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% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}
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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Columbia-191125}
\def\thistalkshortcut{co19}
\def\title{Some Feynman Diagrams in Pure Algebra}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\webdef{{\textgreek{web}$\coloneqq$\url{http://drorbn.net/\thistalkshortcut/}}}
\def\web#1{{\href{http://drorbn.net/\thistalkshortcut/#1}{\textgreek{web}/#1}}}

\def\blue{\color{blue}}
\def\purple{\color{purple}}
\def\red{\color{red}}
\def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}}

\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}
% Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top:
\def\imagetop#1{\vtop{\null\hbox{#1}}}

\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\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\TC{\mathit{TC}}
\def\tr{\operatorname{tr}}
\def\vT{{\mathit v\!T}}
\def\wt{\operatorname{wt}}

\def\bbe{\mathbbm{e}}
\def\bbD{{\mathbb D}}
\def\bbH{{\mathbb H}}
\def\bbN{{\mathbb N}}
\def\bbO{{\mathbb O}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\def\calE{{\mathcal E}}
\def\calF{{\mathcal F}}
\def\calG{{\mathcal G}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calL{{\mathcal L}}
\def\calM{{\mathcal M}}
\def\calO{{\mathcal O}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\fraka{{\mathfrak a}}
\def\frakg{{\mathfrak g}}
\def\tilE{\tilde{E}}

\def\cellscale{0.645}

\begin{document}
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\parpic[r]{\qrcode[height=3.5em,level=L,nolink,tight]{drorbn.net/\thistalkshortcut}}
\parbox[b]{3.6in}{{\footnotesize\navigator}\newline{\Large\bf\red \title}}
\hfill\parbox[b]{0.7in}{\footnotesize
  With Roland\newline van der Veen
}\ \includegraphics[height=12mm]{../../Projects/Gallery/VanDerVeen.jpg}
\hfill\parbox[b]{2.3in}{\footnotesize
  \null\hfill Thanks for allowing me in Columbia U!
  \newline\null\hfill\webdef
  \newline\null\hfill Slides w/ no handout/URL should be banned!
}
\vskip -3mm
\rule{\dimexpr\linewidth-4em}{1pt}
\vspace{-8mm}

\begin{multicols}{2} \raggedcolumns

{\red\bf Abstract.} I will explain how the computation of compositions of
maps of a certain natural class, from one polynomial ring into another,
naturally leads to a certain composition operation of quadratics and to
Feynman diagrams. I will also explain, with very little detail, how this
is used in the construction of some very well-behaved poly-time computable
knot polynomials.

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

{\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}

{\red\bf Gentle Agreement.} Everything converges!

{\red\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$(y,b,a,x)^*=(\eta,\beta,\alpha,\xi)$

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

{\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
$\calG(L\act M) = \left(\calG(L)|_{z_b\to\partial_{\zeta_b}}\calG(M)\right)_{\zeta_b=0}$.

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

{\red\bf Basic Examples.} {\bf 1.}
$\calG(id\colon\bbQ[y,a,x]\to\bbQ[y,a,x]) = \bbe^{\eta y+\alpha a+\xi x}$.

\needspace{1.25cm}
\parpic[r]{$\xymatrix@C=5mm@R=5mm{
  \bbQ[ z]_i\otimes\bbQ[ z]_j
    \ar[r]^<>(0.5){m^{ij}_k} \ar@{=}[d] &
  \bbQ[ z]_k \ar@{=}[d] \\
  \bbQ[ z_i,z_j] \ar[r]^<>(0.5){m^{ij}_k} &
  \bbQ[ z_k]
}$}
{\bf 2.} The standard commutative product  $m^{ij}_k$ of polynomials
is given by $z_i,z_j\to z_k$. Hence $\calG(m^{ij}_k) = m^{ij}_k(\bbe^{\zeta_i z_i+\zeta_j z_j})
= \bbe^{(\zeta_i+\zeta_j)z_k}$.

