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 CNLS Office 
Wednesday, June 10, 2009
10:00 AM - 11:00 AM
CNLS Conference Room (TA-3, Bldg 1690)


From the Fundamental Theorem of Algebra to Astrophysics: a "Harmonious Path"

Dmitry Khavinson
University of South Florida

We shall discuss recently found unexpected connections between the Fundamental Theorem of Algebra and Gravitational Microlensing. The Fundamental Theorem of Algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. We shall discuss the extension of the FTA to some non-analytic harmonic polynomials of degree n. In particular, for somespecific harmonic polynomials of degree n the number of zeros turns out to be linear in n, as was conjectured in the early 90's. In 2004 G. Neumann and D. Khavinson showed that the maximal number of zeros of some rational harmonic functions of degree n, n > 1, is also linear in n. It turned out that this result resolved the conjecture by an astrophysicist S. H. Rhie dealing with the estimate on the maximal number of images of a star if the light from it is deflected by n co-planar masses. The first non-trivial case of one mass was already investigated by A. Einstein around 1912. We shall also discuss the problem of gravitational lensing of a point source of light, e.g., a star, or a quasar, by an elliptic galaxy, more precisely the problem of the maximal number of images that one can observe. Under some more or less “natural physical” assumptions on the mass distribution of gas within the galaxy one can prove (C. Fassnacht, Ch. Keeton and DK (2007) and DK and E. Lundberg (2009)) that the number of visible images is always finite, and, under additional assumptions, can never be more than 4. Interestingly, the latter situation can actually occur and has been observed by astronomers with the help of the Hubble telescope. Still there are much more open questions than there are answers.

Host: Razvan Teodorescu, T-CNLS/T-4