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\author[Emily Peters]{Emily Peters \\ \url{http://math.mit.edu/\~ eep} \\ joint work with Jones, Morrison, Penneys, Snyder, Tener}
\title{Classifying subfactors up to index 5}
\date{ECOAS, Dartmouth, October 24 2010}

\usepackage{multimedia}


\begin{document}

\frame{\titlepage}

\beamertemplatetransparentcovered 

\mode<beamer>{\setbeamercolor{block title}{bg=green!40!black}}

\beamersetuncovermixins 
{\opaqueness<1->{60}} 
{} 

\section{The classification up to index 4}

\begin{frame}
\frametitle{Index less than 4}

\begin{thm}[Jones, Ocneanu, Kawahigashi, Izumi, Bion-Nadal]
The principal graph of a subfactor of index less than 4 is one of

\begin{tabular}{ll}
$A_n = \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (0,0) circle (.5mm) node [above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\node at (3,0) {$\cdots$};
	\filldraw (4,0) circle (.5mm);
	
	\draw (0,0)--(2.5,0);
	\draw (3.5,0)--(4,0);

	\draw[decorate,decoration={brace, mirror}] (0,-.2)--(4,-.2) node[midway, below] {$n$ vertices};
\end{tikzpicture}
$, $n\geq 2$ & index $4 \cos^2(\frac{\pi}{n+1})$ 
\\

$D_{2n} = \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (0,0) circle (.5mm) node [above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\node at (2,0) {$\cdots$};
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,.5) circle (.5mm);
	\filldraw (4,-.5) circle (.5mm);

	\draw (0,0)--(1.5,0);
	\draw (2.5,0)--(3,0)--(4,.5);
	\draw (3,0)--(4,-.5);

	\draw[decorate,decoration={brace, mirror}] (0,-.7)--(4,-.7) node[midway, below] {$2n$ vertices};
\end{tikzpicture}
$, $n\geq 2$ & index $4 \cos^2(\frac{\pi}{4n-2})$
\\

$E_6 = \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (0,0) circle (.5mm) node [above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,0) circle (.5mm);
	\filldraw (2,1) circle (.5mm);

	\draw (0,0)--(4,0);
	\draw (2,0)--(2,1);
\end{tikzpicture}$
&
 index $4 \cos^2(\frac{\pi}{12}) \approx 3.73$ 
 \\

$E_8 = \begin{tikzpicture}[baseline=0, scale=.7]
	\filldraw (-2,0) circle (.5mm) node [above] {$*$};
	\filldraw (-1,0) circle (.5mm);
	\filldraw (0,0) circle (.5mm);
	\filldraw (1,0) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,0) circle (.5mm);
	\filldraw (2,1) circle (.5mm);

	\draw (-2,0)--(4,0);
	\draw (2,0)--(2,1);
\end{tikzpicture}$
\qquad \qquad
& index $4 \cos^2(\frac{\pi}{30}) \approx 3.96$
\end{tabular}
\end{thm}

\end{frame}







\begin{frame}
Suppose $N\subset M$ is a subfactor, ie a unital inclusion of type $II_1$ factors.  


\begin{defn}
The  {\em index} of $N \subset M$ is $[M:N]:= \dim_N L^2(M)$.
\end{defn}


\begin{example}
If $R$ is the hyperfinite $II_1$ factor, and $G$ is a finite group which acts outerly on $R$, then $R \subset R \rtimes G$ is a subfactor of index $\abs{G}$.

\vspace{6pt}
If $H \leq G$, then $R \rtimes H \subset R \rtimes G$ is a subfactor of index $[G:H]$.
\end{example}

\begin{thm}[Jones]
The possible indices for a subfactor are
$$\{4 \cos(\frac{\pi}{n})^2 | n \geq 3 \} \cup [4,\infty].$$
\end{thm}

\end{frame}









\begin{frame}

Let $X= _N \!\!M_M$ and $\overline{X}= _M \!\!(M^{op})_N$, and $\otimes = \otimes_N$ or $\otimes_M$ as needed.

\begin{defn}
The {\em standard invariant} of $N \subset M$ is the (planar) algebra of bimodules generated by $X$:
\begin{align*}
 X \quad , \quad 
 X \otimes \overline{X} \quad , \quad 
 X \otimes \overline{X} \otimes X \quad , \quad 
 X \otimes \overline{X} \otimes X \otimes \overline{X} \quad , \quad\ldots \\
 \overline{X} \quad , \quad
 \overline{X} \otimes X \quad , \quad
 \overline{X} \otimes X \otimes \overline{X} \quad , \quad
 \overline{X} \otimes X \otimes \overline{X} \otimes X \quad , \quad  \ldots
 \end{align*}
\end{defn}

\begin{defn}
The {\em principal graph} of $N \subset M$ has vertices for (isomorphism classes of) irreducible $N$-$N$ and $N$-$M$ bimodules, and an edge from $_{N}Y_N$ to $_{N}Z_M$ if $Z \subset Y \otimes X$ (iff  $Y \subset Z \otimes \overline{X}$).  