\needspace{1.25cm}
\parpic[r]{$\xymatrix@C=5mm@R=5mm{
  \bbQ[ z]_i 
    \ar[r]^<>(0.5){\Delta^i_{jk}} \ar@{=}[d] &
  \bbQ[ z]_j\otimes\bbQ[ z]_k \ar@{=}[d] \\
  \bbQ[ z_i] \ar[r]^<>(0.5){\Delta^i_{jk}} &
  \bbQ[ z_j,z_k]
}$}
{\bf 3.} The standard co-commutative co-product  $\Delta^i_{jk}$ of polynomials
is given by $z_i\to z_j+z_k$. Hence $\calG(\Delta^i_{jk}) = \Delta^i_{jk}(\bbe^{\zeta_i z_i})
= \bbe^{\zeta_i(z_j+z_k)}$.

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

{\red\bf Heisenberg Algebras.} Let $\bbH=\langle x,y\rangle/[x,y]=\hbar$ (with $\hbar$ a scalar),
let $\bbO_i\colon\bbQ[x_i,y_i]\to\bbH_i$ is the ``$x$ before $y$''
PBW ordering map and let $hm^{ij}_k$ be the composition
\[ \begin{CD} \bbQ[x_i,y_i,x_j,y_j] @>\bbO_i\otimes\bbO_j>> \bbH_i\otimes\bbH_j
  @>m^{ij}_k>> \bbH_k @>\bbO_k^{-1}>> \bbQ[x_k,y_k]. \end{CD}
\]
Then $\calG(hm^{ij}_k) = \bbe^{\Lambda_\hbar}$, where
$\Lambda_\hbar=-\hbar\eta_i\xi_j+(\xi_i+\xi_j)x_k+(\eta_i+\eta_j)y_k$.

{\bf Proof 1.} Recall the ``Weyl form of the
CCR'' $\bbe^{\eta y}\bbe^{\xi x} = \bbe^{-\hbar\eta\xi}\bbe^{\xi x}\bbe^{\eta
y}$, and compute
\begin{multline*}
  \calG(hm^{ij}_k)
  = \bbe^{\xi_ix_i+\eta_iy_i+\xi_jx_j+\eta_jy_j}
    \act \bbO_i\otimes\bbO_j \act m^{ij}_k \act \bbO_k^{-1} \\
  = \bbe^{\xi_ix_i}\bbe^{\eta_iy_i}\bbe^{\xi_jx_j}\bbe^{\eta_jy_j} \act m^{ij}_k \act \bbO_k^{-1}
  = \bbe^{\xi_ix_k}\bbe^{\eta_iy_k}\bbe^{\xi_jx_k}\bbe^{\eta_jy_k} \act \bbO_k^{-1} \\
  = \bbe^{-\hbar\eta_i\xi_j}\bbe^{(\xi_i+\xi_j)x_k}\bbe^{(\eta_i+\eta_j)y_k} \act \bbO_k^{-1}
  = \bbe^{\Lambda_\hbar}.
\end{multline*}
  
{\bf Proof 2.} We compute in a faithful 3D representation $\rho$ of $\bbH$:
\newline\null\hfill\text{(\web{hm})}

\vskip -3.5mm
\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{../UCLA-191101/Snips/FDAExamples/hm-\the\snip.pdf}
\ifnum \snip<4 \repeat

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

{\red\bf A Real DoPeGDO Example} (DoPeGDO$\coloneqq$Docile Perturbed Gaussian Differential Operators).
Let
$sl_{2+}^\epsilon \coloneqq L\langle y,b,a,x\rangle$ 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 if $\exists \epsilon^{-1}$, $sl_{2+}^\epsilon\cong
sl_2\oplus\langle t\rangle$. Let $CU\coloneqq\calU(sl_{2+}^\epsilon)$, and let $cm^{ij}_k$
be the composition below, where $\bbO_i\colon\bbQ[y_i,b_i,a_i,x_i]\to CU_i$ be the PBW
ordering map in the order $ybax$:
\[ \xymatrix@C=9mm@R=5mm{
  CU_i\otimes CU_j \ar[r]^<>(0.5){m^{ij}_k} &
  CU_k \\
  \bbQ[ y_i,b_i,a_i,x_i,y_j,b_j,a_j,x_j ]
    \ar[r]^<>(0.5){cm^{ij}_k} \ar[u]_<>(0.4){\bbO_{i,j}} &
  \bbQ[ y_k,b_k,a_k,x_k ]
    \ar[u]_<>(0.4){\bbO_k} &
} \]

