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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{LearningSeminarOnCategorification-2006}
\def\title{The Alexander Polynomial is a Quantum Invariant in a Different Way}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to speak in \magenta my basement!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/cat20}{http://drorbn.net/cat20/}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}

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\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}

\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}

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\def\ad{\operatorname{ad}}
\def\Ad{\operatorname{Ad}}
\def\aft{$\overrightarrow{\text{4T}}$}
\def\AS{\mathit{AS}}
\def\bbZZ{{\mathbb Z\mathbb Z}}
\def\CW{\text{\it CW}}
\def\diag{\operatorname{diag}}
\def\eps{\epsilon}
\def\FL{\text{\it FL}}
\def\Hom{\operatorname{Hom}}
\def\IHX{\mathit{IHX}}
\def\mor{\operatorname{mor}}
\def\PvT{{\mathit P\!v\!T}}
\def\remove{\!\setminus\!}
\def\STU{\mathit{STU}}
\def\SW{\text{\it SW}}
\def\TC{\mathit{TC}}
\def\tr{\operatorname{tr}}
\def\vT{{\mathit v\!T}}

\def\bara{{\bar a}}
\def\barb{{\bar b}}
\def\barT{{\bar T}}
\def\bbE{{\mathbb E}}
\def\bbe{\mathbbm{e}}
\def\bbN{{\mathbb N}}
\def\bbO{{\mathbb O}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\bcA{{\bar{\mathcal A}}}
\def\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\def\calF{{\mathcal F}}
\def\calG{{\mathcal G}}
\def\calH{{\mathcal H}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calL{{\mathcal L}}
\def\calM{{\mathcal M}}
\def\calO{{\mathcal O}}
\def\calP{{\mathcal P}}
\def\calR{{\mathcal R}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\fraka{{\mathfrak a}}
\def\frakb{{\mathfrak b}}
\def\frakg{{\mathfrak g}}
\def\frakh{{\mathfrak h}}
\def\tilE{\tilde{E}}
\def\tilq{\tilde{q}}

\def\tDelta{\tilde{\Delta}}
\def\tf{\tilde{f}}
\def\tF{\tilde{F}}
\def\tg{\tilde{g}}
\def\tI{\tilde{I}}
\def\tm{\tilde{m}}
\def\tR{\tilde{R}}
\def\tsigma{\tilde{\sigma}}
\def\tS{\tilde{S}}
\def\tSW{\widetilde{\SW}}

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%%%

\def\Sidebar{{\raisebox{-1mm}{\parbox[t]{2.51in}{
\setlength{\parindent}{3mm}
\hyphenpenalty=10000 \tolerance=2000 \emergencystretch=20pt

\noindent\ \href{http://drorbn.net/AcademicPensieve/Talks/LearningSeminarOnCategorification-2006/OpeningCrawl@.mp4}{\red \faPlay}
On a chat window here I saw a comment ``Alexander is the quantum
$gl(1|1)$ invariant''. I have an opinion about this, and I'd like to
share it. First, some stories.

I left the wonderful subject of Categorification nearly 15 years
ago. It got crowded, lots of very smart people had things to say, and
I feared I will have nothing to add. Also, clearly the next step was to
categorify all other ``quantum invariants''. Except it was not clear what
``categorify'' means. Worse, I felt that I (perhaps ``we all'') didn't
understand ``quantum invariants'' well enough to try to categorify them,
whatever that might mean.

I still feel that way! I learned a lot since 2006, yet I'm still
not comfortable with quantum algebra, quantum groups, and quantum
invariants. I still don't feel that I know what God had in mind when
She created this topic.

Yet I'm not here to rant about my philosophical quandaries, but only
about things that I learned about the Alexander polynomial after 2006.

Yes, the Alexander polynomial fits within the Dogma, ``one invariant
for every Lie algebra and representation'' (it's $gl(1|1)$, I hear). But
it's better to think of it as a quantum invariant arising by other means,
outside the Dogma.

