\documentclass[10pt,notitlepage]{article} \usepackage[english,greek]{babel} \usepackage{amsmath,graphicx,amssymb,datetime,amscd,dbnsymb} \usepackage[usenames,dvipsnames]{xcolor} \usepackage{txfonts} % for the likes of \coloneqq. \usepackage[all]{xy} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref} \hypersetup{colorlinks, linkcolor={blue!50!black}, citecolor={blue!50!black}, urlcolor=blue } % Cyrillic letters following http://tex.stackexchange.com/questions/14633/what-packages-will-let-me-use-cyrillic-characters-in-math-mode: %\usepackage[T2A,T1]{fontenc} %\newcommand{\Zhe}{{\mbox{\usefont{T2A}{\rmdefault}{m}{n}\CYRZH}}} \usepackage[T2A,T1]{fontenc} \makeatletter \def\easycyrsymbol#1{\mathord{\mathchoice {\mbox{\fontsize\tf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}} {\mbox{\fontsize\tf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}} {\mbox{\fontsize\sf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}} {\mbox{\fontsize\ssf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}} }} \makeatother \newcommand{\Zhe}{{\easycyrsymbol{\CYRZH}}} \newcommand{\zhe}{{\easycyrsymbol{\cyrzh}}} \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\j{} \def\navigator{{Dror Bar-Natan: Talks: Louvain-1506:}} \def\w#1{{\href{http://www.math.toronto.edu/drorbn/Talks/Louvain-1506/#1}{$\omega$/#1}}} \def\webdef{{$\omega:=$\url{http://www.math.toronto.edu/~drorbn/Talks/Louvain-1506}}} \def\webnote{{Handout, video, and links at \w{}}} \def\blue{\color{blue}} \def\dgreen{\color{ForestGreen}} \def\red{\color{red}} \def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}} \def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:#1}}} \newcommand{\ad}{\operatorname{ad}} \def\CW{\text{\it CW}} \def\FA{\text{\it FA}} \def\FL{\text{\it FL}} \newcommand{\Fun}{\operatorname{Fun}} \def\gr{{\operatorname{gr}}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calK{{\mathcal K}} \def\calO{{\mathcal O}} \def\calS{{\mathcal S}} \def\calU{{\mathcal U}} \def\frakg{{\mathfrak g}} \def\vK{{\mathit v\mathcal K}} \def\wK{{\mathit w\mathcal K}} \def\aft{{\overrightarrow{4T}}} \def\aAS{{\overrightarrow{AS}}} \def\aSTU{{\overrightarrow{STU}}} \def\aIHX{{\overrightarrow{IHX}}} %%% \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red Abstract.} We will repeat the 3D story of the previous 3 talks one dimension up, in 4D. Surprisingly, there's more room in 4D, and things get easier, at least when we restrict our attention to "w-knots", or to "simply-knotted 2-knots". But even then there are intricacies, and we try to go beyond simply-knotted, we are completely confused. }}}} \def\Recall{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red Recall.} \[ \begin{CD} \underset{\text{\red Topology}}{\calK} @>\text{$Z$: \color{red}high algebra}>\parbox{1.2in}{\scriptsize solving finitely many equations in finitely many unknowns }> \underset{\text{\red Combinatorics}}{\calA\coloneqq\gr\calK} @>\parbox{0.8in}{\centering\scriptsize given a ``Lie'' algebra $\frakg$ }>\parbox{0.82in}{\scriptsize {\color{red}low algebra:} pictures represent formulas }> \text{``$\calU(\frakg)$''} \end{CD} \] }}}} \def\BigOpen{{\raisebox{2.5mm}{\parbox[t]{3.96in}{ {\red A Big Open Problem.} $\delta$ maps w-knots onto simple 2-knots. To what extent is it a bijection? What other relations are required? In other words, {\bf find a simple description of simple 2-knots}. Kawauchi~\cite{Kawauchi:Ribbon} may already know the answer. }}}} \def\BracketRise{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}The Bracket-Rise Theorem.