Gartside, Paul; Matematični kolokvij maj 2009

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Hilbert's 13th Problem

Paul Gartside

University of Pittsburgh, ZDA

14. maj 2009

The 13th Problem from Hilbert's famous list asks: can every continuous function of three variables be written as a composition of functions of two variables? Hilbert expected the answer to be ‘no’.

In 1954 Vitushkin proved that there are continuously differentiable functions of 3 variables which can not be written as a composition of continuously differentiable functions of 2 variables. Thereby giving the kind of negative result Hilbert was expecting.

But three years later Kolmogorov gave a remarkable positive solution — in fact every continuous function of n variables can be written as a composition of functions of just one variable plus addition.

In this talk the speaker will outline a proof of Kolmogorov's theorem and of Vitushkin's result, and then discuss ongoing work of the speaker and Feng Ziqin surrounding Hilbert's 13th Problem.

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