\vspace{6pt}
Ditto for the {\em dual principal graph}, with $M$-$M$ and $M$-$N$ bimodules.
\end{defn}

\end{frame}











\begin{frame}

\frametitle{Example:  $R \rtimes H \subset R \rtimes G$}

Again, let $G$ be a finite group with subgroup $H$, and act outerly on $R$.  Consider $N = R \rtimes H \subset R \rtimes G = M$.

The irreducible $M$-$M$ bimodules are of the form $R \otimes V$ where $V$ is an irreducible $G$ representation.
The irreducible $M$-$N$ bimodules are of the form $R \otimes W$ where $W$ is an $H$ irrep.

The dual principal graph of $N \subset M$ is the induction-restriction graph for irreps of $H$ and $G$.

\begin{example}[$S_3 \leq S_4$]
\begin{tikzpicture}[yscale=.5]
	\node (MtrivM) at (0,0) {trivial};
	\node (MstndM) at (2,0) {standard};
	\node (MVM) at (4,0) {$V$};
	\node (MtensM) at (6.5,0) {sign$\otimes$standard};
	\node (MsignM) at (9,0) {sign};
	
	\node (MtrivN) at (1,2) {trivial};
	\node (MstndN) at (4,2) {standard};
	\node (MsignN) at (8,2) {sign};
	
	\draw (MtrivM) -- (MtrivN) -- (MstndM) -- (MstndN)--(MtensM)--(MsignN)--(MsignM);
	\draw (MstndN)--(MVM);
\end{tikzpicture}
\end{example}

(The principal graph is an  induction-restriction graph too, for $H$ and various subgroups of $H$.)

\end{frame}










\begin{frame}

\frametitle{Index 4}

\begin{thm}[Popa]

The principal graphs of a subfactor of index 4 are extended Dynkin diagram:


%
${A}_n^{(1)} = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (0,0) circle (.5mm) node [above] {$*$};
\filldraw (1,.5) circle (.5mm);
\filldraw (2,.5) circle (.5mm);
\node at (3,.5) {$\cdots$};
\filldraw (1,-.5) circle (.5mm);
\filldraw (2,-.5) circle (.5mm);
\node at (3,-.5) {$\cdots$};
\filldraw (4,.5) circle (.5mm);
\filldraw (4,-.5) circle (.5mm);
\filldraw (5,0) circle (.5mm);

\draw (0,0)--(1,.5)--(2.5,.5);
\draw (0,0)--(1,-.5)--(2.5,-.5);

\draw (3.5,.5)--(4,.5)--(5,0)--(4,-.5)--(3.5,-.5);
	
\draw[decorate,decoration={brace, mirror}] (0,-.7)--(5,-.7) node[midway, below] {$n+1$ vertices};
\end{tikzpicture}
$, $n\geq 1$,
%
${D}_{n}^{(1)} = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (0,-.5) circle (.5mm) node [above] {$*$};
\filldraw (1,0) circle (.5mm);
\filldraw (0,.5) circle (.5mm);
\filldraw (2,0) circle (.5mm);
\node at (3,0) {$\cdots$};
\filldraw (4,0) circle (.5mm);
\filldraw (5,.5) circle (.5mm);
\filldraw (5,-.5) circle (.5mm);
\draw (0,.5)--(1,0)--(2.5,0);
\draw (1,0)--(0,-.5);
\draw (3.5,0)--(4,0)--(5,.5);
\draw (4,0)--(5,-.5);
\draw[decorate,decoration={brace, mirror}] (0,-.7)--(5,-.7) node[midway, below] {$n+1$ vertices};
\end{tikzpicture}
$, $n\geq 3$,

