polya's "Mathematical methods in science". A rotating vat of fluid is shown to take the shape of a paraboloid of revolution. If the fluid is mercury, you have a parabolic mirror.
A special case of Sharkovsky's Theorem was proved.