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Dror Bar-Natan:
Talks:
# Kolmogorov's Solution of Hilbert's 13th Problem

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The Graduate Student Seminar

Department of Mathematics, University of Toronto

September 30, 2009

Are there any truly new continuous functions of two variables, beyond
addition and functions of one variable? Well, multiplication isn't one,
for multiplication is a simple composition of addition and of functions
of just one variable: *xy* = exp(log *x* + log
*y*). Powers and logarithms wouldn't do either, as
*x*^{y} = exp(exp(log *y* + log log
*x*)) and log_{x} *y* = exp(log log
*y* + (- log log *x*)). Trig functions won't do, they
are functions of one variable. Maybe Bessel functions? Anybody has a
clue what they are? Anyway, according to Kolmogorov
and Arnol'd, we need not
worry, for the following amazing theorem holds true:
**Theorem.** Any real-valued continuous function on a compact set
in **R**^{n} is a finite composition of (several
instances of) the binary function "+" and of single-variable continuous
functions.

On top of being a beauty, this theorem also resolves the 13th of Hilbert's
famed 23
problems, in which he presented a certain specific function of
three variables and asked if it can be re-written as a composition of
continuous functions of two or less variables. Well, our theorem does a
lot better.

Ok, if you can't make it to the lecture or if you want to do some
work in advance, just do some
exercises!

More under Academic
Pensieve: 2009-09: Mathematica Notebooks: Hilbert13th.