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\parbox[b]{4.2in}{
  \footnotesize\url{\myurl/Talks/MUGS-1501/}
  \newline\bf
  \href{\myurl/}{Dror Bar-Natan}:
  \href{\myurl/Talks/}{Talks}:
  \href{\myurl/Talks/MUGS-1501/}{MUGS-1501}:
}
\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}

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

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{\bf The Princess Bride, 1987.}

{\bf Inigo:}  You are using Bonetti's defense against me, uh?
\newline {\bf Man in Black:}  I thought it fitting, considering the rocky terrain.
\newline {\bf Inigo:}  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 {\bf Inigo:}  Unless the enemy has studied his Agrippa, which I have! You are wonderful!
\newline {\bf MiB:}  Thank you. I've worked hard to become so.
\newline {\bf Inigo:}  I admit it, you are better than I am.
\newline {\bf MiB:}  Then why are you smiling?
\newline {\bf Inigo:}  Because I know something you don't know.
\newline {\bf MiB:}  And what is that?
\hfill{\bf Inigo:} I am not left-handed.
\newline {\bf MiB:}  You're amazing!
\hfill {\bf Inigo:}  I ought to be after twenty years.
\newline {\bf MiB:}  There is something I ought to tell you.
\hfill {\bf Inigo:}  Tell me.
\newline {\bf MiB:}  I'm not left-handed either.
\hfill {\bf Inigo:}  Who are you?
\newline {\bf MiB:}  No one of consequence.
\hfill {\bf Inigo:}  I must know.
\newline {\bf MiB:}  Get used to disappointment.
\hfill {\bf Inigo:} Okay.
\newline {\bf Inigo:}  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
\newline {\bf MiB:}  Please understand I hold you in the highest respect.

\columnbreak

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

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{\bf Yes, Prime Minister, 1986.}

{\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?
\newline\null\hfill{\bf W:} Yes
\newline{\bf H:} Do you think there is a lack of discipline in our Comprehensive schools?
\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.
\newline\null\hfill{\bf W:} How?
\newline{\bf H:} Mr. Woolley, are you worried about the danger of war?
\newline\null\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?
\hfill{\bf W:} Yes
\newline{\bf H:} Would you oppose the reintroduction of National Service?
\newline\null\hfill{\bf W:} Yes
\newline{\bf H:} There you are, you see Bernard. The perfect balanced sample.

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{\bf References.}\hfill(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}.

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