{\bf Claim.} Let\hfill\text{\footnotesize(all brawn and no brains)}
\begin{multline*}
  \Lambda = \left(\eta _i+\frac{e^{-\alpha _i-\epsilon  \beta _i} \eta _j}{1+\epsilon  \eta _j \xi
_i}\right) y_k+\left(\beta _i+\beta
   _j+\frac{\log \left(1+\epsilon  \eta _j \xi _i\right)}{\epsilon }\right) b_k+ \\
  \left(\alpha
_i+\alpha _j+\log \left(1+\epsilon  \eta
   _j \xi _i\right)\right) a_k+\left(\frac{e^{-\alpha _j-\epsilon  \beta _j} \xi _i}{1+\epsilon
\eta _j \xi _i}+\xi _j\right) x_k
\end{multline*}
Then
$\bbe^{\eta_iy_i+\beta_ib_i+\alpha_ia_i+\xi_ix_i +
\eta_jy_j+\beta_jb_j+\alpha_ja_j+\xi_jx_j} \act \bbO_{i,j} \act cm^{ij}_k
= \bbe^\Lambda \act \bbO_k$, and hence $\calG(cm^{ij}_k)=\bbe^\Lambda$.

\needspace{3\baselineskip}
{\bf Proof.} We compute in a faithful 2D representation $\rho$ of $CU$:
\newline\null\hfill\text{(\web{sl2})}

\vskip -3.5mm
\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{../UCLA-191101/Snips/FDAExamples/sl2-\the\snip.pdf}
\ifnum \snip<6 \repeat

{\bf Note 1.} If the lower half of the alphabet ($a,b,\alpha,\beta$) is regarded as constants, then
$\Lambda=C+Q+\sum_{k\geq 1}\epsilon^kP^{(k)}$ is a docile perturbed Gaussian relative to the upper half of the
alphabet ($x,y,\xi,\eta$): $C$ is a scalar, $Q$ is a quadratic, and $\deg P^{(k)}\leq 2k+2$.

{\bf Note 2.} $\wt(x,y,\xi,\eta;a,b,\alpha,\beta;\epsilon)=(1,1,1,1;2,0,0,2;-2)$.

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

{\bf\red Quadratic Casimirs.} If $t\in\frakg\otimes\frakg$ is the quadratic Casimir of a
semi-simple Lie algebra $\frakg$, then $\bbe^t$, regarded by PBW
as an element of $\calS^{\otimes 2}=\Hom\left(\calS(\frakg)^{\otimes
0}\to\calS(\frakg)^{\otimes 2}\right)$, has a latin-latin dominant Gaussian factor. Likewise for $R$-matrices.

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

(Baby) {\red\bf DoPeGDO} $\coloneqq$ The category with objects finite sets$^{\dagger 1}$ and
\[ \mor(A\to B) = \left\{\calL =\omega\exp(Q+P) \right\} \subset \bbQ\llbracket\zeta_A,z_B,\epsilon\rrbracket, \]
where:
$\bullet$~$\omega$ is a scalar.$^{\dagger 2}$
$\bullet$~$Q$ is a ``small'' $\epsilon$-free quadratic in $\zeta_A\cup z_B$.$^{\dagger 3}$
$\bullet$~$P$ is a ``docile perturbation'': $P=\sum_{k\geq 1}\epsilon^kP^{(k)}$, where
  $\deg P^{(k)}\leq 2k+2$.$^{\dagger 4}$
$\bullet$~Compositions:$^{\dagger 6}$
$\calL\act \calM \coloneqq \left( \calL|_{z_i\to\partial_{\zeta_i}}\calM \right)_{\zeta_i=0}$.

