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\def\calA{{\mathcal A}}
\def\calK{{\mathcal K}}
\def\frakg{{\mathfrak g}}
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\def\web#1{{\href{http://www.math.toronto.edu/drorbn/Talks/#1}{$\omega$/#1}}}
\begin{document}
\noindent{\Large\bf Dror Bar-Natan: Statement of Recent Work (September 2012)}
\vskip 3mm
\textbf{Abstract Preliminaries.} A \emph{filtered vector space} is a vector space $V=V_0$ (usually infinite dimensional, usually one with elements whose study is hard yet desirable) along with a decreasing sequence of subspaces $V=V_0\supset V_1\supset V_2\supset\dots$. I like to think of $V$ as if it contains information that could be studied inductively, and of $V_n$ as ``that part of $V$ that is irrelevant until day $n$ of the study''. Thus the quotient $V/V_n$ is what we could or should have studied before day $n$. On day $n$ we may study $V_n$, though we still don't care about $V_{n+1}$, so the quotient $V_n/V_{n+1}$ is precisely what we need to study on that day. Thus the direct sum
$\gr V := \bigoplus_n V_n/V_{n+1}$
(also known as ``the \emph{associated graded space of $V$}'') can be viewed as ``$V$, sliced out for an inductive study''. An \emph{expansion} for $V$ is a map $Z:V\to\gr V$ satisfying a simple non-degeneracy condition. An expansion may be thought of as a machine that breaks any element of $V$ into a sequence of easier parts, with part $n$ ready for study on day $n$. Often times the spaces $V_n/V_{n+1}$ are much simpler than $V$, and so there is great interest in finding ``good'' expansions.
What's ``good''? Often times $V$ will come with various algebraic operations --- various composition laws, or various other ways to transform its elements. Under mild conditions, these operations will themselves be amenable to inductive study --- meaning that they induce similar operations on $\gr V$. A \emph{homomorphic} (``good'') expansion is an expansion $Z:V\to\gr V$ that intertwines the operations of $V$ with the operations of $\gr V$. A homomorphic expansion allows for an even better inductive probing of $V$ --- not only the elements of $V$ can be studied in the simplified inductive context, but so can the relations between the elements of $V$ and the operations applied to these elements.
Generally speaking, mere expansions are cheap and easy to get. Homomorphic expansions, on the other hand, are expensive and valuable. They don't always exist, when they exist they are often hard to construct, yet when they are available, they are often very useful.
\textbf{Knots.} The set $\calK$ of knots (``\textbf{topology}'', below), for example, can be made into a vector space by allowing formal linear combinations and can then be filtered using the ``Vassiliev filtration'' which will not be recalled here except by its essence, the formula ``$\doublepoint\to\overcrossing-\undercrossing$'' (I was amongst the earliest contributors; see \cite{Bar-Natan:OnVassiliev}). The resulting ``$\gr V$'' is the space $\calA$ of chord diagrams (``\textbf{combinatorics}'', below). Slightly generalizing to ``parenthesized tangles'', the construction of a homomorphic expansion $Z$ turns out to depend mostly on the choice of one very special element $\Phi=Z(\Associator)$ of $\calA$, which has to satisfy some complicated equations whose origin is in category theory --- mostly the ``pentagon'' and ``hexagon'' equations (``\textbf{high algebra}'', below; I had a role in that too --- see \cite{Bar-Natan:NAT, Bar-Natan:Associators}). Further, it turns out that the space $\calA$ can be re-interpreted as a space of formulas that make sense in any appropriate Lie algebra (``\textbf{low algebra}'', below, \cite{Bar-Natan:OnVassiliev}), and hence much that is done with and about $Z$ and $\Phi$ has a Lie-theoretic and representation-theoretic meaning. (A lovely example is the explanation of the Lie-theoretic Duflo-isomorphism as the knot theoretic ``$1+1=2$'' \cite{Bar-NatanLeThurston:TwoApplications}).
\textbf{u, v, and w.} Renaming ``knots'' to be ``u-knots'' (``u'' for usual), I found that the above pattern of \textbf{topology} leading to \textbf{combinatorics} by means of a natural filtration, leading to \textbf{low algebra} by interpreting the combinatorics as the combinatorics of formulas, and to \textbf{high algebra} by the study of expansions, persists for several other classes of knots. A quick summary is in the table below:
\begin{center} {
\def\uT{\parbox[t]{1.6in}{\small
Ordinary (\underline{u}sual) knotted objects in 3D --- braids,
knots, links, tangles, knotted graphs, etc.