} $\calA^w$ is isomorphic to }}}} \def\BRC{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}Corollaries.} (1) Only wheels and isolated arrows persist: \[ \calA^w(\uparrow_n)\cong\calU(\FL(n)^n_{tb}\ltimes\CW(n)) \quad\text{and}\quad \zeta\coloneqq\log Z\in\FL(n)^n\times\CW(n) \] has completely explicit formulas using natural $\FL/\CW$ operations \cite{KBH}. (2) Related to f.d.\ Lie algebras! }}}} \def\LowAlgebra{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}Low Algebra.} With $(x_i)$ and $(\varphi^j)$ dual bases of $\frakg$ and $\frakg^\ast$ and with $[x_i,x_j]=\sum b_{ij}^kx_k$, we have $\calA^w\to\calU$ via }}}} \def\P{{$\displaystyle \sum_{i,j,k,l,m,n=1}^{\dim\frakg} \hspace{-4mm} b_{ij}^kb_{kl}^m \varphi^i\varphi^jx_nx_m\varphi^l\in\calU(I\frakg:=\frakg^\ast\rtimes\frakg) $}} \def\TooEasy{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\color{red}Too easy so far!} Yet once you add ``foam vertices'', it gets related to the Kashiwara-Vergne problem \cite{KashiwaraVergne:Conjecture} as told by Alekseev-Torossian \cite{AT}: }}}} \def\wJacobi{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}w-Jacobi diagrams and $\calA$.} $\calA^w(Y\uparrow)\cong\calA^w(\uparrow\uparrow\uparrow)$ is }}}} \def\degdef{{$\scriptstyle\deg=\frac12\#\{\text{vertices}\}=6$}} \def\samerels{{same relations, plus}} \def\wJacobiActions{{\raisebox{0mm}{\parbox[t]{3.96in}{ $\Delta$ acts by double and sum, $S$ by reverse and negate. }}}} \def\sKnotTheoreticStatement{{\raisebox{3mm}{\parbox[t]{2.7in}{ \parshape 3 0in 2.65in 0in 2.6in 0in 2.5in {\color{red}Knot-Theoretic statement (simplified).} There exists a homomorphic expansion $Z$ for trivalent w-tangles. In particular, $Z$ should respect $R4$. }}}} \def\sDiagrammaticStatement{{\raisebox{3mm}{\parbox[t]{1.5in}{ {\color{red}Diagrammatic statement (simplified).} Let $R=\exp\rightarrowdiagram\in\calA^w(\uparrow\uparrow)$. There exist $V\in\calA^w(\uparrow\uparrow)$ so that: }}}} \def\sAlgebraicStatement{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\color{red}Algebraic statement (simplified).} With $r\in\frakg^\ast\otimes\frakg$ the identity element and with $R=e^r\in\hat\calU(I\frakg)\otimes\hat\calU(\frakg)$ there exist $V\in\hat\calU(I\frakg)^{\otimes 2}$ so that $V(\Delta\otimes 1)(R) = R^{13}R^{23}V$ in $\hat\calU(I\frakg)^{\otimes 2}\otimes\hat\calU(\frakg)$ }}}} \def\sUnitaryStatement{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\color{red}Unitary statement (simplified).} There exists a {\j unitary} tangential differential operator $V$ defined on $\Fun(\frakg_x\times\frakg_y)$ so that $V\widehat{e^{x+y}} = \widehat{e^x}\widehat{e^y}V$ (allowing $\hat\calU(\frakg)$-valued functions) }}}} \def\sGroupAlgebraStatement{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}Group-Algebra statement (simplified).} For every $\phi,\psi\in\Fun(\frakg)^G$ (with small support), the following holds in $\hat\calU(\frakg)$: \[ \iint\limits_{\frakg\times\frakg}\phi(x)\psi(y)e^{x+y} = \iint\limits_{\frakg\times\frakg}\phi(x)\psi(y)e^xe^y. \] \vskip -4mm \rightline{\scriptsize(shhh, this is Duflo)} }}}} \def\sUnitaryToGroupAlgebra{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}Unitary $\Longrightarrow$ Group-Algebra.