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В четверг, 15 июня 2006 года, в 15:40 в конференц-зале НМУ, Б. Власьевский 11, состоится доклад: The full converse to the four vertex theorem. Лектор – Herman Gluck (University of Pennsylvania).

In this talk I will describe Bjorn Dahlberg’s proof of the full converse to the Four Vertex Theorem: any continuous real-valued function on the circle which is either constant or has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane.

The necessity of this condition was proved by Syamadas Mukhopadhyaya in 1909 for strictly positive curvature, and by Adolf Kneser in 1912 in the general case.

I proved the sufficiency for strictly positive curvature in 1971, as a special case of the existence of n-spheres in (n+1)-space with preassigned strictly positive Gaussian curvature. Dahlberg proved the sufficiency in general in 1997, but died in January 1998. His paper was recovered by his students after his death, edited, and only appeared last year.

In the talk I will review this background, and then focus on the new ideas introduced by Dahlberg.