}}
\def\vT{\parbox[t]{1.8in}{\small
\underline{V}irtual knotted objects --- ``algebraic'' knotted objects,
or ``not specifically embedded'' knotted objects; knots drawn on a
surface, modulo stabilization.
}}
\def\wT{\parbox[t]{1.875in}{\small
Ribbon knotted objects in 4D; ``flying rings''. Like v, but also with
``overcrossings commute''.
}}
\def\uC{\parbox[t]{1.6in}{\small
Chord diagrams and Jacobi diagrams, modulo $4T$, $STU$, $IHX$, etc.
}}
\def\vC{\parbox[t]{1.8in}{\small
Arrow diagrams and v-Jacobi diagrams, modulo $6T$ and various
``directed'' $STU$s and $IHX$s, etc.
}}
\def\wC{\parbox[t]{1.875in}{\small
Like v, but also with ``tails commute''. Only ``two in one out''
internal vertices.
}}
\def\uL{\parbox[t]{1.6in}{\small
Finite dimensional metrized Lie algebras, representations, and
associated spaces.
}}
\def\vL{\parbox[t]{1.8in}{\small
Finite dimensional Lie bi-algebras, representations, and associated
spaces.
}}
\def\wL{\parbox[t]{1.875in}{\small
Finite dimensional co-commutative Lie bi-algebras (i.e.,
$\frakg\ltimes\frakg^\ast$), representations, and associated
spaces.
}}
\def\uH{\parbox[t]{1.6in}{\small
The Drinfel'd theory of associators.
}}
\def\vH{\parbox[t]{1.8in}{\small
Likely, quantum groups and the Etingof-Kazhdan theory of quantization
of Lie bi-algebras.
}}
\def\wH{\parbox[t]{1.875in}{\small
The Kashiwara-Vergne-Alekseev-Torossian theory of convolutions on Lie
groups and Lie algebras.
}}
\begin{tabular}{c|c|c|c}
& u-Knots & v-Knots & w-Knots \\
\hline
\parbox[t]{0.55in}{Topology} & \uT & \vT & \wT \\
\hline
\parbox[t]{0.55in}{Combi-natorics} & \uC & \vC & \wC \\
\hline
\parbox[t]{0.55in}{Low Algebra} & \uL & \vL & \wL \\
\hline
\parbox[t]{0.55in}{High Algebra} & \uH & \vH & \wH \\
\end{tabular}
} \end{center}
A more complete version of the above table would contain a few further rows, for quantum field theory, for configuration space integrals, for graph homology, and perhaps more. It may also contain some further columns (deformation quantization of Poisson structures ``p'' has entries in all rows except the topology row. I much want to know what ``p-topology'' would be). Also, the table fails to indicate that there are maps between the entities in the topology row --- specifically, there are maps $u\to v\to w$. These maps have analogs or implications in all other rows, serving to explain the otherwise mysterious connections that exists between, say, Drinfel'd associators and solutions of the Kashiwara-Vergne problem \cite{AlekseevTorossian:KashiwaraVergne, Bar-NatanDancso:WKO}.
Thus my most significant scientific work over the last 5 years had been the assembly of the above table. Much of it is still unwritten. The written parts include:
\begin{itemize}
\item My paper and series of videos \cite{Bar-NatanDancso:WKO} (with Dancso) contain the complete ``w-story'' from the topology of 2-dimensional knots in 4-dimensional space to the high algebra of Kashiwara-Vergne and Alekseev-Torossian and its relationship with Drinfel'd associators.
\item \cite{Bar-NatanDancso:KTGs} is about the u-column, describing the relationship between knotted trivalent graphs and associators.
\item \cite{Bar-NatanDancso:Furusho} lives at u cross high algebra, explaining (following Furusho) why under certain conditions, the pentagon equation implies the hexagon equation.
\item \cite{Bar-NatanHalachevaLeungRoukema:v-Dims} contains strong computational evidence that some 18 variants of the combinatorics entry of the v column are ``correct''.
