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% Following http://tex.stackexchange.com/a/847/22475:
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% Following http://tex.stackexchange.com/questions/59340/how-to-highlight-an-entire-paragraph
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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{DaNang-1905}
\def\title{Everything around $sl_{2+}^\epsilon$ is {\bf DoPeGDO}. So what?}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to \magenta Da Nang!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/v19}{http://drorbn.net/v19/}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}
\def\more{More at \web{talks}}

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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{$\Box$}}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\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\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\barT{{\bar T}}
\def\bbe{\mathbbm{e}}
\def\bbD{{\mathbb D}}
\def\bbE{{\mathbb E}}
\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\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\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}}

% From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes
\DeclareMathOperator*{\midotimes}{\text{\raisebox{0.25ex}{\scalebox{0.8}{$\bigotimes$}}}}

\def\cellscale{0.645}

%%%

\def\credits{{\raisebox{0.75mm}{\parbox[t]{1.3in}{
Continues Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC} and \text{Overbay}
\cite{Overbay:Thesis}, joint with van der Veen.
}}}}

\def\AbstractA{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Abstract.} I'll explain what ``everything around'' means:
classical and quantum $m$, $\Delta$, $S$, $tr$, $R$, $C$, and $\theta$,
as well as $P$, $\Phi$, $J$, $\bbD$, and more, and all of their
compositions. What {\bf DoPeGDO} means: the category of Docile Perturbed
Gaussian Differential Operators. And what $sl_{2+}^\epsilon$ means:
a solvable approximation of the semi-simple Lie algebra $sl_2$.
}}}}

\def\AbstractB{{\raisebox{-1mm}{\parbox[t]{3.9in}{
Knot theorists should rejoice \text{because} all this leads to very powerful and
well-behaved poly-time-computable knot \text{invariants}. Quantum algebraists
should rejoice because it's a realistic playground for testing complicated
\text{equations} and theories.
}}}}

\def\maximm{{\thickmuskip=0mu $m\colon U \otimes U \to U$}}
\def\maximDelta{{\thickmuskip=0mu $\Delta\colon U \to U \otimes U$}}
\def\maximS{{\thickmuskip=0mu $S\colon U \to U$}}
\def\maximtr{{\thickmuskip=-2mu $\tr\colon U \to U/xw=wx$}}
\def\maximR{{\thickmuskip=0mu $R\in U\otimes U$}}
\def\maximC{{\thickmuskip=0mu $C\in U$}}
\def\maximPhi{{\thickmuskip=0mu $\Phi\in U^{\otimes 3}$}}
\def\maximJ{{\thickmuskip=0mu $J\in U\otimes U$}}

\def\Udef{{\raisebox{2mm}{\parbox[t]{1.625in}{
$sl_{2+}^\epsilon \coloneqq L\langle y,b,a,x\rangle$ subject to ${[a,x]=x}$,
$[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon x$, and $[x,y]=\epsilon a+b$. So
$t\coloneqq\epsilon a-b$ is central and $sl_{2+}^\epsilon/\langle t\rangle\cong sl_2$.

$U$ is either $CU=\calU(sl_{2+}^\epsilon)$ or $QU=\calU_\hbar(sl_{2+}^\epsilon) = A\langle y,b,a,x\rangle$ with
$[a,x]=x$, $[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon x$, and $xy-qyx=(1-AB)/\hbar$,
where $q=\bbe^{\hbar\epsilon}$, $A=\bbe^{-\hbar\epsilon a}$, and $B=\bbe^{-\hbar b}$. Set
$T=A^{-1}B=\bbe^{\hbar t}$.
}}}}

\def\zdef{{\raisebox{2mm}{\parbox[t]{2.96in}{
Conventions. For a set $A$, $z_A\coloneqq\{z_i\}_{i\in A}$ and $\zeta_A\coloneqq\{z^*_i=\zeta_i\}_{i\in A}$.
Always, at least one of $\{z_i,\zeta_i\}$ is ``heavy''.

$\text{\bf DoPeGDO} \coloneqq$ The category with objects finite sets and
$\mor(A\to B)$:
\[ \left\{ F=\omega\exp\left(Q+\sum_{k\geq 1}\epsilon^kP^{(k)}\right) \right\} \]
$\bullet$~$\omega$ is a scalar. $\bullet$ $Q$ is a ``small'' quadratic in $\zeta_A\cup z_B$.
$\bullet$~The $P_k$ are ``perturbation polynomials''; the ``heavy degree'' of $P_k$ is $\leq
k+1$. $\bullet$~Compositions:
\[ F\act G = G\circ F \coloneqq
  \left( G|_{\zeta_i\to\partial_{z_i}}F \right)_{z_i=0}
  = \left( F|_{z_i\to\partial_{\zeta_i}}G \right)_{\zeta_i=0}.
\]
{\bf Cool!} $(V^*)^{\otimes A}\otimes V^{\otimes B}$ explodes; the ranks of quadratics and
fixed-degree polynomials grow slowly!
}}}}

\def\QEFG{{In $\mor(A\to B)$, $Q=\hspace{-2mm}\sum\limits_{i\in A,j\in B}\hspace{-2mm}E_{ij}\zeta_iz_j + \frac12\hspace{-1mm}\sum\limits_{i,j\in
A}\hspace{-1mm}F_{ij}\zeta_i\zeta_j + \frac12\hspace{-1mm}\sum\limits_{i,j\in B}\hspace{-1mm}G_{ij}z_iz_j $}}

\def\Compositions{{\raisebox{1mm}{\parbox[t]{3.96in}{
Where
$\bullet$\quad $\omega=\omega_1\omega_2\det(I-F_2G_1)^{-1}$.
\newline$\bullet$\quad $E=E_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\quad $F=F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T$.
\newline$\bullet$\quad $G=G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\quad $P$ is computed using ``connected Feynman diagrams'' or as the solution of a
messy PDE (yet we're still in algebra!).
\newline\null\hfill\text{\mpink \ldots a presentation / implementation dead end.}
}}}}