Alexander comes from (or in) practically any non-Abelian Lie algebra. Foremost
from the not-even-semi-simple 2D ``$ax+b$'' algebra. You get a
polynomially-sized extension to tangles using some lovely formulas
(can you categorify them?). It generalizes to higher dimensions and
it has an organized family of siblings. (There are some questions too,
beyond categorification).

I note the spectacular existing categorification of Alexander by Ozsv\'ath
and Szab\'o. The theorems are proven and a lot they say, the programs run
and fast they run. Yet if that's where the story ends, She has abandoned
us. Or at least abandoned me: a simpleton will never be able to catch up.

\red If you care only about categorification, the take-home from my talk
will be a challenge: Categorify what I believe is the best Alexander
invariant for tangles.
}}}}

\def\C{$C^{-1}$}
\def\ai{$a_i$} \def\bi{$b_i$}
\def\aj{$\bara_j$} \def\bj{$\barb_j$}
\def\ak{$\bara_k$} \def\bk{$\barb_k$}

\def\technique{{\raisebox{2mm}{\parbox[t]{4.1875in}{
{\red\bf The Yang-Baxter Technique.} Given an algebra $U$ (typically
some $\hat\calU(\frakg)$ or $\hat\calU_q(\frakg)$) and suitable elements $R$, $C$,
\[
  R \!=\!\! \sum a_i\otimes b_i\in U\otimes U
  \quad\text{with}\quad R^{-1} \!=\!\! \sum\bara_i\otimes\barb_i
  \quad\text{and}\quad C,C^{-1}\in U,
\]
form\hfill
$\displaystyle Z(K) = \sum_{i,j,k} a_iC^{-1}\barb_k\bara_jb_i \otimes \barb_j\bara_k$. \hfill\null
\par{\red Problem.} Extract information from $Z$.
\par{\red The Dogma.} Use representation theory. In principle finite, but {\em
slow}.
}}}}

\def\example{{\raisebox{2mm}{\parbox[t]{5.375in}{
\parshape 3 0in 3.8in 0in 3.8in 0in 5.375in
{\red\bf Example 1.} Let $\fraka \coloneqq L\langle a,x\rangle/([a,x]=x)$,
$\frakb \coloneqq \fraka^\star=\langle b,y\rangle$, and $\frakg \coloneqq
\frakb\rtimes\fraka=\frakb\oplus\fraka$ with $[a,x]=x$, $[a,y]=-y$, $[b,\cdot]=0$, and $[x,y]=b$ and with
$\deg(y,b,a,x)=(1,1,0,0)$. Let $U=\hat\calU(\frakg)$ and
\[ R \coloneqq \bbe^{b\otimes a+y\otimes x} \in U\otimes U 
  \quad\text{or better}\quad
  R_{ij} \coloneqq \bbe^{b_ia_j+y_ix_j} \in U_i\otimes U_j,
  \quad\text{and}\quad
  C_i=\bbe^{-b_i/2}.
\]
}}}}

\def\scalars{{\raisebox{2mm}{\parbox[t]{5.375in}{
{\red\bf Theorem 1.} With ``scalars''$\coloneqq$power series
in $\{b_i\}$ which are rational functions in $\{b_i\}$ and
$\{B_i\coloneqq\bbe^{b_i}\}$,
}}}}

\def\thmone{\resizebox{5.4in}{!}{$\displaystyle
  Z(K) = \bbO_{\mathit ybax}\left(\omega^{-1}\bbe^{l^{ij}b_ia_j+q^{ij}y_ix_j}(1{\gray+\eps P_1+\eps^2P_2+\ldots})\right)
$}}

\def\examtwo{{\raisebox{2mm}{\parbox[t]{2.6875in}{
{\red\bf Example 2.} Let $\frakh \coloneqq A\langle p,x\rangle/([p,x]=1)$ be the Heisenberg algebra, with
$C_i=\bbe^{t/2}$ and $R_{ij}=\bbe^{t/2}\bbe^{t(p_i-p_j)x_j}$.
\hfill\parbox{1.2in}{\tiny I just told you the whole Alexander story! Everything else is details.}