\end{itemize}
Many of the not-yet-written parts are available online as videotaped lectures accompanied by detailed summary handouts. With $\omega:=$\url{http://www.math.toronto.edu/~drorbn/Talks/}, these include \web{Aarhus-0706} on potential applications to topology, \web{Trieste-0905}, \web{Goettingen-1004}, and \web{Caen-1206} on the whole table, \web{MSRI-0808}, \web{Bonn-0908}, and \web{Montpellier-1006} on the w column, \web{Toronto-11011} on the relationship between u and the Grothendieck-Teichmuller group, \web{Tennessee-1103} on the ``configuration spaces'' row that was omitted above, \web{SwissKnots-1105} on my hopes for the v column, \web{Sandbjerg-0810} on a ``penultimate'' Alexander invariant, \web{Chicago-1009} on the relationship between w and especially the ``$ax+b$'' Lie algebra and the Alexander polynomial, \web{Regina-1206} on an Alexander polynomial offshoot of the w story, which appears to be an ``ultimate'' Alexander invariant, and \web{Hamburg-1208} on a non-commutative generalization of the Alexander polynomial and its relationship with the BF quantum field theory.
\begin{thebibliography}{BLT}
\bibitem[AT]{AlekseevTorossian:KashiwaraVergne} A.~Alekseev and
C.~Torossian,
{\em The Kashiwara-Vergne conjecture and Drinfeld's associators,}
\arXiv{0802.4300}.
\bibitem[BN1]{Bar-Natan:OnVassiliev} D.~Bar-Natan,
\href{http://www.math.toronto.edu/~drorbn/LOP.html#OnVassiliev}{{\em
On the Vassiliev knot invariants,}}
Topology {\bf 34} (1995) 423--472.
\bibitem[BN2]{Bar-Natan:NAT} D.~Bar-Natan,
\href{http://www.math.toronto.edu/~drorbn/LOP.html#NAT}{{\em
Non-associative tangles,}}
in {\em Geometric topology} (proceedings of the Georgia international
topology conference), (W.~H.~Kazez, ed.), 139--183, Amer.{} Math.{}
Soc.{} and International Press, Providence, 1997.
\bibitem[BN3]{Bar-Natan:Associators} D.~Bar-Natan,
\href{http://www.math.toronto.edu/~drorbn/LOP.html#Associators}{{\em On
Associators and the Grothendieck-Teichmuller Group I,}}
Selecta Mathematica, New Series {\bf 4} (1998) 183--212.
\bibitem[BD1]{Bar-NatanDancso:Furusho} D.~Bar-Natan and Z.~Dancso,
{\em Pentagon and Hexagon Equations Following Furusho,}
Proceedings of the American Mathematical Society {\bf 140-4} (2012) 1243--1250, \arXiv{1010.0754}.
\bibitem[BD2]{Bar-NatanDancso:KTGs} D.~Bar-Natan and Z.~Dancso,
\href{http://www.math.toronto.edu/~drorbn/papers/ktgs/}{{\em Homomorphic Expansions for Knotted Trivalent Graphs,}}
Journal of Knot Theory and its Ramifications, to appear, \arXiv{1103.1896}.
\bibitem[BD3]{Bar-NatanDancso:WKO} D.~Bar-Natan and Z.~Dancso,
{\em Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne,}
in preparation, see \url{http://drorbn.net/index.php?title=WKO}.
\bibitem[BHLR]{Bar-NatanHalachevaLeungRoukema:v-Dims} D.~Bar-Natan,
I.~Halacheva, L.~Leung, and F.~Roukema,
\href{http://www.math.toronto.edu/~drorbn/papers/v-Dims}{{\em Some
Dimensions of Spaces of Finite Type Invariants of Virtual Knots,}}
Experimental Mathematics {\bf 20-3} (2011) 282--287,\arXiv{0909-5169}.
\bibitem[BLT]{Bar-NatanLeThurston:TwoApplications} D.~Bar-Natan,
T.~Q.~T.~Le, and D.~P.~Thurston,
{\em Two applications of elementary knot theory to Lie algebras and
Vassiliev invariants,}
Geometry and Topology {\bf 7-1} (2003) 1--31, \arXiv{math.QA/0204311}.
\end{thebibliography}
\end{document}