\def\oneup{{\raisebox{1mm}{\parbox[t]{1.08in}{
One abstraction level up from tangles!
}}}}

\def\todo{{\raisebox{1mm}{\parbox[t]{3.96in}{
{\bf\red To do.}
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt]
\item Finish the objects matrix.
\item Insert the implementation of naive DoPeGDO.
\item Some examples how parts of the Portfolio are {DoPeGDO} / {DoPeGDO}$_2$.
\item Completing the definition of {DoPeGDO}$_2$.
\item A menagerie of extras.
\end{itemize}
}}}}

\def\OverratedOne{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\bf\red Categories are overrated (1)!} $\bullet$~Tangles are artificially made to have a
``top'' and a ``bottom''. $\bullet$~Tangles are accessed by their ends and not by their strands;
crossings are named by their position and not by the strands involved:
}}}}

\def\TwoReidemeister{{\raisebox{1mm}{\parbox[t]{2.15in}{
Is this $\sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2$
\newline or $\sigma_{12}\sigma_{13}\sigma_{23}=\sigma_{23}\sigma_{13}\sigma_{12}$?

\vskip 3mm
$\bullet$~Easier to talk about ``skein theory''.
\newline $\bullet$~Harder to talk about ``universal quantum invariants''.
}}}}

\def\refs{{\raisebox{3mm}{\parbox[t]{3.95in}{
%{\red\bf References.}
{\footnotesize
%\par\vspace{-3mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}

\bibitem[BNG]{Bar-NatanGaroufalidis:MMR} D.~Bar-Natan and S.~Garoufalidis,
  {\em On the Melvin-\newline Morton-Rozansky conjecture,}
  Invent.\ Math.\ {\bf 125} (1996) 103--133.

\bibitem[BV]{PP1} D.~Bar-Natan and R.~van~der~Veen,
  {\em A Polynomial Time Knot Polynomial,}
  \arXiv{1708.04853}.

\bibitem[Fa]{Faddeev:ModularDouble} L.~Faddeev,
  {\em Modular Double of a Quantum Group,}
  \arXiv{math/9912078}.

\bibitem[GR]{GaroufalidisRozansky:LoopExpansion} S.~Garoufalidis and L.~Rozansky,
  {\em The Loop Exapnsion of the Kontsevich Integral, the Null-Move, and $S$-Equivalence,}
  \arXiv{math.GT/0003187}.

\bibitem[MM]{MM} P.~M.~Melvin and H.~R.~Morton,
  {\em The coloured Jones function,}
  Commun.\ Math.\ Phys.\ {\bf 169} (1995) 501--520.

\bibitem[Ov]{Overbay:Thesis} A.~Overbay,
  {\em Perturbative Expansion of the Colored Jones Polynomial,}
  University of North Carolina PhD thesis, \web{Ov}.

\bibitem[Qu]{Quesne:Jackson} C.~Quesne,
  {\em Jackson's $q$-Exponential as the Exponential of a Series,}
  \arXiv{math-ph/0305003}.

\bibitem[Ro1]{Ro} L.~Rozansky,
  {\em A contribution of the trivial flat connection to the Jones
polynomial and Witten's invariant of 3d manifolds, I,}
  Comm.\ Math.\ Phys.\ {\bf 175-2} (1996) 275--296, \arXiv{hep-th/9401061}.

\bibitem[Ro2]{Rozansky:Burau} L.~Rozansky,
  {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton
    Expansion of the Colored Jones Polynomial,}
  Adv.\ Math.\ {\bf 134-1} (1998) 1--31, \arXiv{q-alg/9604005}.

\bibitem[Ro3]{Rozansky:U1RCC} L.~Rozansky,
  {\em A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture,}
  \arXiv{math/0201139}.

\bibitem[Za]{Zagier:Dilogarithm} D.~Zagier,
  {\em The Dilogarithm Function,}
  in Cartier, Moussa, Julia, and Vanhove (eds) {\em Frontiers in Number
  Theory, Physics, and Geometry II.} Springer, Berlin, Heidelberg,
  and \web{Za}.

\end{thebibliography}}
}}}}

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\begin{document} \latintext
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\begin{center}
\null\vfill\input{DoPeGDO1.pdftex_t}\vfill\null
\null\vfill\input{DoPeGDO2.pdftex_t}\vfill\null
\end{center}

\eject
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\newpage

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\def\gY{\textcolor{ForestGreen}{\ding{52}}}
\def\oY{\textcolor{Orange}{\ding{52}}}
\def\neg{\hspace{-6pt}}
\def\headcell{
  \neg diagram\neg & \neg\parbox{2in}{{\blue $n^t_k$}\quad Alexander's $\omega^+$
    \hfill genus / \textcolor{ForestGreen}{ribbon} \newline
  {\red Today's $\rho_1^+$}
    \hfill unknotting \# / \textcolor{Orange}{amphi?}}
}
\def\rolcell#1#2#3#4#5#6#7#8{
  \neg\raisebox{-3pt}{\includegraphics[height=21pt]{../UNC-1610/KnotFigs/#1.pdf}}\neg
& \neg\parbox[b]{2in}{
    {\blue $#2$}\quad $#3$\hfill $#5$ / #7 \\
    {\red $#4$} \hfill $#6$ / #8
}}

\vskip -7mm

{\footnotesize \begin{longtable}{|cl|cl|cl|}
\hline \headcell & \headcell & \headcell \\ \endhead
\hline
\input ../Toronto-1811/table.tex
\end{longtable}}

\end{document}

\endinput