Kummer Surfaces: 200 Years of Study Igor Dolgachev The fascinating story about the Kummer surface starts from material where double refraction occurs: a ray of light the discovery by Augustin-Jean Fresnel in 1822 of the equa- splits into two, traveling at the same speed along different tion describing the propagation of light in an optically paths. The speed of light may depend on the coordinates biaxial crystal [Fre]. Biaxial crystals are an example of 푥 = (푥1, 푥2, 푥3) of a point and the unit direction vector 휉 = (휉1, 휉2, 휉3). The propagation of light is described by Igor Dolgachev is a professor of mathematics, emeritus, at the University of the speed 푣(푥, 휉) at 푥 in the direction 휉. We say that the Michigan. His email address is
[email protected]. matter is homogeneous if 푣 does not depend on 푥 and we The article is based on the author’s Oliver Club talk at Cornell University de- livered on October 10, 2019, exactly 121 years since the first meeting of the say that it is isotropic if it does not depend on 휉. For exam- club, at which then the faculty member John Hutchinson spoke. As we will see, ple, while a student, James Maxwell described a lens that Hutchinson contributed significantly to the theory of Kummer surfaces. reminded him of the eyes of a fish. Through his fish eye, Communicated by Notices Associate Editor Angela Gibney. he found that light is inhomogeneous but isotropic, bend- For permission to reprint this article, please contact: ing in arcs whose shape depends on where they start and
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