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\def\navigator{{Video and more @ Dror Bar-Natan: Talks: Vienna-1402:}}
\def\webdef{{\url{http://www.math.toronto.edu/~drorbn/Talks/Vienna-1402}}}
%\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Vienna-1402/#1}{$\omega$/#1}}}

\def\blue{\color{blue}}
\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:#1}}}

\def\Ad{\operatorname{Ad}}
\def\CW{\text{\it CW}}
\def\FL{\text{\it FL}}

\def\bbR{{\mathbb R}}
\def\calD{{\mathcal D}}
\def\calG{{\mathcal G}}
\def\calO{{\mathcal O}}
\def\frakg{{\mathfrak g}}

%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Abstract.} I will describe a
\cbox{yellow}{semi-rigorous} reduction of perturbative BF theory
(Cattaneo-Rossi~\cite{CattaneoRossi:WilsonSurfaces}) to computable
combinatorics, in the case of ribbon 2-links. Also, I will explain
how and why my approach may or may not work in the non-ribbon
case. \cbox{yellow}{Weak} this result is, and at least partially
already known (Watanabe~\cite{Watanabe:CSI}). Yet in the ribbon case,
the resulting invariant is a universal finite type invariant, a gadget
that significantly generalizes and clarifies the Alexander polynomial
and that is closely related to the Kashiwara-Vergne problem. I cannot
rule out the possibility that the corresponding gadget in the non-ribbon
case will be as interesting.
\hfill\raisebox{2pt}{\text{\footnotesize
  (good news in \cbox{yellow}{highlight})
}}
}}}}

\def\doublebubble{\url{http://youtu.be/mHyTOcfF99o}}
\def\trefoilbubble{\url{http://youtu.be/YCA0VIExVhge}}

\def\BFBox{{\raisebox{2mm}{\parbox[t]{3.4in}{
{\red BF Following~\cite{CattaneoRossi:WilsonSurfaces}.}
$A\in\Omega^1(M=\bbR^4,\frakg)$, $B\in\Omega^2(M,\frakg^*)$,
\[ S(A,B)\coloneqq\int_M\langle B,F_A\rangle. \]
With $f\colon(S=\bbR^2)\to M$, $\xi\in\Omega^0(S,\frakg)$,
$\beta\in\Omega^1(S,\frakg^*)$, set
\[
  \calO(A,B,f)\coloneqq\int\calD\xi\calD\beta\exp\left(
    \frac{i}{\hbar}\int_S\left\langle\xi,d_{f^*A}\beta+f^*B\right\rangle
  \right).
\]
}}}}

\def\spuntref{{\raisebox{0.5mm}{\parbox[t]{1.75in}{
A {\red 2-twist spun trefoil} by Carter-Kamada-Saito~\cite{CKS}.
}}}}

\def\FeynmanRules{{\raisebox{2mm}{\parbox[t]{4in}{
\parshape 6 0in 1.75in 0in 1.75in 0in 1.75in 0in 1.75in 0in 1.75in 0in 4in
{\red The BF Feynman Rules.} For an edge $e$, let $\Phi_e$ be its
direction, in $S^3$ or $S^1$. Let $\omega_3$ and $\omega_1$ be volume forms
on $S^3$ and $S_1$. Then for a 2-link $(f_t)_{t\in T}$,
\[ \def\neg{{\hspace{-1mm}}}
  \zeta=\log\neg\sum_{\text{diagrams}\atop D}\neg
  \frac{D}{|\text{Aut}(D)|}
  \underbrace{\int_{\bbR^2}\neg\!\!\cdots\neg\int_{\bbR^2}}_{S\text{-vertices}}
  \underbrace{\int_{\bbR^4}\neg\!\!\cdots\neg\int_{\bbR^4}}_{M\text{-vertices}}
  \prod_{\text{red}\atop e\in D}\Phi_e^\ast\omega_3
  \prod_{\text{black}\atop e\in D}\Phi_e^\ast\omega_1
\]
is an invariant in $\CW(\FL(T))\to\CW(T)$, ``cyclic words in $T$''.
}}}}

\def\sumklm{$\displaystyle\sum_{k,l,m\geq 0}\frac{(+)^k(-)^l(+)^m}{k!l!m!}$}
\def\sumkm{$\displaystyle\sum_{k,m\geq 0}\frac{(+)^k(-)^m}{k!m!}$}

\def\Theorem{{\raisebox{3mm}{\parbox[t]{3.95in}{
{\red Theorem 1 (with Cattaneo (credit, no blame)).} In the ribbon case,
$e^\zeta$ can be computed as follows:
}}}}

\def\Combinatorics{{\raisebox{1mm}{\parbox[t]{3.95in}{
{\red Theorem 2.} Using Gauss diagrams to represent knots and
$T$-component pure tangles, the above formulas define an invariant in
$\CW(\FL(T))\to\CW(T)$, ``cyclic words in $T$''.

