\documentclass[10pt,notitlepage]{article}
\usepackage[english,greek]{babel}
\usepackage{amsmath,graphicx,amssymb,datetime,picins}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{txfonts}	% for the likes of \coloneqq.
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}
\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}

\newenvironment{myitemize}{
  \begin{list}{$\bullet$}{
    \setlength{\leftmargin}{10pt}
    \setlength{\labelwidth}{10pt}
    \setlength{\labelsep}{4pt}
    \setlength{\parsep}{0pt}
    \setlength{\topsep}{0pt}
    \setlength{\itemsep}{0pt}
  }
}{
  \end{list}
}

\def\navigator{{Video, links, and more @ Dror Bar-Natan: Talks: Georgetown-1503:}}
\def\webdef{{{\greektext web}$\coloneqq$\url{http://www.math.toronto.edu/~drorbn/Talks/Georgetown-1503}}}
\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Georgetown-1503/#1}{{\greektext web}/#1}}}

\def\blue{\color{blue}}
\def\red{\color{red}}

\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:#1}}}

\newcommand{\gr}{\operatorname{gr}}

\def\bbC{{\mathbb C}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}

\def\PB{{\mathit P\!B}}
\def\PvB{{\mathit P\!v\!B}}
\def\PvT{{\mathit P\!v\!T}}
\def\PwB{{\mathit P\!w\!B}}
\def\PwT{{\mathit P\!w\!T}}
\def\PwTT{{\mathit P\!w\!T\!T}}

\def\sumint{\setbox0=\hbox{$\displaystyle\sum$}\mathop{\rlap{\copy0}%
  \kern0pt \hbox to \wd 0{\hss$\displaystyle\int$\hss}}}

%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{
\parshape 9 0in 2.96in 0in 2.96in 0in 2.96in 0in 2.96in 0in 2.96in 0in 2.96in 0in 2.96in 0in 2.96in 0in 3.96in
{\red Abstract.} It is insufficiently well known that the good old Taylor
expansion has a completely algebraic characterization, which generalizes to
arbitrary groups (and even far beyond). Thus one may ask: Does the braid
group have a Taylor expansion? (Yes, using iterated integrals and/or
associators). Do braids on a torus (``elliptic braids'') have Taylor
expansions? (Yes, using more sophisticated iterated integrals /
associators). Do virtual braids have Taylor expansions? (No, yet for nearby
objects the deep answer is Probably Yes). Do groups of flying rings (braid
groups one dimension up) have Taylor expansions? (Yes, easily, yet the link
to TQFT is yet to be fully explored). 
}}}}

\def\Disclaimer{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Disclaimer.} I'm asked to talk in a meeting on ``iterated
integrals'', and that's my best. Many of you may think it all trivial.
Sorry.
}}}}

\def\Expansions{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Expansions for Groups.} Let $G$ be a group, $\calK=\bbQ G=\{\sum
a_ig_i\colon
a_i\in\bbQ,\,g_i\in G\}$ its group-ring, $\calI=\{\sum a_ig_i\colon\sum
a_i=0\}$ its augmentation ideal. Let
\newline\null$\displaystyle\quad
  \calA=\gr\calK:=\widehat\bigoplus_{m\geq 0}\,\calI^m/\calI^{m+1}.
$\newline
Note that $\calA$ inherits a product from $G$.

\parshape 6 0in 2.45in 0in 2.45in 0in 2.45in 0in 2.45in 0in 2.45in 0in 3.96in
{\red Definition.} A linear $Z\colon\calK\to\calA$
is an ``expansion'' if for any $\gamma\in\calI^m$,
$Z(\gamma)=(0,\ldots,0,\gamma/\calI^{m+1},\ast,\ldots)$, a
``multiplicative expansion'' if in addition it preserves the product,
and a ``Taylor expansion'' if it also preserves the co-product, induced
from the diagonal map $G\to G\times G$.

{\red Example.} Let $\calK=C^\infty(\bbR^n)$ and $\calI=\{f\colon
f(0)=0\}$. Then $\calI^m=\{f\colon f\text{ vanishes like }|x|^m\}$
so $\calI^m/\calI^{m+1}$ degree $m$ homogeneous polynomials and
$\calA=\{\text{power series}\}$. The Taylor series is the unique Taylor
expansion!

