\documentclass[11pt,notitlepage]{article}
\usepackage{amsmath,graphicx,amssymb,stmaryrd,amscd,multicol,enumitem,txfonts,picins}
\usepackage[setpagesize=false]{hyperref}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}
% Modified from http://tex.stackexchange.com/questions/37297/how-to-get-element-position-in-latex:
\usepackage{zref-user, zref-abspos}
\newwrite\posfile
\openout\posfile=\jobname.pos\relax
\makeatletter
\newcommand\dimtoin[1]{%
    \strip@pt\dimexpr 0.013837\dimexpr#1\relax\relax%
}
\makeatother

\def\myurl{http://www.math.toronto.edu/~drorbn}

\paperwidth 8in
\paperheight 10.5in
\textwidth 8in
\textheight 10.5in
\oddsidemargin -0.75in
\evensidemargin \oddsidemargin
\topmargin -0.75in
\headheight 0in
\headsep 0in
\footskip 0in
\parindent 0in
\setlength{\topsep}{0pt}
\pagestyle{empty}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\bbC{{\mathbb C}}
\def\calD{{\mathcal D}}

\begin{document}\thispagestyle{empty}
\setlength{\jot}{0ex}
\setlength{\abovedisplayskip}{0.5ex}
\setlength{\belowdisplayskip}{0.5ex}
\setlength{\abovedisplayshortskip}{0ex}
\setlength{\belowdisplayshortskip}{0ex}

\parbox[b]{4.2in}{
  \footnotesize\url{\myurl/Talks/CMU-1504/}
  \newline\bf
  \href{\myurl/}{Dror Bar-Natan}:
  \href{\myurl/Talks/}{Talks}:
  \href{\myurl/Talks/CMU-1504/}{CMU-1504}:
}
\hfill\parbox[b]{2.4in}{
  \null\hfill{\Huge\bf Commutators}
}

\begin{multicols}{2}

{\bf Abstract.} The commutator of two elements $x$ and $y$ in a group $G$ is $xyx^{-1}y^{-1}$. That is, $x$ followed by $y$ followed by the inverse of $x$ followed by the inverse of $y$. In my talk I will tell you how commutators are related to the following four riddles:

\begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=0pt,topsep=0pt]
\item Can you send a secure message to a person you have never communicated with before (neither privately nor publicly), using a messenger you do not trust?
\item Can you hang a picture on a string on the wall using $n$ nails, so that if you remove any one of them, the picture will fall?
\item Can you draw an $n$-component link (a knot made of $n$ non-intersecting circles) so that if you remove any one of those $n$ components, the remaining $(n-1)$ will fall apart?
\item Can you solve the quintic in radicals? Is there a formula for the zeros of a degree $5$ polynomial in terms of its coefficients, using only the operations on a scientific calculator?
\end{enumerate}

\rule{\columnwidth}{0.5pt}

\textbf{Definition.} The commutator of two elements $x$ and $y$ in a group $G$ is $[x,y] \coloneqq xyx^{-1} y^{-1}$.

\textbf{Example 1.} In $S_3$, $[(12), (23)] = (12) (23) (12)^{-1} (23)^{-1}=(123)$ and in general in $S_{\geq 3}$,
\[ [(ij),(jk)]=(ijk). \]

\textbf{Example 2.} In $S_{\geq 4}$,
\[ [(ijk), (jkl)] = (ijk) (jkl) (ijk)^{-1} (jkl)^{-1}=(il)(jk). \]

\textbf{Example 3.} In $S_{\geq 5}$,
\[ [(ijk), (klm)] = (ijk) (klm) (ijk)^{-1} (klm)^{-1} =(jkm). \]

\textbf{Example 4.} So, in fact, in $S_5$, $(123) = [(412),(253)] = [[(341),(152)],[(125),(543)]]
= [[[(234),(451)],[(315),(542)]],[[(312),(245)],[(154),(423)]]]
= [\ 
  [[[(123),(354)],[(245),(531)]],[[(231),(145)],[(154),(432)]]],\allowbreak
  [[[(431),(152)],[(124),(435)]],[[(215),(534)],[(142),(253)]]]\ 
]$.

\rule{\columnwidth}{0.5pt}

\textbf{Solving the Quadratic}, $ax^2+bx+c=0$: $\delta =\sqrt{\Delta}$; $\Delta =b^2-4 a c$; $r=\frac{\delta -b}{2 a}$.

