\documentclass[10pt,notitlepage]{article} \usepackage{amsmath,graphicx,amssymb,datetime,stmaryrd} \usepackage[setpagesize=false]{hyperref} \usepackage[all]{xy} \usepackage[usenames,dvipsnames]{xcolor} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks, linkcolor={blue!50!black}, citecolor={blue!50!black}, urlcolor=blue } \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: McMaster-180921:}} \def\w#1{{\href{http://drorbn.net//McMaster-180921/#1}{$\omega$/#1}}} \def\webdef{{$\omega:=$\url{http://drorbn.net/McMaster-180921}}} \def\webnote{{Handout, video, and links at \w{}}} \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red Abstract.} I will describe the general ``expansions'' machine whose inputs are topics in topology (and more) 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 \[ x+y-\log e^y e^x = (1-e^{-\ad x})F + (e^{\ad y}-1)G \quad\text{\scriptsize in $F\!L$} \] 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]{3.96in}{ {\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 $\calI^m/\calI^{m+1}$ is 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{{$=\left(\parbox{0.85in}{\scriptsize The set of all 2D projections of reality\hfill$(=\bbQ^3\bbR^2)$ }\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\bbQ^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]{3.96in}{ 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]{1.46in}{ {\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!} % associators (!) and elliptic associators (?), Grothendieck-Teichm\"uller % groups (!), quantization of Lie bi-algebras (!?), deformation-quantization % of Poisson structures (??),$\ldots$ }}}} \def\uvw{\raisebox{1mm}{\scalebox{0.5}{$\xymatrix@C=12.5mm@R=6mm{ \text{\red Why} & u\calK \ar[ld] \ar[dd] \ar[rr]^{\text{$Z$: the Kontsevich}}_{\text{integral}} & & \calA^u \ar[dd] \ar[rd] & & \operatorname{ker}_\frakg\text{\red\ and more!} \ar[ld] \\ v\calK \ar[rd] \ar'[r]'[rrr]^{\text{$Z$: hard work}}_{\text{Etingof-Kazhdan}} [rrrr] & & & & \calA^v \ar[ld] \ar[rd] & \text{(given $\frakg$)} \\ \text{\red care?} & w\calK \ar[rr]^{\text{$Z$: today's}}_{\text{work}} & & \calA^w & & \text{Around }\operatorname{Rep}(\frakg) }$}}} \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