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\par\noindent{\large\bf A Missing Link Between 2D and 4D \hfill Proposed Research at the University of Sydney, 2019}
\vskip 2mm

\begin{multicols}{2}

\noindent{\bf Executive Summary.} In~\cite{WKO} Zsuzsanna Dancso of the University of Sydney and I found a surprising relation between the topology of certain knotted objects in $\bbR^4$ and the ``Kashiwara-Vergne'' (KV) equations, which first arose in the context of harmonic analysis on Lie groups~\cite{KashiwaraVergne:Conjecture}. In~\cite{AlekseevKawazumiKunoNaef:GTandKV}, building on the same basic principle of ``homomorphic expansions'' yet applying that principle to completely different objects, Alekseev, Kawazumi, Kuno, and Naef found that the same KV equations are also related to the ``Goldman-Turaev Lie bialgebra'', a decidedly 2D construct. There has to be a direct way, not via the complicated Kashiwara-Vergne equations, to see that the Goldman-Turaev Lie bialgebra is related to knotted objects in $\bbR^4$, but as of yet, we don't know what that direct way is. I hope that a long visit by myself to Sydney will allow Dr.~Dancso and myself to uncover that missing link.

\noindent{\bf In More Words.} A recurring theme in mathematics is that finding a filtration on an object makes it easier to study that object. If $K$ is a vector space, for example, a descending filtration on $K$ is a sequence of vector spaces $K=K_0\supset K_1\supset K_2\supset K_3\cdots$. We think of $K_3$ as ``that part of $K$ whose study we are willing to postpone by 3 days'', and of $K_{14}$ as ``that part of $K$ whose study we are willing to delay by a whole fortnight''. With this in mind we end up spreading the study of $K$ over our whole sabbatical and perhaps more, and in day $n$ we only need to study $A_n\coloneqq K_n/K_{n+1}$. In good cases, even if $K$ is a space of intricate things, each $A_n$ is a space of simpler things, and our process of systematic procrastination pays off. One famous example is the case of homology --- in that case $K$ is additionally equipped with a differential $d$ that is appropriately compatible with the filtration, and careful procrastination -- venturing also towards $K_n/K_{n+r}$ -- leads one to the study of ``spectral sequences''.

The $K$'s we care about are a bit different --- they are spaces of linear combinations of topological things, like braids or knots or 2-knots or curves on surfaces. They carry some algebraic structures that arises from operations defined on topological things: concatenations, doubling operations, intersections, etc. They carry filtrations that resemble the filtration of a group ring by powers of its augmentation ideal. And the following question always arises:

\begin{quote} To what extent does the study of $K$ resemble the sum total of all of our daily chores, namely the study of the ``associated graded'' space $A\coloneqq\prod A_n=\prod K_n/K_{n+1}$?
\end{quote}

A more precise formulation is ``is there a homomorphic expansion, an appropriately non-degenerate map $Z\colon K\to A$ that preserves all available algebraic structure''? (Fully precise formulations are available too; for example, in~\cite{WKO}). Let us highlight two specific examples:

\noindent{\bf Knotted Objects in 4D.} Here $K$ is a certain space of knotted surfaces in $\bbR^4$ (some conditions apply, some singularities allowed). The corresponding associated graded space is a space whose basic ingredients are the free Lie algebra $\FL(S)$ and the vector space of cyclic words $\CW(S)$ on some alphabet $S$, and the construction of a homomorphic expansion turns out to be equivalent to a solution of the Kashiwara-Vergne problem~\cite{KashiwaraVergne:Conjecture}: Find $f,g\in\FL(x,y)$ so that
\begin{equation} \label{eq:KV1}
  x + y - \log e^ye^x = (1-e^{-\ad x})f + (e^{\ad y}-1)g,
\end{equation}
\begin{multline} \label{eq:KV2}
  \frac12\tr_u\left(\left(
    \frac{\ad x}{e^{\ad x}-1} + \frac{\ad x}{e^{\ad x}-1}
    - \frac{\ad \BCH(x,y)}{e^{\ad \BCH(x,y)}-1}
  \right)(u)\right) \\
  = \atdiv_xf + \atdiv_y g.
\end{multline}
The details are in~\cite{WKO}. (The KV statement is about convolutions on Lie groups and algebras, even if this is not visible above).
%(which in itself implies the equality of certain convolutions computed on a Lie group with certain convolutions computed on its Lie algebra)