\textbf{Solving the Cubic}, $ax^3+bx^2+cx+d=0$: $\Delta =27 a^2 d^2-18 a b c d+4 a c^3+4 b^3 d-b^2 c^2$; $\delta =\sqrt{\Delta }$; $\Gamma =27 a^2 d-9 a b c+3 \sqrt{3} a \delta +2 b^3$; $\gamma =\sqrt[3]{\frac{\Gamma }{2}}$; $r=-\frac{\frac{b^2-3 a c}{\gamma }+b+\gamma }{3 a}$.

\textbf{Solving the Quartic}, $ax^4+bx^3+cx^2+dx+e=0$: $\Delta _0=12 a e-3 b d+c^2$; $\Delta _1=-72 a c e+27 a d^2+27 b^2 e-9 b c d+2 c^3$; $\Delta _2=\frac{1}{27} \left(\Delta _1^2-4 \Delta _0^3\right)$; $u=\frac{8 a c-3 b^2}{8 a^2}$; $v=\frac{8 a^2 d-4 a b c+b^3}{8 a^3}$; $\delta _2=\sqrt{\Delta _2}$; $Q=\frac{1}{2} \left(3 \sqrt{3} \delta _2+\Delta _1\right)$; $q=\sqrt[3]{Q}$; $S=\frac{\frac{\Delta _0}{q}+q}{12 a}-\frac{u}{6}$; $s=\sqrt{S}$; $\Gamma =-\frac{v}{s}-4 S-2 u$; $\gamma =\sqrt{\Gamma }$; $r=-\frac{b}{4 a}+\frac{\gamma }{2}+s$.

\end{multicols}

\rule{\columnwidth}{0.5pt}

{\bf Theorem.} The is no general formula, using only the basic arithmetic operations and taking roots, for the solution of the quintic equation $ax^5+bx^4+cx^3+dx^2+ex+f=0$.

{\bf Key Point.} The ``persistent root'' of a closed path (path lift, in topological language) may not be closed, yet the persistent root of a commutators of closed paths is always closed.

\zsavepos{proof-start}\write\posfile{List[\dimtoin{\zposy{proof-start}sp},}

{\bf Proof.} Suppose there was a formula, and consider the corresponding ``composition of machines'' picture:

\[ \input{proof.pdf_t} \]

Now if $\gamma^{(1)}_1$,  $\gamma^{(1)}_2$, \ldots, $\gamma^{(1)}_{16}$, are paths in $X_0$ that induce permutations of the roots and we set $\gamma^{(2)}_1\coloneqq[\gamma^{(1)}_1,\gamma^{(1)}_2]$, $\gamma^{(2)}_2\coloneqq[\gamma^{(1)}_3,\gamma^{(1)}_4]$, \ldots, $\gamma^{(2)}_8\coloneqq[\gamma^{(1)}_{15},\gamma^{(1)}_{16}]$, $\gamma^{(3)}_1\coloneqq[\gamma^{(2)}_1,\gamma^{(2)}_2]$, \ldots, $\gamma^{(3)}_4\coloneqq[\gamma^{(2)}_7,\gamma^{(2)}_8]$, $\gamma^{(4)}_1\coloneqq[\gamma^{(3)}_1,\gamma^{(3)}_2]$, $\gamma^{(4)}_2\coloneqq[\gamma^{(3)}_3,\gamma^{(3)}_4]$, and finally $\gamma^{(5)}\coloneqq[\gamma^{(4)}_1,\gamma^{(4)}_2]$ (all of those, commutators of ``long paths''; I don't know the word ``homotopy''), then $\gamma^{(5)}\act C\act P_1\act R_1\act\cdots\act R_4$ is a closed path. Indeed,

\parpic[r]{\includegraphics[width=0.75in]{../../Projects/Gallery/Arnold.jpg}~V.I.~Arnold\qquad}
$\bullet$ In $X_0$, none of the paths is necessarily closed.

$\bullet$ After $C$, all of the paths are closed.

$\bullet$ After $P_1$, all of the paths are still closed.

$\bullet$ After $R_1$, the $\gamma^{(1)}$'s may open up, but the $\gamma^{(2)}$'s remain closed.

$\cdots$

$\bullet$ At the end, after $R_4$, $\gamma^{(4)}$'s may open up, but $\gamma^{(5)}$ remains closed.

But if the paths are chosen as in Example 4, $\gamma^{(5)}\act C\act P_1\act R_1\act\cdots\act R_4$ is not a closed path. \hfill $\Box$

\zsavepos{proof-end}\write\posfile{\dimtoin{\zposy{proof-end}sp}]}

\rule{\columnwidth}{0.5pt}

\parpic[r]{\includegraphics[width=0.7in]{QRCode.png}}
{\bf References.}
V.I. Arnold, 1960s, hard to locate.