{\red Comment.} Unlike lower central series constructions, this generalizes
effortlessly to arbitrary algebraic structures.
}}}}

\def\FT{{\raisebox{1mm}{\parbox[t]{1.6in}{\footnotesize
{\red P.S.} $(\calK/\calI^{m+1})^\ast$ is Vassiliev / finite-type /
polynomial invariants.
}}}}

\def\Braids{{\raisebox{2mm}{\parbox[t]{3.96in}{
\parshape 8 0in 3.125in 0in 3.125in 0in 3.125in 0in 2in 0in 2in 0in 2in 0in 2in 0in 3.96in
{\red Pure Braids.} Take $G = \PB_n =
\pi_1(C_n=\bbC^n\setminus\text{diags})$. It is generated by the
love-behind-the-bars braids $\sigma_{ij}$, modulo ``Reidemeister
moves''. $\calI$ is generated by $\{\sigma_{ij}-1\}$ and
$\calA$ by $\{t_{ij}\}$, the classes of the $\sigma_{ij}-1$ in
$\calA_1=\calI/\calI^2$. Reidemeister becomes

\ $\displaystyle [t_{ij}+t_{ik},t_{jk}]=0 \ \text{and}\ [t_{ij},t_{kl}]=0$.

\vskip -2mm
\parpic[r]{$\displaystyle Z(\gamma)=
  \hspace{-10mm}\sumint_{\substack{
    m\geq 0 \\
    0<t_1<\ldots<t_m<1 \\
    1\leq i_1<j_1, i_2<j_2,\ldots,i_m<j_m\leq n
  }}\hspace{-10mm}
  \prod_{\alpha=1}^m\frac{t_{i_\alpha j_\alpha}}{2\pi i}
    d\log(z_{i_\alpha}-z_{j_\alpha}),
$}
{\red Theorem.} For $\gamma\colon[0,1]\to C_n$, with $z_i$ its $i$th
coordinate, the iterated integral formula on the right
defines a Taylor expansion for $\PB_n$.

\parshape 1 0in 2.75in
{\red Comments.} $\bullet$ I don't know a combinatorial/algebraic proof
that $\PB_n$ has a Taylor expansion.
$\bullet$ Generic ``partial expansion'' do not extend!
$\bullet$ This is the seed for the Drinfel'd theory of associators!
$\bullet$ Confession: I don't know a clean derivation of a presentation
  of $\PB_n$.
}}}}

\def\EllipticBraids{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red Elliptic Braids.} $\PB^1_n \coloneqq \pi_1(C^1_n)$ is generated
by $\sigma_{ij}$, $X_i$, $Y_j$, with $\PB_n$ relations and $(X_i,X_j) =
1 = (Y_i,Y_j)$, $(X_i,Y_j) = \sigma_{ij}^{-1}$, $(X_iX_j,\sigma_{ij})
=$ $ 1 =$ $ (Y_iY_j,\sigma_{ij})$, and $\prod X_i$ and $\prod Y_j$
are central. \cite{Bezrukavnikov:KoszulDG} implies $\calA(\PB^1_n)
= \left\langle x_i,y_j\right\rangle$ $/$ $\left( [x_i,x_j] =\right.$
$[y_i,y_j] =$ $[x_i+x_j,[x_i,y_j]] =$ ${[y_i+y_j,[x_i,y_j]]} =$ $[x_i,\sum
y_j]=$ $[y_j,\sum x_i]=$ $ 0,$ $\left.[x_i,y_j] = [x_j,y_i]\right)$, and
\cite{CalaqueEnriquezEtingof:KZB} construct a Taylor expansion using {\em
sophisticated} iterated integrals. \cite{Enriquez:EllipticAssociators}
relates this to {\em Elliptic Associators}.
}}}}

\def\VirtualBraids{{\raisebox{2mm}{\parbox[t]{3.96in}{
\parshape 1 0in 2.75in
{\red Virtual Braids.} $\PvB_n$ is given by the ``braids for dummies''
presentation:

$\displaystyle
  \left\langle \sigma_{ij} \mid
    \sigma_{ij}\sigma_{ik}\sigma_{jk} = \sigma_{jk}\sigma_{ik}\sigma_{ij},
    \ \sigma_{ij}\sigma_{kl} = \sigma_{kl}\sigma_{ij}
  \right\rangle
$

\parshape 1 0in 2.75in
(every quantum invariant extends to $\PvB_n$!). By~\cite{LeeP:Quadratic},
$\calA(\PvB_n)$ is

$\displaystyle
  \left\langle a_{ij} \mid
    [a_{ij},a_{ik}] + [a_{ij},a_{jk}] + [a_{ik},a_{jk}]
      = 0 = [a_{ij},a_{kl}]
  \right\rangle
$

\parshape 1 0in 3.25in
{\red Theorem~\cite{LeeP:Quadratic}.} While quadratic, $\PvB_n$ does not
have a Taylor expansion.