\vskip 0.5mm{\red Claim.} $R_{ij}=\bbO_{px}\left(\bbe^{(\bbe^t-1)(p_i-p_j)x_j}\right)$.
}}}}

\def\thmtwo{{\raisebox{3mm}{\parbox[t]{2.6875in}{
{\red\bf Theorem 2.} $Z(K) = \bbO_{\mathit px}\left(\omega^{-1}\bbe^{q^{ij}p_ix_j}\right)$ where $\omega$ and the
$q^{ij}$ are rational functions in $T=\bbe^t$. In fact
$\omega$ and $\omega q^{ij}$ are Laurent polynomials ({\red categorify us!}). When $K$ is a long knot,
$\omega$ is the Alexander polynomial.
}}}}

\def\packaging{{\raisebox{2mm}{\parbox[t]{2.6875in}{
{\red\bf Packaging.} Write $\bbO_{\mathit px}\left(\omega^{-1}\bbe^{q^{ij}p_ix_j}\right)$ as
\[ \bbE_{p_1,\ldots,x_1,\ldots}[\omega,Q]\leftrightarrow\begin{array}{c|ccc}
  \omega & x_1 & x_2 & \cdots \\
  \hline
  p_1 & q^{11} & q^{12} & \cdots \\
  p_2 & q^{21} & q^{22} & \cdots \\
  \vdots & \vdots & \vdots & \ddots \\
\end{array} \]
%\[ \bbE_{p_1,\ldots,x_1,\ldots}[\omega,Q]\leftrightarrow\begin{pmatrix}
%  \omega & x_1 & x_2 & \cdots \\
%  p_1 & q^{11} & q^{12} & \cdots \\
%  p_2 & q^{21} & q^{22} & \cdots \\
%  \vdots & \vdots & \vdots & \ddots \\
%\end{pmatrix}_\frakh \]
}}}}

\def\ft{{\raisebox{2mm}{\parbox[t]{2.6875in}{
{\red\bf The ``First Tangle''.} \quad $Z(K)=$
\vskip 2mm
\par\quad$\bbE_{12}\left[\frac{2T-1}{T}, \frac{(T-1)(p_1-p_2)(Tx_1-x_2)}{2T-1}\right]$
\vskip 1mm
\par\qquad$\def\arraystretch{1.35}
  =\begin{array}{c|cc}
    {\scriptstyle 2-T^{-1}} & x_1 & x_2 \\
    \hline
    p_1 & \frac{T(T-1)}{2T-1} & \frac{1-T}{2T-1} \\
    p_2 & \frac{T(1-T)}{2T-1} & \frac{T-1}{2T-1} \\
  \end{array}
$
%\par\qquad$\def\arraystretch{1.35}
%  =\begin{pmatrix}
%    \frac{1-2T}{T^2} & x_1 & x_2 \\
%    p_1 & \frac{T(T-1)}{2T-1} & \frac{1-T}{2T-1} \\
%    p_2 & \frac{T(1-T)}{2T-1} & \frac{T-1}{2T-1} \\
%  \end{pmatrix}_\frakh
%$
}}}}