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

{\red\bf So What?} If $V$ is a representation, then
$V^{\otimes n}$ explodes as a function of $n$, while in {\bf DoPeGDO}
up to a fixed power of $\epsilon$, the ranks of $\mor(A\to B)$
grow polynomially as a function of $|A|$ and $|B|$.

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

{\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{./}{Compositions.pdftex_t}
}
\parpic[r]{\scalebox{0.9}{\import{../UCLA-191101/}{FeynmanDiagrams.pdftex_t}}}
where $\bullet$\ $E=E_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\ $F=F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T$.
\newline$\bullet$\ $G=G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\ $\omega=\omega_1\omega_2\det(I-F_2G_1)^{-1}$.
\newline$\bullet$\ $P$ is computed as the solution of a
  messy PDE or using ``connected Feynman diagrams'' (yet we're still in pure algebra!). Docility is preserved.

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

{\bf\red DoPeGDO Footnotes.} Each variable has a
  ``weight''$\in\{0,1,2\}$, and always $\wt z_i+\wt\zeta_i=2$.
\par$\dagger 1$.~Really, ``weight-graded finite sets'' $A=A_0\sqcup A_1\sqcup A_2$.
\par$\dagger 2$.~Really, a power series in the weight-0 variables$^{\dagger 5}$.
\par\hangindent=6mm$\dagger 3$.~The weight of $Q$ must be 2, so it
  decomposes as $Q=Q_{20}+Q_{11}$. The coefficients of $Q_{20}$ are rational
  numbers while the coefficients of $Q_{11}$ may be weight-0 power series$^{\dagger 5}$.
\par\hangindent=6mm$\dagger 4$.~Setting $\wt\epsilon=-2$, the weight of $P$
  is $\leq 2$ (so the powers of the weight-0 variables are not
  constrained)$^{\dagger 5}$.
\par\hangindent=6mm$\dagger 5$.~In the knot-theoretic case, all weight-0 power series are rational functions of
  bounded degree in the exponentials of the weight-0 variables.
\par\hangindent=6mm$\dagger 6$.~There's also an obvious product
\[ \mor(A_1\to B_1)\times\mor(A_2\to B_2)\to\mor(A_1\sqcup A_2\to B_1\sqcup B_2). \]

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

\def\linalg{\parbox{1.0in}{\footnotesize
{\red\bf Analog.} Solve
\newline\null\hfill$Ax=a,\ B(x)y=b$\hfill\null
}}
\parpic[r]{\import{../UCLA-191101/}{FullDoPeGDO.pdftex_t}}
{\bf\red Full DoPeGDO.} Compute compositions in two phases:

$\bullet$~A 1-1 phase over the ring of power series in the weight-0 variables, in which the weight-2 variables
are spectators.

$\bullet$~A (slightly modified) 2-0 phase over $\bbQ$, in which the weight-1 variables are spectators.

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

{\bf\red Questions.} $\bullet$~Are there QFT precedents for ``two-step Gaussian integration''?

$\bullet$~In QFT, one saves even more by considering ``one-particle-irreducible'' diagrams and ``effective
actions''. Does this mean anything here?

$\bullet$~Understanding $\Hom(\bbQ[z_A]\to\bbQ[z_B])$ seems like a good cause. Can you find other applications for
the technology here?

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

\vskip -3mm
\[ \left(\parbox{3.75in}{\footnotesize
$QU=\calU_\hbar(sl_{2+}^\epsilon) = A\langle y,b,a,x\rangle\llbracket\hbar\rrbracket$ with
$[a,x]=x$, $[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon
x$, and $xy-qyx=(1-AB)/\hbar$, where $q=\bbe^{\hbar\epsilon}$,
$A=\bbe^{-\hbar\epsilon a}$, and $B=\bbe^{-\hbar b}$. Also
$\Delta(y,b,a,x) = (y_1+B_1y_2, b_1+b_2, a_1+a_2, x_1+A_1x_2)$,
$S(y,b,a,x) = (-B^{-1}y, -b, -a, -A^{-1}x)$,
and $R=\sum\hbar^{j+k}y^kb^j\otimes a^jx^k/j![k]_q!$.
}\right) \]
\vskip 1mm

