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

###
The Graduate Student Seminar

Department of Mathematics, University of Toronto

October 10, 2002, 12:00 Noon, SS 5017B

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!

This page is http://www.math.toronto.edu/~drorbn/Talks/UofT-021010/.

A chocolate tablet from

http://www.vantagehouse.com/hans/hb_pralines.htm