\def\thmthree{{\raisebox{2mm}{\parbox[t]{2.625in}{
{\red\bf Theorem 3.} Full evaluation via
\vskip 1mm\par$\displaystyle
  \left(\tensor[_i]{\overcrossing}{_j},\tensor[_j]{\undercrossing}{_i}\right)
  \to
  \begin{array}{c|cc} 1 & x_i & x_j \\ \hline p_i & 0 & T^{\pm 1}-1 \\ p_j & 0 & 1-T^{\pm 1} \end{array}
$\hfill\text{(1)$\Box$\quad}
\vskip 1mm\par$\displaystyle
  K_1\sqcup K_2
  \to
  \begin{array}{c|cc} \omega_1\omega_2 & X_1 & X_2 \\ \hline P_1 & A_1 & 0 \\ P_2 & 0 & A_2 \end{array}
$\hfill\text{(2)$\Box$\quad}
\vskip 1mm\par$\displaystyle
  \begin{array}{c|ccc}
    \omega & x_i & x_j & \cdots \\
    \hline
    p_i & \alpha & \beta & \theta \\
    p_j & \gamma & \delta & \epsilon \\
    \vdots & \phi & \psi & \Xi
  \end{array}
  \overset{hm^{ij}_k}{\longrightarrow}
$\hfill\text{(3)\quad}
\vskip 1mm\par\null\hfill$\displaystyle
  \begin{array}{c|cc}
    (1+\gamma)\omega & x_k & \cdots \\
    \hline
    p_k & 1+\beta-\frac{(1-\alpha)(1-\delta)}{1+\gamma} & \theta+\frac{(1-\alpha)\epsilon}{1+\gamma} \\
    \vdots & \psi+\frac{(1-\delta)\phi}{1+\gamma} & \Xi-\frac{\phi\epsilon}{1+\gamma}
  \end{array}
$
\vskip 1mm \par
``$\Gamma$-calculus'' relates via $A\leftrightarrow I-A^T$ and has slightly simpler formulas:
$\omega\to(1-\beta)\omega$,
\[
  \begin{pmatrix}
    \alpha & \beta & \theta \\ \gamma & \delta & \epsilon \\ \phi & \psi & \Xi
  \end{pmatrix}
  \to
  \begin{pmatrix}
    \gamma + \frac{\alpha\delta}{1-\beta} & \epsilon + \frac{\delta\theta}{1-\beta} \\
    \phi + \frac{\alpha\psi}{1-\beta} & \Xi + \frac{\psi\theta}{1-\beta}
  \end{pmatrix}
\]
}}}}

\def\WhyCategorify{{\raisebox{2mm}{\parbox[t]{2.625in}{
\hyphenpenalty=10000 \tolerance=2000 \emergencystretch=20pt
{\red\bf Why Should You Categorify This?} The simplest and fastest
Alexander for tangles, easily generalizes to the multi-variable case,
generalizes to v-tangles and w-tangles, generalizes to other Lie
algebras. In fact, it's in almost any Lie algebra, and you don't even
need to know what is $gl(1|1)$! {\red But \text{you'll} have to deal
with denominators and/or divisions!}
}}}}

\def\hg{{\raisebox{2.5mm}{\parbox[t]{2.625in}{\small
{\red\bf Note.} Example 1 $\leftrightsquigarrow$ Example 2 via
$\frakg\hookrightarrow\frakh(t)$ via $(y,b,a,x)\mapsto(-tp,t,px,x)$.
}}}}

\pagestyle{empty}

\begin{document} \latintext
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\begin{center}
\null\vfill\input{AQDW1.pdftex_t}\vfill\null
%\null\vfill\input{AQDW2.pdftex_t}\vfill\null
\end{center}

\eject
\newgeometry{textwidth=8in,textheight=10.5in}
\begin{multicols}{2}

{\red\bf The PBW Principle} Lots of algebras are isomorphic as vector
spaces to polynomial algebras. So we want to understand arbitrary linear
maps between polynomial algebras.

\vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm}

{\bf Convention.} For a finite set $A$, let $z_A\coloneqq\{z_i\}_{i\in A}$ and let
${\zeta_A\coloneqq\{z^*_i=\zeta_i\}_{i\in A}}$.
\hfill$(p,x)^*=(\pi,\xi)$

{\bf\red The Generating Series $\calG\colon\Hom(\bbQ[z_A]\!\to\!\bbQ[z_B]) \to \bbQ\llbracket\zeta_A,z_B
\rrbracket$.}

{\bf Claim.} $L\in\Hom(\bbQ[z_A]\to\bbQ[z_B])
\xrightarrow[\calG]{\raisebox{-0.75ex}[0ex][0ex]{$\sim$}} \bbQ[z_B]\llbracket\zeta_A\rrbracket\ni \calL$ via
\[ \calG(L)
  \coloneqq \sum_{n\in\bbN^A}\frac{\zeta_A^n}{n!}L(z_A^n)
  = L \left(\bbe^{\sum_{a\in A}\zeta_a z_a}\right)
  = \calL
  = \tensor[_{\text{greek}}]{\calL}{_{\text{latin}}},
\]
\[ \calG^{-1}(\calL)(p) = \left(\left.p\right|_{z_a\to\partial_{\zeta_a}}\calL\right)_{\zeta_a=0}
  \quad\text{for $p\in\bbQ[ z_A]$}.
\]

