Pure Math Research Sheds Light on Gravitational Lensing

June 20, 2008

Two mathematicians were trying to extend the fundamental theorem of algebra to harmonic polynomials. At the same time, an astrophysicist was working out the number of images generated by a gravitational lens. Taken together, their results completed the mathematicians' proof and confirmed the astrophysicist's conjecture.

An intervening gravitational lens can split the light from a distant galaxy so as to produce multiple images of the light source. Several years ago, Sun Hong Rhie, then at the University of Notre Dame, showed that a particular configuration of four stars would generate 15 images. She conjectured that a lens made up of n stars would produce 5n – 5 images.

Meanwhile, Dmitry Khavinson of the University of South Florida and Genevra Neumann of the University of Northern Iowa were trying to determine the number of zeros for a certain class of rational harmonic functions, where n is the degree of the harmonic polynomial. They discovered that there could never be more than 5n – 5 solutions, but they couldn’t prove that this result represented the tightest possible limit.

Mathematician Jeff Rabin of the University of California, San Diego put two and two together when he saw a preprint describing the astrophysicist's work. Counting the number of images of a light source for the particular gravitational lensing systems studied by Rhie is equivalent to counting the number of zeroes of a rational harmonic function.

Rabin passed the reference on to the mathematicians, telling them that their work had confirmed Rhie's conjecture. The mathematicians, in turn, could complete their proof, setting 5n – 5 as a firm limit.

"This kind of exchange of ideas between math and physics is important to both fields," Rabin told New Scientist.

After hearing about Rhie's work, Khavinson and Neumann contacted other mathematicians and astrophysicists who worked on similar problems, and they received feedback that they used to revise their own paper. These interactions led Khavinson into other collaborations with astrophysicists on related questions.

"I find this kind of interdisciplinary collaboration extremely exciting and simulating," Khavinson says. "I just hope that I will be able to continue these collaborations."

An article by Khavinson and Neumann, titled "From the Fundamental Theorem of Algebra to Astrophysics: A 'Harmonious' Path," appears in the June/July Notices of the American Mathematical Society.

Source: New Scientist, June 5, 2008; American Mathematical Society, June 5, 2008.