Algebra meets astrophysics

June 8, 2008, 8:26 pm

Abstract algebra and astrophysics don’t have much to do with each other, right? Well, perhaps not, after all. Here’s a story about the results from a researcher in gravitational lensing being used to prove an extension of the Fundamental Theorem of Algebra to rational harmonic functions. Snippet:

In 2004, [mathematicians Dmitry Khavinson and Genevra Neumann] proved that for one simple class of rational harmonic functions, there could never be more than 5n – 5 solutions. But they couldn’t prove that this was the tightest possible limit; the true limit could have been lower.

It turned out that Khavinson and Neumann were working on the same problem as [astrophysicist Sun Hong Rhie]. To calculate the position of images in a gravitational lens, you must solve an equation containing a rational harmonic function.

When mathematician Jeff Rabin of the University of California, San Diego, US, pointed out a preprint describing Rhie’s work, the two pieces fell into place. Rhie’s lens completes the mathematicians’ proof, and their work confirms her conjecture. So 5n – 5 is the true upper limit for lensed images.

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

Indeed, and very cool. The paper that Khavinson and Neumann wrote, with an update that addresses the relevance of Rhie’s result on gravitational lensing, is here.

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