{\bf Claim.} If $L\in\Hom(\bbQ[z_A]\to\bbQ[ z_B])$, $M\in\Hom(\bbQ[z_B]\to\bbQ[ z_C])$, then
$\calG(L\act M) = \left(\calG(L)|_{z_b\to\partial_{\zeta_b}}\calG(M)\right)_{\zeta_b=0}$.

{\bf Examples.} $\bullet$ $\calG(id\colon\bbQ[p,x]\to\bbQ[p,x]) = \bbe^{\pi p+\xi x}$.

\par$\bullet$ Consider
$R_{ij}\in(\frakh_i\otimes\frakh_j)\llbracket t\rrbracket
\cong \Hom\left(\bbQ[]\to\bbQ[p_i,x_i,p_j,x_j]\right)\llbracket
t\rrbracket$. Then $\calG(R_{ij}) = \bbe^{(\bbe^t-1)(p_i-p_j)x_j} = \bbe^{(T-1)(p_i-p_j)x_j}$.

\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm

{\red\bf Heisenberg Algebras.} Let $\frakh=A\langle p,x\rangle/([p,x]=1)$,
let $\bbO_i\colon\bbQ[p_i,x_i]\to\frakh_i$ is the ``$p$ before $x$''
PBW normal ordering map and let $hm^{ij}_k$ be the composition
\[ \begin{CD} \bbQ[p_i,x_i,p_j,x_j] @>\bbO_i\otimes\bbO_j>> \frakh_i\otimes\frakh_j
  @>m^{ij}_k>> \frakh_k @>\bbO_k^{-1}>> \bbQ[p_k,x_k]. \end{CD}
\]
Then $\calG(hm^{ij}_k) = \bbe^{-\xi_i\pi_j+(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}$.

{\bf Proof.} Recall the ``Weyl CCR'' $\bbe^{\xi x}\bbe^{\pi p} =
\bbe^{-\xi\pi}\bbe^{\pi p}\bbe^{\xi x}$, and find
\begin{multline*}
  \calG(hm^{ij}_k)
  = \bbe^{\pi_ip_i+\xi_ix_i+\pi_jp_j+\xi_jx_j}
    \act \bbO_i\otimes\bbO_j \act m^{ij}_k \act \bbO_k^{-1} \\
  = \bbe^{\pi_ip_i}\bbe^{\xi_ix_i}\bbe^{\pi_jp_j}\bbe^{\xi_jx_j} \act m^{ij}_k \act \bbO_k^{-1}
  = \bbe^{\pi_ip_k}\bbe^{\xi_ix_k}\bbe^{\pi_jp_k}\bbe^{\xi_jx_k} \act \bbO_k^{-1} \\
  = \bbe^{-\xi_i\pi_j}\bbe^{(\pi_i+\pi_j)p_k}\bbe^{(\xi_i+\xi_j)x_k} \act \bbO_k^{-1}
  = \bbe^{-\xi_i\pi_j+(\pi_i+\pi_j)p_k+(\xi_i+\xi_j)x_k}.
\end{multline*}

\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm}

{\red\bf GDO} $\coloneqq$ The category with objects finite sets and
\[ \mor(A\to B) = \left\{\calL =\omega\bbe^Q \right\} \subset \bbQ\llbracket\zeta_A,z_B\rrbracket, \]
where:
$\bullet$~$\omega$ is a scalar.
$\bullet$~$Q$ is a ``small'' quadratic in $\zeta_A\cup z_B$.
$\bullet$~Compositions:
$\calL\act \calM \coloneqq \left( \calL|_{z_i\to\partial_{\zeta_i}}\calM \right)_{\zeta_i=0}$.