} $\iint e^{x+y}\phi(x)\psi(y) = \left\langle 1, e^{x+y}\phi(x)\psi(y) \right\rangle = \left\langle V1, Ve^{x+y}\phi(x)\psi(y) \right\rangle = \left\langle 1, e^xe^yV\phi(x)\psi(y)\right\rangle = \left\langle 1, e^xe^y\phi(x)\psi(y) \right\rangle = \iint e^xe^y\phi(x)\psi(y). $ }}}} \def\ConvolutionsAndGroupAlgebras{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}Convolutions and Group Algebras} (ignoring all Jacobians). If $G$ is finite, $A$ is an algebra, $\tau:G\to A$ is multiplicative then $(\Fun(G),\star)\to(A,\cdot)$ via $L:f\mapsto\sum f(a)\tau(a)$. For Lie $(G,\frakg)$, \[ \begin{array}{ccc} \xymatrix@L=2pt@M=1mm@C=15mm{ (\frakg,+)\ni x \ar[r]^{\tau_0=\exp_\calS} \ar[rd]^{\exp_\calU} \ar[d]^{\exp_G} & e^x\in\hat\calS(\frakg) \ar[d]^\chi \\ (G,\cdot)\ni e^x \ar[r]^{\tau_1} & e^x\in\hat\calU(\frakg) } & \raisebox{-7.5mm}{\text{so}} & \xymatrix@L=2pt@M=1mm{ \Fun(\frakg) \ar[r]^{L_0} \ar[d]^{\Phi^{-1}} & \hat\calS(\frakg) \ar[d]^\chi \\ \Fun(G) \ar[r]^{L_1} & \hat\calU(\frakg) } \end{array} \] with $L_0\psi=\int\psi(x)e^xdx\in\hat\calS(\frakg)$ and $L_1\Phi^{-1}\psi=\int\psi(x)e^x\in\hat\calU(\frakg)$. Given $\psi_i\in\Fun(\frakg)$ compare $\Phi^{-1}(\psi_1)\star\Phi^{-1}(\psi_2)$ and $\Phi^{-1}(\psi_1\star\psi_2)$ in $\hat\calU(\frakg)$: \[ \star\text{ in }G:\ \iint\psi_1(x)\psi_2(y)e^xe^y \qquad \star\text{ in }\frakg:\ \iint\psi_1(x)\psi_2(y)e^{x+y} \] }}}} \def\sUnitaryAndAlgebraic{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}Differential Ops.} We can also interpret $\hat\calU(I\frakg)$ as tangential differential operators on $\Fun(\frakg)$: $\varphi\in\frakg^\ast$ becomes a multiplication operator, and $x\in\frakg$ becomes a tangential derivation, in the direction of the action of $\ad x$: $(x\varphi)(y):=\varphi([x,y])$. }}}} \def\uw{{\raisebox{3mm}{\parbox[t]{2.25in}{ {\color{red}$u\leftrightarrow w$} The diagram on the right explains the relationship between associators and solutions of the Kashiwara-Vergne problem. }}}} \def\uwd{$\displaystyle\xymatrix@L=2pt@M=1mm@C=14mm{ {\mathit K\!T\!G} \ar[r]^a \ar[d]^{Z^u} & {\mathit w\!T\!F} \ar[d]^{Z^w} \\ \calA^u \ar[r]^\alpha & \calA^w }$} \def\sConvolutionsStatement{{\raisebox{3mm}{\parbox[t]{3.96in}{ {\color{red}Convolutions statement (Kashiwara-Vergne, simplified).} Convolutions of invariant functions on a Lie group agree with convolutions of invariant functions on its Lie algebra. More accurately, let $G$ be a finite dimensional Lie group and let $\frakg$ be its Lie algebra, and let $\Phi:\Fun(G)\to\Fun(\frakg)$ be given by $\Phi(f)(x):=f(\exp x)$. Then if $f,g\in\Fun(G)$ are Ad-invariant and supported near the identity, then $\Phi(f)\star\Phi(g) = \Phi(f\star g)$. }}}} \def\BigQ{{\raisebox{3.5mm}{\parbox[t]{3.96in}{ {\red Question.} Does it all extend to arbitrary 2-knots (not necessarily ``simple'')? To arbitrary codimension-2 knots? }}}} \def\BFBox{{\raisebox{2mm}{\parbox[t]{3.4in}{ {\red BF Following~\cite{CattaneoRossi:WilsonSurfaces}.} $A\in\Omega^1(M=\bbR^4,\frakg)$, $B\in\Omega^2(M,\frakg^*)$, \[ S(A,B)\coloneqq\int_M\langle B,F_A\rangle. \] With $\kappa\colon(S=\bbR^2)\to M$, $\beta\in\Omega^0(S,\frakg)$, $\alpha\in\Omega^1(S,\frakg^*)$, set \[ \calO(A,B,\kappa)\coloneqq\int\calD\beta\calD\alpha\exp\left( \frac{i}{\hbar} \int_S\left\langle\beta,d_{\kappa^*A}\alpha+\kappa^*B\right\rangle \right). \] }}}} \def\FeynmanRules{{\raisebox{2mm}{\parbox[t]{3.96in}{ \parshape 4 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.96in {\red The BF Feynman Rules.