{\bf\red Theorem.} Everything of value regrading $U=CU$ and/or its quantization $U=QU$ is {\bf DoPeGDO}:

\vskip -1mm
\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 $\tr\colon U \to U/wx=xw$}}
\def\maximR{{\thickmuskip=0mu $R\in QU\otimes QU$}}
\def\maximC{{\thickmuskip=0mu $C^{\pm 1}\in QU$}}
\def\maximPhi{{\thickmuskip=0mu $\Phi\in CU^{\otimes 3}$}}
\def\maximJ{{\thickmuskip=0mu $J\in CU\otimes CU$}}
\[ \import{../UCLA-191101/}{Grid.pdftex_t} \]
also Cartan's $\theta$, the $\bbD$equantizator, and more, and all of their compositions.

\rule{\linewidth}{1pt}\vspace{0mm}

\parpic[r]{\scalebox{1}{\import{../UCLA-191101/}{Puddle.pdftex_t}}}
{\red\bf Solvable Approximation.} In $sl_n$, half is enough! Indeed
${sl_n\oplus\fraka_{n-1}} = \calD(\uppertriang,b,\delta)$.  Now define
$sl^\epsilon_{n+}\coloneqq\calD(\uppertriang,b,\epsilon\delta)$.
Schematically, this is ${[\uppertriang,\uppertriang]=\uppertriang}$,
$[\lowertriang,\lowertriang]=\epsilon\lowertriang$, and
$[\uppertriang,\lowertriang]=\lowertriang+\epsilon\uppertriang$. The same
process works for all semi-simple Lie algebras, and at $\epsilon^{k+1}=0$
always yields a solvable Lie algebra.

\vskip -3mm
\def\bracket{$b({\red\uppertriang})=b\colon{\red\uppertriang}\otimes\!{\red\uppertriang}
  \to{\red\uppertriang}$}
\def\cobracket{$b({\blue\lowertriang})\leadsto\delta\colon{\red\uppertriang}
  \to{\red\uppertriang}\otimes\!{\red\uppertriang}$}
\[ \hspace{-35mm}\import{../UCLA-191101/}{Double.pdftex_t} \]

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

{\bf\red Conclusion.} There are lots of poly-time-computable well-behaved
near-Alexander knot invariants: $\bullet$~They extend to tangles with appropriate
multiplicative behaviour. $\bullet$~They have cabling and strand reversal
formulas.\hfill\web{akt}

The invariant for $sl_{2+}^\epsilon/(\epsilon^2=0)$ (prior art: \web{Ov})
attains 2,883 distinct values on the 2,978 prime knots with $\leq 12$
crossings. HOMFLY-PT and Khovanov homology together attain only 2,786
distinct values.

%\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm}
%
%{{\red References.}\footnotesize
%\par\vspace{-3mm}
%\renewcommand{\section}[2]{}%
%\begin{thebibliography}{}
%\setlength{\parskip}{0pt}
%\setlength{\itemsep}{0pt plus 0.3ex}
%
%\bibitem[BN]{KBH} D.~Bar-Natan,
%  {\em Balloons and Hoops and their Universal Finite Type Invariant, BF
%    Theory, and an Ultimate Alexander Invariant,}
%  \web{KBH}, \arXiv{1308.1721}.
%
%\end{thebibliography}}

\end{multicols}

\vskip -5mm

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}}

\setlength{\outerboxwidth}{2.45in} \setlength{\innerboxwidth}{2.05in}
{\footnotesize \begin{longtable}{|c|c|c|}
\hline \headcell & \headcell & \headcell \\ \endhead
\hline
\import{../QMUL-1908/}{table1.tex}
\end{longtable}}

\setlength{\outerboxwidth}{3.76in} \setlength{\innerboxwidth}{3.36in}
{\footnotesize \begin{longtable}{|c|c|c|}
\hline \headcell & \headcell \\ \endhead
\hline
\import{../QMUL-1908/}{table2.tex}
\end{longtable}}

\end{document}

\endinput