\noindent{\bf The ``2D'' Goldman Turaev Lie Bialgebra.} Here $K$ is the space of formal linear combinations of homotopy classes of curves on a surface $\Sigma$, or alternatively, of conjugacy classes in $\pi_1(\Sigma)$. This space carries the ``Goldman bracket'', defined on pairs of curves by mapping such a pair to the signed sum -- over all of their intersection points -- of the ``smoothing'' corresponding to such an intersection: $\doublepoint\mapsto\pm\smoothing$. A similar definition using self-intersections defines the ``Turaev co-bracket'' on $K$ (some intricacies suppressed). The corresponding associated graded space is now related to the free associative algebra $\FA(S)$ and the space of cyclic words $\CW(S)$, and by what may appear to be a miracle, the problem of finding a homomorphic expansion is equivalent to the same Kashiwara-Vergne problem, \eqref{eq:KV1} and \eqref{eq:KV2}. The details are in~\cite{AlekseevKawazumiKunoNaef:GTandKV}.

\parpic[r]{$\xymatrix{
2D \ar@{<.>}[rr]^{\text{Sydney!}} \ar@{<->}[dr]_{\text{\cite{AlekseevKawazumiKunoNaef:GTandKV}}} & &
4D \ar@{<->}[dl]^{\text{\cite{WKO}}} \\
& KV &
}$}
\noindent{\bf What We Plan.} We don't believe in miracles, and so we believe that there must be a direct explanation for the appearance of the Kashiwara-Vergne equations in both 2D and 4D topology. There must be some direct link completing the triangle on the right. Zsuzsanna Dancso and I hope to find it during my visit to Sydney in September-October 2019.

{\renewcommand{\section}[2]{}%
\begin{thebibliography}{}

\bibitem[AKKN]{AlekseevKawazumiKunoNaef:GTandKV} A.~Alekseev, N.~Kawazumi, Y.~Kuno, and F.~Naef,
  {The Goldman-Turaev Lie bialgebra in Genus Zero and the Kashiwara-Vergne Problem,}
  Adv.\ in Math.\ {\bf 326} (2018) 1--53, \arXiv{1703.05813}.

\bibitem[KV]{KashiwaraVergne:Conjecture} M.~Kashiwara and M.~Vergne,
  \href{http://www.springerlink.com/content/v73014gx14084624/}{{\em The
    Campbell-Hausdorff Formula and Invariant Hyperfunctions,}}
  Invent.\ Math.\ {\bf 47} (1978) 249--272.
  
\bibitem[WKO]{WKO} D.~Bar-Natan and Z.~Dancso,
  \href
    {http://drorbn.net/AcademicPensieve/Projects/WKO1}
    {{\em Finite Type Invariants of W-Knotted Objects I: W-Knots and the
      Alexander Polynomial,}}
  Alg.\ and Geom.\ Top.\ {\bf 16-2} (2016) 1063--1133,
  \arXiv{1405.1956}.

%\bibitem[BND2]{WKO2}
  D.~Bar-Natan and Z.~Dancso,
  \href
    {http://drorbn.net/AcademicPensieve/Projects/WKO2}
    {{\em Finite Type Invariants of W-Knotted Objects II: Tangles and
      the Kashiwara-Vergne Problem,}}
  Math.\ Ann.\ {\bf 367} (2017) 1517--1586,
  \arXiv{1405.1955}.

%\bibitem[BND3]{WKO3}
  D.~Bar-Natan and Z.~Dancso,
  {\em Finite Type Invariants of W-Knotted Objects III: Double Tree
    Construction,}
  in preparation, \web{wko3}.

%\bibitem[BN]{WKO4}
  D.~Bar-Natan,
  {\em Finite Type Invariants of W-Knotted Objects IV: Some Computations,}
  in preparation, \web{wko4}, \arXiv{1511.05624}.

\end{thebibliography}}

\end{multicols}

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