} For an edge $e$, let $\Phi_e$ be its direction, in $S^3$ or $S^1$. Let $\omega_3$ and $\omega_1$ be volume forms on $S^3$ and $S_1$. Then \[ \def\neg{{\hspace{-1mm}}} Z_{BF}=\neg\sum_{\text{diagrams}\atop D}\neg \frac{[D]}{|\text{Aut}(D)|} \underbrace{\int_{\bbR^2}\neg\!\!\cdots\neg\int_{\bbR^2}}_{S\text{-vertices}} \underbrace{\int_{\bbR^4}\neg\!\!\cdots\neg\int_{\bbR^4}}_{M\text{-vertices}} \prod_{\text{red}\atop e\in D}\Phi_e^\ast\omega_3 \prod_{\text{black}\atop e\in D}\Phi_e^\ast\omega_1 \] (modulo some IHX-like relations).\hfill See also~\cite{Watanabe:CSI} }}}} \def\BFIssues{{\raisebox{3.5mm}{\parbox[t]{3.96in}{ {\red Issues.} $\bullet$ Signs don't quite work out, and BF seems to reproduce only ``half'' of the wheels invariant on simple 2-knots. \newline$\bullet$ There are many more configuration space integrals than BF Feynman diagrams and than just trees and wheels. \newline$\bullet$ I don't know how to define / analyze ``finite type'' for general 2-knots. \newline$\bullet$ I don't know how to reduce $Z_{BF}$ to combinatorics / algebra. }}}} \def\Riddles{{\raisebox{0mm}{\parbox[t]{3.96in}{ {\bf Riddles}, in case you are bored. \begin{myitemize} \item Can you find uncountably many distinct subsets $\{A_\alpha\}$ of $\bbZ$ such that whenever $\alpha\neq\beta$ either $A_\alpha\subset A_\beta$ or $A_\beta\subset A_\alpha$? \item Can you find uncountably many distinct subsets $\{B_\alpha\}$ of $\bbZ$ such that whenever $\alpha\neq\beta$ the intersection $B_\alpha\cap B_\beta$ is finite? \end{myitemize} }}}} \def\refs{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red References.} \small \par\vspace{-2mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \bibitem[AT]{AT} A.~Alekseev and C.~Torossian, {\em The Kashiwara-Vergne conjecture and Drinfeld's associators,} Annals of Mathematics {\bf 175} (2012) 415--463, \arXiv{0802.4300}. \bibitem[BN]{KBH} D.~Bar-Natan, {\em Balloons and Hoops and their Universal Finite Type Invariant, BF Theory, and an Ultimate Alexander Invariant,} \w{KBH}, \arXiv{1308.1721}. \bibitem[BND1]{WKO1} D.~Bar-Natan and Z.~Dancso, {\em Finite Type Invariants of W-Knotted Objects I: W-Knots and the Alexander Polynomial,} \w{WKO1}, \arXiv{1405.1956}. \bibitem[BND2]{WKO2} D.~Bar-Natan and Z.~Dancso, {\em Finite Type Invariants of W-Knotted Objects II: Tangles and the Kashiwara-Vergne Problem,} \w{WKO2}, \arXiv{1405.1955}. \bibitem[CS]{CS} J.~S.~Carter and M.~Saito, {\em Knotted surfaces and their diagrams,} Math.\ Surv.\ and Mono.\ {\bf 55}, Amer.\ Math.\ Soc., Providence 1998. \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[KV]{KashiwaraVergne:Conjecture} M.~Kashiwara and M.~Vergne, {\em The Campbell-Hausdorff Formula and Invariant Hyperfunctions,} Invent.\ Math.\ {\bf 47} (1978) 249--272. \bibitem[Ka]{Kawauchi:Ribbon} A.~Kawauchi, {\em A Chord Diagram of a Ribbon Surface-Link,} \url{http://www.sci.osaka-cu.ac.jp/~kawauchi/}. \bibitem[Wa]{Watanabe:CSI} T.~Watanabe, {\em Configuration Space Integrals for Long $n$-Knots, the Alexander Polynomial and Knot Space Cohomology,} Alg.\ and Geom.\ Top.\ {\bf 7} (2007) 47--92, \arXiv{math/0609742}. \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{Louvain5-1.pstex_t} \vfill\null \newpage \null\vfill \input{Louvain5-2.pstex_t} \vfill\null \end{center} \end{document} \endinput