Neumann, Genevra

Several months ago, D. Khavinson and I extended the complex dynamics approach of Khavinson and G. Swiatek for finding a bound on the number of distinct zeros of $f(z) = p(z) - \overline{z}$ to the case of $f(z) = p(z)/q(z) - \overline{z}$, where $p$ and $q$ are analytic polynomials. J. Rabin pointed out that our result settles a conjecture of S. H. Rhie concerning the maximum number of lensed images due to an $n$-point gravitational lens and that Rhie has constructed examples that settle our question concerning the sharpness of the bound. A preprint of this paper that includes applications to gravitational lensing can be found at http://arxiv.org/abs/math.CV/0401188.

My research concerns complex-valued harmonic functions. Let $f(z)$ be a complex-valued harmonic function defined on an open subset of the complex plane (or on the entire complex plane). A main result in my thesis (2003) is that the cluster set combined with the image of the critical set partitions the complex plane into regions where each point has the same number of distinct preimages. The cluster set consists of those finite values $w$ such that there exists a sequence $\{z_n\}$ in the domain of $f$ where $z_n$ approaches the boundary of the domain (or $\infty$) and $f(z_n) \rightarrow w$. The critical set consists of those points where the function is not locally $1 - 1$. Graphs of caustics in gravitational lensing are similar to graphs of the image of the critical set of complex-valued harmonic functions.

I want to learn more about gravitational lensing. I am hoping that complex analysis will be useful in other lensing problems and also that mathematical techniques used in gravitational lensing could be applied to questions concerning the behavior of complex-valued harmonic functions.

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