\documentclass[10pt,notitlepage]{article}
\usepackage{amsmath,graphicx,amssymb,color,datetime,stmaryrd}
\usepackage[setpagesize=false]{hyperref}
\usepackage[all]{xy}

\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}

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

\newcommand{\ad}{\operatorname{ad}}
\newcommand{\Fun}{\operatorname{Fun}}
\newcommand{\gr}{\operatorname{gr}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\tr}{\operatorname{tr}}

\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\calA{{\mathcal A}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\frakg{{\mathfrak g}}

\def\navigator{{Dror Bar-Natan: Talks: Bern-131104:}}

\def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Bern-131104/#1}{$\omega$/#1}}}
\def\webdef{{$\omega:=$\url{http://www.math.toronto.edu/~drorbn/Talks/Bern-131104}}}
\def\webnote{{Video, handout, links at \w{}}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{4in}{
{\red Abstract.} I will describe a general machine, a close cousin of
Taylor's theorem, whose inputs are topics in topology and whose outputs are
problems in algebra. There are many inputs the machine can take, and many
outputs it produces, but I will concentrate on just one input/output pair.
When fed with a certain class of knotted 2-dimensional objects in
4-dimensional space, it outputs the Kashiwara-Vergne Problem (1978 \w{KV},
solved Alekseev-Meinrenken 2006 \w{AM}, elucidated Alekseev-Torossian
2008-2012 \w{AT}), a problem about convolutions on Lie groups and Lie
algebras.
}}}}

\def\KV{{\raisebox{2mm}{\parbox[t]{3.2in}{
{\red The Kashiwara-Vergne Conjecture.} There exist two series $F$ and
$G$ in the completed free Lie algebra $F\!L$ in generators $x$ and $y$ so that
\[ x+y-\log e^y e^x = (1-e^{-\ad x})F + (e^{\ad y}-1)G \text{ in }F\!L \]
and so that with $z=\log e^x e^y$,
\begin{multline*}
  \tr(\ad x)\partial_x F + \tr(\ad y)\partial_y G
  \quad\text{\scriptsize in cyclic words} \\
  = \frac12\tr\left(
    \frac{\ad x}{e^{\ad x}-1} + \frac{\ad y}{e^{\ad y}-1}
    - \frac{\ad z}{e^{\ad z}-1} - 1
  \right)
\end{multline*}
Implies the loosely-stated {\color{red}convolutions statement}:
Convolutions of invariant functions on a Lie group agree with convolutions
of invariant functions on its Lie algebra.
}}}}

\def\Machine{{\raisebox{2mm}{\parbox[t]{4in}{
{\red The Machine.} Let $G$ be a group, $\calK=\bbQ G=\{\sum a_ig_i\colon
a_i\in\bbQ,\,g_i\in G\}$ its group-ring, $\calI=\{\sum a_ig_i\colon\sum
a_i=0\}\subset\calK$ 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 $G$.

{\red Definition.} A linear $Z\colon\calK\to\calA$ is an ``expansion'' if
for any $\gamma\in\calI^m$,
$Z(\gamma)=(0,\ldots,0,\gamma/\calI^{m+1},\ast,\ldots)$, and a
``homomorphic expansion'' if in addition it preserves the product.

{\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 a homomorphic
expansion!
}}}}

\def\FT{{\raisebox{1mm}{\parbox[t]{1.5in}{\footnotesize
{\red P.S.} $(\calK/\calI^{m+1})^\ast$ is Vassiliev / finite-type /
polynomial invariants.
}}}}

\def\eKa{{$\calK=$}}
\def\eKb{{$=\left(\parbox{0.75in}{\scriptsize
  The set of all 2D projections of reality
}\right)$}}
\def\eKc{{\parbox{1.5in}{\scriptsize
  An expansion {\color{red}$Z$} is a choice of a {\color{red}``progressive
  scan''} algorithm.
}}}
\def\fK{{$\scriptstyle\calK/\calK_1$}}
\def\fKa{{$\scriptstyle\calK/\calK_2$}}
\def\fKb{{$\scriptstyle\calK/\calK_3$}}
\def\fKc{{$\scriptstyle\calK/\calK_4$}}
\def\gK{{$\scriptstyle\calK/\calK_1$}}
\def\gKa{{$\scriptstyle\calK_1/\calK_2$}}
\def\gKb{{$\scriptstyle\calK_2/\calK_3$}}
\def\gKc{{$\scriptstyle\calK_3/\calK_4$}}
\def\gKd{{$\scriptstyle\calK_4/\calK_5$}}
\def\gKe{{$\scriptstyle\calK_5/\calK_6$}}
\def\gKf{{$\cdots$}}
\def\hK{{$\scriptstyle\bbR^3$}}
\def\hKc{{$\scriptstyle\ker(\calK/\calK_4\to\calK/\calK_3)$}}
\def\op{{$\ \scriptstyle\oplus$}}
\def\la{{$\ \scriptstyle\leftarrow$}}

\def\FindingZ{{\raisebox{0mm}{\parbox[t]{4in}{
In the finitely presented case, finding $Z$ amounts to solving a system of
equations in a graded space.
}}}}

\def\PA{{\raisebox{0mm}{\parbox[t]{2.2in}{\small
\parshape 4 0in 0.8in 0in 0.8in 0in 0.8in 0in 2.2in
``Planar Algebra'': The objects are ``tiles'' that can be composed in
arbitrary planar ways to make bigger tiles.
}}}}

\def\GM{{\raisebox{2mm}{\parbox[t]{2.5in}{
{\red The Machine} generalizes to arbitrary algebraic structures!
}}}}

\def\Theorem{{\raisebox{2mm}{\parbox[t]{3.2in}{
{\red Theorem} (with Zsuzsanna Dancso, \w{WKO}).
There is a bijection between the set of homomorphic expansions
for $w\calK$ and the set of solutions of the Kashiwara-Vergne problem.
{\red This is the tip of a major iceberg.}
}}}}

\begin{document}
\setlength{\jot}{3pt}
\setlength{\abovedisplayskip}{0.5ex}
\setlength{\belowdisplayskip}{0.5ex}
\setlength{\abovedisplayshortskip}{0ex}
\setlength{\belowdisplayshortskip}{0ex}

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

\end{document}

\endinput