\parshape 3 0in 3.25in 0in 3.25in 0in 3.96in
{\red Comment.} By the tough theory of quantization of solutions of
the classical Young-Baxter equation \cite{EtingofKazhdan:BialgebrasI,
Enriquez:Quantization}, $\PvT_n$ does have a Taylor expansion. But
$\PvT_n$ is not a group.
}}}}

\def\FlyingRings{{\raisebox{2mm}{\parbox[t]{3.5in}{
{\red Flying Rings.} $\PwB_n =
\PvB_n/(\sigma_{ij}\sigma_{ik}=\sigma_{ik}\sigma_{ij})$ is $\pi_1($flying
rings in $\bbR^3)$. $\calA(\PwB_n) = \calA(\PvB_n)/[a_{ij},a_{ik}]=0$, and
$Z$ is as easy as it gets: $Z(\sigma_{ij})=e^{a_{ij}}$
\cite{BerceanuPapadima:BraidPermutation, WKO}. Indeed,
$Z(\sigma_{ij}\sigma_{ik}\sigma_{jk}) = e^{a_{ij}}e^{a_{ik}}e^{a_{jk}} =
e^{a_{ij}+a_{ik}}e^{a_{jk}} = e^{a_{ij}+a_{ik}+a_{jk}} =
Z(\sigma_{jk}\sigma_{ik}\sigma_{ij})$.

{\red Comments.}
$\bullet$ Extends to $\PwT$ and generalizes the Alexander polynomial, and
even to $\PwTT$ and interprets the Kashiwara-Vergne problem \cite{WKO}.
$\bullet$ I don't know an iterated-integral derivation, or
any TQFT derivation, though BF theory probably comes close
\cite{CattaneoRossi:WilsonSurfaces}.
}}}}

\def\refs{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red References.}\small\hfill Paper in Progress: \w{ExQu}
\par\vspace{-2mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}

\bibitem[BND]{WKO} D.~Bar-Natan and Z.~Dancso,
  {\em Finite Type Invariants of W-Knotted Objects I: Braids,
  Knots and the Alexander Polynomial,}
  \w{WKO1}, \arXiv{1405.1956}; and 
  {\em II: Tangles and the Kashiwara-Vergne Problem,}
  \w{WKO2}, \arXiv{1405.1955}.

\bibitem[BP]{BerceanuPapadima:BraidPermutation} B.~Berceanu and S.~Papadima,
  {\em Universal Representations of Braid and Braid-Permutation Groups,}
  J.~of Knot Theory and its Ramifications {\bf 18-7} (2009) 973--983,
  \arXiv{0708.0634}.

\bibitem[Bez]{Bezrukavnikov:KoszulDG} R.~Bezrukavnikov,
  {\em Koszul DG-Algebras Arising from Configuration Spaces,}
  Geom.\ Func.\ Anal.\ {\bf 4-2} (1994) 119--135.

\bibitem[CEE]{CalaqueEnriquezEtingof:KZB} D.~Calaque, B.~Enriquez, and
    P.~Etingof,
  {\em Universal KZB Equations I: The Elliptic Case,}
  Prog.\ in Math.\ {\bf 269} (2009) 165--266, \arXiv{math/0702670}.

\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[En1]{Enriquez:Quantization} B.~Enriquez,
  {\em A Cohomological Construction of Quantization Functors of Lie
    Bialgebras,}
  Adv.\ in Math.\ {\bf 197-2} (2005) 430–-479,
  \arXiv{math/0212325}.

\bibitem[En2]{Enriquez:EllipticAssociators} B.~Enriquez,
  {\em Elliptic Associators,}
  Selecta Mathematica {\bf 20} (2014) 491--584, \arXiv{1003.1012}.

\bibitem[EK]{EtingofKazhdan:BialgebrasI} P.~Etingof and D.~Kazhdan,
  {\em Quantization of Lie Bialgebras, I,}
  Selecta Mathematica {\bf 2} (1996) 1--41, \arXiv{q-alg/9506005}.

\bibitem[Lee]{LeeP:Quadratic} P.~Lee,
  {\em The Pure Virtual Braid Group Is Quadratic,}
  Selecta Mathematica {\bf 19-2} (2013) 461--508, \arXiv{1110.2356}.

\end{thebibliography}
}}}}

\paperwidth 8.5in
\paperheight 11in
\textwidth 8.5in
\textheight 11in
\oddsidemargin -1in
\evensidemargin \oddsidemargin
\topmargin -1in
\headheight 0in
\headsep 0in
\footskip 0in
\parindent 0in
\setlength{\topsep}{0pt}
\pagestyle{empty}

\begin{document} \latintext
\setlength{\jot}{0ex}
\setlength{\abovedisplayskip}{0.5ex}
\setlength{\belowdisplayskip}{0.5ex}
\setlength{\abovedisplayshortskip}{0ex}
\setlength{\belowdisplayshortskip}{0ex}

\begin{center}
\null\vfill \input{Taylor.pstex_t} \vfill\null
\end{center}

\end{document}

\endinput