V.B. Alekseev, {\em Abel's Theorem in Problems and Solutions, Based on the Lecture of Professor V.I. Arnold,} Kluwer 2004.

A. Khovanskii, {\em Topological Galois Theory, Solvability and Unsolvability of Equations in Finite Terms,} Springer 2014.

B. Katz, {\em Short Proof of Abel's Theorem that 5th Degree Polynomial Equations Cannot be Solved,} YouTube video, \url{http://youtu.be/RhpVSV6iCko}.

\newpage

\begin{multicols}{2}

{\bf The Princess Bride, 1987.}

{\footnotesize
\hfill{\bf Inigo Montoya:} You are using Bonetti's defense against me, uh?
\newline {\bf Man in Black:}  I thought it fitting, considering the rocky terrain.
\newline\null\hfill{\bf IM:}  Naturally, you must expect me to attack with Capo Ferro.
\newline {\bf MiB:}  Naturally, but I find that Thibault cancels out Capo Ferro, don't you?
\newline\null\hfill{\bf IM:} Unless the enemy has studied his Agrippa, which I have!
\newline\null\hfill You are wonderful!
\newline {\bf MiB:}  Thank you. I've worked hard to become so.
\newline\null\hfill{\bf IM:}  I admit it, you are better than I am.
\newline {\bf MiB:}  Then why are you smiling?
\newline\null\hfill{\bf IM:}  Because I know something you don't know.
\newline {\bf MiB:}  And what is that?
\hfill{\bf IM:} I am not left-handed.
\newline {\bf MiB:}  You're amazing!
\hfill {\bf IM:}  I ought to be after twenty years.
\newline {\bf MiB:}  There is something I ought to tell you.
\hfill {\bf IM:}  Tell me.
\newline {\bf MiB:}  I'm not left-handed either.
\hfill {\bf IM:}  Who are you?
\newline {\bf MiB:}  No one of consequence.
\hfill {\bf IM:}  I must know.
\newline {\bf MiB:}  Get used to disappointment.
\hfill {\bf IM:} Okay.\quad Kill me quickly.
\newline {\bf MiB:} I would as soon destroy a stained-glass window as an artist like yourself. However, since I can't have you following me either\ldots\qquad Please understand I hold you in the highest respect.
}

\rule{\columnwidth}{0.5pt}

{\bf Yes, Prime Minister, 1986.}

{\footnotesize
{\bf Sir Humphrey:} You know what happens: nice young lady comes up to you. Obviously you want to create a good impression, you don't want to look a fool, do you? So she starts asking you some questions: Mr. Woolley, are you worried about the number of young people without jobs?
\hfill{\bf Bernard Woolley:} Yes
\newline{\bf H:} Are you worried about the rise in crime among teenagers?
\hfill{\bf W:} Yes
\newline{\bf H:} Do you think there is a lack of discipline in our Comprehensive schools?
\newline\null\hfill{\bf W:} Yes
\newline{\bf H:} Do you think young people welcome some authority and leadership in their lives?
\hfill{\bf W:} Yes
\newline{\bf H:} Do you think they respond to a challenge?
\hfill{\bf W:} Yes
\newline{\bf H:} Would you be in favour of reintroducing National Service?
\newline\null\hfill{\bf W:} Oh...well, I suppose I might be.
\newline{\bf H:} Yes or no?
\hfill{\bf W:} Yes
\newline{\bf H:} Of course you would, Bernard. After all you told me can't say no to that. So they don't mention the first five questions and they publish the last one.
\hfill{\bf W:} Is that really what they do?
\newline{\bf H:} Well, not the reputable ones no, but there aren't many of those. So alternatively the young lady can get the opposite result.
\hfill{\bf W:} How?
\newline{\bf H:} Mr. Woolley, are you worried about the danger of war?
\hfill{\bf W:} Yes
\newline{\bf H:} Are you worried about the growth of armaments?
\hfill{\bf W:} Yes
\newline{\bf H:} Do you think there is a danger in giving young people guns and teaching them how to kill?
\hfill{\bf W:} Yes
\newline{\bf H:} Do you think it is wrong to force people to take up arms against their will?
\newline\null\hfill{\bf W:} Yes
\newline{\bf H:} Would you oppose the reintroduction of National Service?
\hfill{\bf W:} Yes
\newline{\bf H:} There you are, you see Bernard. The perfect balanced sample.
}
\end{multicols}

\end{document}
\endinput