$\bullet$ Agrees with BN-Dancso~\cite{WKO} and with~\cite{KBH}.
$\bullet$ In-practice computable!
$\bullet$ Vanishes on braids.
$\bullet$ Extends to w.
$\bullet$ Contains Alexander.
$\bullet$ The ``missing factor'' in Levine's
  factorization~\cite{Levine:Factorization} (the rest
  of~\cite{Levine:Factorization} also fits, hence contains the MVA).
$\bullet$ Related to / extends
  Farber's~\cite{Farber:NoncommutativeRationalFunctions}?
$\bullet$ Should be summed and categorified.
}}}}

\def\Sketch{{\raisebox{2mm}{\parbox[t]{3in}{
\parshape 4 0in 2in 0in 2in 0in 2in 0in 3in
{\red Sketch of Proof.} In $4D$ axial gauge, only ``drop down'' red
propagators, hence in the ribbon case, no $M$-trivalent vertices. $S$
integrals are $\pm 1$ iff ``ground pieces'' run on nested curves as
below, and exponentials arise when several propagators compete for the
same double curve. And then the combinatorics is obvious\ldots
}}}}

\def\BFCS{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Chern-Simons.} When the domain of BF is restricted to ribbon knots,
and the target of CS is restricted to trees and wheels, they agree. Why?
}}}}

\def\veeinvariant{{\raisebox{2mm}{\parbox[t]{1.3333333in}{
{\red Is this all?} What about the $\vee$-invariant?
\par (the ``true'' triple linking number)
}}}}

\def\gnots{{\raisebox{2mm}{\parbox[t]{2.9in}{
{\red Gnots.} In 3D, a generic immersion of $S^1$ is an embedding, a knot.
In 4D, a generic immersion of a surface has finitely-many double points (a
gnot?). Perhaps we should be studying these?
}}}}

\def\vft{{\raisebox{2mm}{\parbox[t]{2in}{
{\red Finite type.} What are finite-type invariants for 2-knots? What would
be ``chord diagrams''?
}}}}

\def\bubblewrap{{\raisebox{2mm}{\parbox[t]{1.92in}{
{\red Bubble-wrap-finite-type.} There's an alternative definition of finite
type in 3D, due to Goussarov (see~\cite{Bar-Natan:Bracelets}). The obvious
parallel in 4D involves ``bubble wraps''. Is it any good?
}}}}

\def\shielded{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Shielded tangles.} In 3D, one can't zoom in and compute ``the
Chern-Simons invariant of a tangle''. Yet there are well-defined invariants
of ``shielded tangles'', and rules for their compositions. What would the
4D analog be?
}}}}

\def\KV{{\raisebox{0mm}{\parbox[t]{4in}{
Will the relationship with the Kashiwara-Vergne problem~\cite{WKO}
necessarily arise here?
}}}}

\def\planecurves{{\raisebox{2mm}{\parbox[t]{3.45in}{
{\red Plane curves.} Shouldn't we understand integral / finite type
invariants of plane curves, in the style of Arnold's $J^+$, $J^-$, and
{\it St}~\cite{Arnold:PlaneCurves}, a bit better?
}}}}

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

\bibitem[Ar]{Arnold:PlaneCurves} V.~I.~Arnold,
  {\em Topological Invariants of Plane Curves and Caustics,}
  University Lecture Series {\bf 5}, American Mathematical Society 1994.

\bibitem[BN1]{Bar-Natan:Bracelets} D.~Bar-Natan,
  {\em Bracelets and the Goussarov filtration of the space of knots,}
  {\em Invariants of knots and 3-manifolds (Kyoto 2001),} Geometry and
  Topology Monographs {\bf 4} 1--12, \arXiv{math.GT/0111267}.

\bibitem[BN2]{KBH} D.~Bar-Natan,
  {\em Balloons and Hoops and their Universal Finite Type Invariant, BF
    Theory, and an Ultimate Alexander Invariant,}
  \url{http://www.math.toronto.edu/~drorbn/papers/KBH/}, \arXiv{1308.1721}.

\bibitem[BND]{WKO} D.~Bar-Natan and Z.~Dancso,
  {\em Finite Type Invariants of W-Knotted Objects: From Alexander to
    Kashiwara and Vergne},
  \url{http://www.math.toronto.edu/~drorbn/papers/WKO/}.

\bibitem[CKS]{CKS} J.~S.~Carter, S.~Kamada, and M.~Saito,
  {\em Diagrammatic Computations for Quandles and Cocycle Knot Invariants,}
  Contemp.\ Math.\ {\bf 318} (2003) 51--74.

\bibitem[CS]{CS} J.~S.~Carter and M.~Saito,
  {\em Knotted surfaces and their diagrams,}
  Mathematical Surveys and Monographs {\bf 55}, American Mathematical
  Society, Providence 1998.

\bibitem[Da]{Da} E.~Dalvit, \url{http://science.unitn.it/~dalvit/}.

\bibitem[CR]{CattaneoRossi:WilsonSurfaces} A.~S.~Cattaneo and C.~A.~Rossi,
  {\em Wilson Surfaces and Higher Dimensional Knot Invariants,}
  Commun.\ in Math.\ Phys.\ {\bf 256-3} (2005) 513--537,
  \arXiv{math-ph/0210037}.

\bibitem[Fa]{Farber:NoncommutativeRationalFunctions} M.~Farber,
  {\em Noncommutative Rational Functions and Boundary Links,}
  Math.\ Ann.\ {\bf 293} (1992) 543--568.

\bibitem[Le]{Levine:Factorization} J.~Levine,
  {\em A Factorization of the Conway Polynomial,}
  Comment.\ Math.\ Helv.\ {\bf 74} (1999) 27--53, \arXiv{q-alg/9711007}.

\bibitem[Ro]{Ro} D.~Roseman,
  {\em Reidemeister-Type Moves for Surfaces in Four-Dimensional Space,}
  Knot Theory, Banach Center Publications {\bf 42} (1998) 347--380.

\bibitem[Wa]{Watanabe:CSI} T.~Watanabe,
  {\em Configuration Space Integrals for Long $n$-Knots, the Alexander
    Polynomial and Knot Space Cohomology,}
  Alg.\ and Geom.\ Top.\ {\bf 7} (2007) 47-–92, \arXiv{math/0609742}.

\end{thebibliography}
}}}}

\def\issues{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Issues.}\begin{itemize}
\item 
\end{itemize}
}}}}

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