${E}_6^{(1)} = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (0,0) circle (.5mm) node [above] {$*$};
\filldraw (1,0) circle (.5mm);
\filldraw (2,0) circle (.5mm);
\filldraw (3,0) circle (.5mm);
\filldraw (4,0) circle (.5mm);
\filldraw (2,1) circle (.5mm);
\filldraw (2,2) circle (.5mm);
\draw (0,0)--(4,0);
\draw (2,0)--(2,2);
\end{tikzpicture}$, \qquad
%
${E}_7^{(1)} = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (-1,0) circle (.5mm) node [above] {$*$};
\filldraw (0,0) circle (.5mm);
\filldraw (1,0) circle (.5mm);
\filldraw (2,0) circle (.5mm);
\filldraw (3,0) circle (.5mm);
\filldraw (4,0) circle (.5mm);
\filldraw (5,0) circle (.5mm);
\filldraw (2,1) circle (.5mm);
\draw (-1,0)--(5,0);
\draw (2,0)--(2,1);
\end{tikzpicture}$,

${E}_8^{(1)} = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (-3,0) circle (.5mm) node [above] {$*$};
\filldraw (-2,0) circle (.5mm);
\filldraw (-1,0) circle (.5mm);
\filldraw (0,0) circle (.5mm);
\filldraw (1,0) circle (.5mm);
\filldraw (2,0) circle (.5mm);
\filldraw (3,0) circle (.5mm);
\filldraw (4,0) circle (.5mm);
\filldraw (2,1) circle (.5mm);
\draw (-3,0)--(4,0);
\draw (2,0)--(2,1);
\end{tikzpicture}$, \quad
%
${A}_\infty = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (0,0) circle (.5mm) node [above] {$*$};
\filldraw (1,0) circle (.5mm);
\filldraw (2,0) circle (.5mm);
\filldraw (3,0) circle (.5mm);
\node at (4,0) {$\cdots$};

\draw (0,0)--(3.5,0);
\end{tikzpicture}$, \quad
%
${A}_{\infty}^{(1)} = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (0,0) circle (.5mm) node [above] {$*$};
\filldraw (1,.5) circle (.5mm);
\filldraw (2,.5) circle (.5mm);
\filldraw (3,.5) circle (.5mm);
\node at (4,.5) {$\cdots$};
\filldraw (1,-.5) circle (.5mm);
\filldraw (2,-.5) circle (.5mm);
\filldraw (3,-.5) circle (.5mm);
\node at (4,-.5) {$\cdots$};

\draw (0,0)--(1,.5)--(3.5,.5);
\draw (0,0)--(1,-.5)--(3.5,-.5);
\end{tikzpicture}
$, \quad
%
${D}_\infty = \begin{tikzpicture}[baseline=0, scale=.5]
\filldraw (0,-.5) circle (.5mm) node [above] {$*$};
\filldraw (1,0) circle (.5mm);
\filldraw (0,.5) circle (.5mm);
\filldraw (2,0) circle (.5mm);
\filldraw (3,0) circle (.5mm);
\node at (4,0) {$\cdots$};
\draw (0,.5)--(1,0)--(3.5,0);
\draw (1,0)--(0,-.5);
\end{tikzpicture}
$

There are multiple subfactors for some of these principal graphs (eg, $n-1$ non-isomorphic hyperfinite subfactors for $D_n^{(1)}$).

