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\def\bbR{{\mathbb R}}

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\def\Abstract{{\raisebox{3mm}{\parbox[t]{3.7in}{\small
{\color{red}Abstract.} To break a week of deep thinking with a nice
colourful light dessert, we will present the Kolmogorov-Arnold solution
of Hilbert's 13th problem with lots of computer-generated rainbow-painted
3D pictures.

In short, Hilbert asked if a certain specific function of three variables
can be written as a multiple (yet finite) composition of continuous
functions of just two variables. Kolmogorov and Arnold showed him silly
(ok, it took about 60 years, so it was a bit tricky) by showing that {\bf
any} continuous function $f$ of any finite number of variables is a finite
composition of continuous functions of a single variable and several
instances of the binary function ``$+$'' (addition). For $f(x,y)=xy$,
this may be $xy=\exp(\log x+\log y)$. For $f(x,y,z)=x^y/z$, this may be
$\exp(\exp(\log y+\log\log x)+(-\log z))$.  What might it be for (say)
the real part of the Riemann zeta function?

The only original material in this talk will be the pictures; the
math was known since around 1957.
}}}}

\def\Zeta{{\raisebox{0mm}{\parbox[t]{1.9in}{\small
$\frac13\operatorname{Re}(\zeta(x+iy))$ on $[0,1]\times[13,17]$
}}}}

\def\Statement{{\raisebox{0mm}{\parbox[t]{3.7in}{
Fix an irrational $\lambda>0$, say $\lambda=(\sqrt{5}-1)/2$. All functions
are continuous.
\par\noindent{\color{red}Theorem.} There exist five
$\phi_i:[0,1]\to[0,1]$ $(1\leq i\leq 5)$ so that for every
$f:[0,1]\times[0,1]\to\bbR$ there exists a
$g:[0,1+\lambda]\to\bbR$ so that
\[ f(x,y) = \sum_{i=1}^5 g(\phi_i(x)+\lambda\phi_i(y)) \]
for every $x,y\in[0,1]$.
}}}}

\def\StepOne{{\raisebox{3mm}{\parbox[t]{3.7in}{
{\color{red}Step 1.} If $\epsilon>0$ and $f:[0,1]\times[0,1]\to\bbR$,
then there exists $\phi:[0,1]\to[0,1]$
and $g:[0,1+\lambda]\to\bbR$ so that
$|f(x,y)-g(\phi(x)+\lambda\phi(y))|<\epsilon$ on at least 98\% of the
area of $[0,1]\times[0,1]$.
\par\noindent{\color{red}The key.} ``Poorify'' chocolate bars.
}}}}

\def\StepTwo{{\raisebox{3mm}{\parbox[t]{4.25in}{
{\color{red}Step 2.} There exists
$\phi:[0,1]\to[0,1]$ so that for every $\epsilon>0$ and every
$f:[0,1]\times[0,1]\to\bbR$ there
exists a $g:[0,1+\lambda]\to\bbR$ so that
$|f(x,y)-g(\phi(x)+\lambda\phi(y))|<\epsilon$ on a set of area at least
$1-\epsilon$ in $[0,1]\times[0,1]$.
\par\noindent{\color{red}The key.} ``Iterated poorification''.
}}}}

\def\StepThree{{\raisebox{3mm}{\parbox[t]{3.7in}{
{\color{red}Step 3.} There exist $\phi_i:[0,1]\to[0,1]$ $(1\leq i\leq
5)$ so that for every $\epsilon>0$ and every $f:[0,1]\times[0,1]\to\bbR$
there exists a $g:[0,1+\lambda]\to\bbR$ so that
\[
  |f(x,y)-\sum_{i=1}^5 g(\phi_i(x)+\lambda\phi_i(y))|
  < \left(\frac23+\epsilon\right)\left\|f\right\|_\infty
\]
for every $x,y\in[0,1]$.
\par\noindent{\color{red}The key.} ``Shift the chocolates''\ldots
}}}}

\def\StepFour{{\raisebox{3mm}{\parbox[t]{3.7in}{
{\color{red}Step 4.} We are done.
\par\noindent{\color{red}The key.} Learn from the artillery!
\vskip 1mm
\par\noindent\small
Set $Tg:=\sum_{i=1}^5 g(\phi_i(x)+\lambda\phi_i(y))$,
$f_1:=f$, $M:=\left\|f\right\|$, and iterate ``shooting
and adjusting''.  Find $g_1$ with $\left\|g_1\right\|\leq
M$ and $\left\|f_2:=f_1-Tg_1\right\|\leq\frac34M$. Find
$g_2$ with $\left\|g_2\right\|\leq \frac34M$ and
$\left\|f_3:=f_2-Tg_2\right\|\leq(\frac34)^2M$. Find $g_3$
with $\left\|g_3\right\|\leq(\frac34)^2M$ and
$\left\|f_4:=f_3-Tg_3\right\|\leq(\frac34)^3M$. Continue to
eternity. When done, set $g=\sum g_k$ and note that $f=Tg$ as required.
}}}}

\def\Exercises{{\raisebox{3mm}{\parbox[t]{3.7in}{
{\color{red}Exercise 1.} Do the $m$-dimensional case.
\par\noindent{\color{red}Exercise 2.} Do $\bbR^m$ instead of just $I^m$.
}}}}

\def\Politics{{\raisebox{3mm}{\parbox[t]{3.7in}{\small
{\color{red}Propaganda.} I love handouts!
$\bullet$ I have nothing to hide and you can take what you want, forwards,
backwards, here and at home.
$\bullet$ What doesn't fit on one sheet can't be done in one hour.
$\bullet$ It takes learning and many hours and a few pennies. The audience's
worth it!
$\bullet$ There's real math in the handout viewer!
}}}}

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