\vskip -1mm\rule{\linewidth}{1pt}\vspace{-1mm}

\parpic[r]{\parbox{0.5in}{
  \includegraphics[width=0.5in]{../../Projects/Gallery/Feynman.jpg}
  \tiny R.~Feynman
}}
{\bf\red Compositions.} In $\mor(A\!\to\!B)$,
\[ Q = \sum_{i\in A,j\in B}E_{ij}\zeta_iz_j
  + \frac12\sum_{i,j\in A}F_{ij}\zeta_i\zeta_j
  + \frac12\sum_{i,j\in B}G_{ij}z_iz_j,
\]
and so\hfill\text{(remember, $e^x=1+x+xx/2+xxx/6+\ldots$)}
\vskip 1mm
{
  \def\E{{\parbox[t]{1in}{
    $E_1E_2+E_1F_2G_1E_2$
    \newline$+E_1F_2G_1F_2G_1E_2$
    \newline$+\ldots$
    \newline$=\sum\limits_{r=0}^\infty E_1(F_2G_1)^rE_2$
  }}}
  \import{./}{GDOComposition.pdftex_t}
}
\par where $\bullet$\ $E=E_1(I-F_2G_1)^{-1}E_2$
\hfill\text{$\bullet$\ $F=F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T$}
\newline$\bullet$\ $G=G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2$
\hfill$\bullet$\ $\omega=\omega_1\omega_2\det(I-F_2G_1)^{-1/2}$

\vskip -3mm\rule{\linewidth}{1pt}\vspace{-1mm}

{\red\bf Proof of Claim in Example 2.} Let $\Phi_1 \coloneqq
\bbe^{t(p_i-p_j)x_j}$ and $\Phi_2 \coloneqq
\bbO_{p_jx_j}\left(\bbe^{(\bbe^t-1)(p_i-p_j)x_j}\right)
\eqqcolon \bbO(\Psi)$. We show that $\Phi_1 \!=\! \Phi_2$ in
$(\frakh_i\otimes\frakh_j)\llbracket t\rrbracket$ by showing that both
solve the ODE $\partial_t\Phi=(p_i\!-\!p_j)x_j\Phi$ with $\Phi|_{t=0}=1$. For $\Phi_1$
this is trivial. $\Phi_2|_{t=0}=1$ is trivial, and
\[ \partial_t\Phi_2=\bbO(\partial_t\Psi) = \bbO(\bbe^t(p_i-p_j)x_j\Psi) \]
\begin{multline*}
  (p_i\!-\!p_j)x_j\Phi_2 = (p_i\!-\!p_j)x_j\bbO(\Psi) = (p_i\!-\!p_j)\bbO(x_j\Psi-\partial_{p_j}\Psi) \\
  = \bbO\left((p_i\!-\!p_j)(x_j\Psi+(\bbe^t-1)x_j\Psi)\right)
  = \bbO(\bbe^t(p_i\!-\!p_j)x_j\Psi)\qquad\Box
\end{multline*}

%\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm}
\columnbreak

{\red\bf Implementation.}\hfill\text{\footnotesize Without, don't trust!}

\input{AlexanderFromHeisenberg.tex}

{\red\bf References.}\hfill\text{On {\greektext web}$=$\href{http://drorbn.net/cat20}{http://drorbn.net/cat20}}

\newpage
{\red \begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-4pt,topsep=0pt]
\item (2m) Thanks, technicalities.
\item (4m) Read the sidebar.
\item (4m) Quantum invariants in an algebra and the read-out issue.
\item (2m) The Dogma and the exp-issue.
\item (5m) For $ax+b$, get Gaussians! (these are easily computable as we shall see),
\item (3m) In general, get ``docile perturbed Gaussians''; the meaning of $\eps$ (still efficiently computable!).
\item (4m) Packaging.
\item (5m) The ``Gold Standard'' theorem.
\item (7m) Ending discussion.
\item (24m) Full computability.
\end{enumerate}}

\end{multicols}

\end{document}

\endinput