\end{thm}

\end{frame}











\section{The classification up to index 5}

\begin{frame}
\beamersetuncovermixins 
{\opaqueness<1->{0}} 
{} 

\frametitle{Haagerup's list}
\begin{itemize}
\item<1-> In 1993 Haagerup classified possible principal graphs for subfactors with index between $4$ and $3+\sqrt{3} \approx 4.73$:
\only<1|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup}, \mathfig{0.225}{graphs/EH}, \mathfig{0.3}{graphs/EEH}, \ldots$,
($\approx  4.30, \, 4.37, \, 4.38, \ldots$)
\item $\mathfig{0.3}{graphs/HA}$, ($\approx 4.56$)
\item \vspace{0.25cm} $\mathfig{0.15}{graphs/hexagon}, \mathfig{0.225}{graphs/Ehexagon}, \ldots$
($\approx  4.62, \, 4.66, \ldots$).
\end{itemize}}
\only<2|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH}, \mathfig{0.3}{graphs/EEH}, \ldots$,
($\approx  4.30, \, 4.37, \, 4.38, \ldots$)
\item $\mathfig{0.3}{graphs/HA-green}$, ($\approx 4.56$)
\item \vspace{0.25cm} $\mathfig{0.15}{graphs/hexagon}, \mathfig{0.225}{graphs/Ehexagon}, \ldots$
($\approx  4.62, \, 4.66, \ldots$).
\end{itemize}}
\only<3|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH}\color{red}{, \mathfig{0.3}{graphs/EEH-red}, \ldots}$,
($\approx  4.30, \, 4.37, \, 4.38, \ldots$)
\item $\mathfig{0.3}{graphs/HA-green}$, ($\approx 4.56$)
\item \vspace{0.25cm} {\color{red} $\mathfig{0.15}{graphs/hexagon-red}, \mathfig{0.225}{graphs/Ehexagon-red}, \ldots$}
($\approx  4.62, \, 4.66, \ldots$).
\end{itemize}}
\only<4>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH-green}\color{red}{, \mathfig{0.3}{graphs/EEH-red}, \ldots}$,
($\approx  4.30, \, 4.37, \, 4.38, \ldots$)
\item $\mathfig{0.3}{graphs/HA-green}$, ($\approx 4.56$)
\item \vspace{0.25cm} {\color{red} $\mathfig{0.15}{graphs/hexagon-red}, \mathfig{0.225}{graphs/Ehexagon-red}, \ldots$}
($\approx  4.62, \, 4.66, \ldots$).
\end{itemize}}
\item<2-> Haagerup and \href{http://dx.doi.org/10.1007/s002200050574}{Asaeda \& Haagerup (1999)} constructed two of these possibilities.
\item<3-> \href{http://www.springerlink.com/content/c4885q02dflfttwm/}{Bisch (1998)} and \href{http://arxiv.org/abs/0711.4144}{Asaeda \& Yasuda (2007)} ruled out infinite families.
\item<4-> Last year we (Bigelow-Morrison-Peters-Snyder) constructed the last missing case. \arxiv{0909.4099}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Extending the classification}
We work with principal graph pairs, meaning both principal and dual principal graphs, and information on which bimodules are dual.
\begin{example}[The Haagerup subfactor's principal graph pair]
$$\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$$
\end{example}
The pair must satisfy an associativity test: $$(X \tensor Y) \tensor X \iso X \tensor (Y \tensor X)$$

\begin{block}{}
We can efficiently enumerate such pairs with index below some number $L$ up to a given rank or depth, obtaining a collection of allowed \emph{vines} and \emph{weeds}.
\end{block}
\end{frame}

\begin{frame}
\begin{defn}
A vine represents an integer family of principal graphs, obtained by \emph{translating} the vine.
\end{defn}

\begin{defn}
A weed represents an infinite family, obtained by either translating or \emph{extending} arbitrarily on the right.
\end{defn}

\begin{block}{}
We can hope that as we keep extending the depth, a weed will turn into a set of vines.
If all the weeds disappear, the enumeration is complete.
This happens in favorable cases (e.g. Haagerup's theorem up to index $3+\sqrt{3}$), but generally we stop with some surviving weeds, and have to rule these out `by hand`.
\end{block}
\end{frame}

\begin{frame}[plain]
\center
\scalebox{0.40}{

\begin{tikzpicture}
[level 1/.style={level distance=46mm, sibling distance=40mm,nodes={draw, fill=white, rectangle, rounded corners}},
level 2/.style={level distance=52mm, sibling distance=20mm},
level 3/.style={level distance=58mm, sibling distance=25mm},
level 4/.style={level distance=64mm},
grow=right]\node[draw, fill=white, rectangle, rounded corners] {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1duals1v1v1x2}\\\bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\end{array}$}
child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x1duals1v1v1x2v1}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1duals1v1v1x2v1x2}\end{array}$}
child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x1v1duals1v1v1x2v1}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v1x1duals1v1v1x2v1x2}\end{array}$}
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child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1p0x1v1x0x0x0p1x0x0x0duals1v1v1x2v2x1x3x4}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1p0x1v0x1p1x0duals1v1v1x2v1x2}\end{array}$}
}}child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x1p0x1duals1v1v1x2v1x2}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1p0x1duals1v1v1x2v1x2x3}\end{array}$}
child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x1p0x1v0x1duals1v1v1x2v1x2}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1p0x1v0x0x1duals1v1v1x2v1x2x3}\end{array}$}
child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x1p0x1v0x1v1duals1v1v1x2v1x2v1}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1p0x1v0x0x1v1duals1v1v1x2v1x2x3v1}\end{array}$}
}}}child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1duals1v1v1x2v2x1x3}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1duals1v1v1x2v1}\end{array}$}
child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0duals1v1v1x2v2x1x3}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1duals1v1v1x2v1}\end{array}$}
}child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1p0x0x1duals1v1v1x2v2x1x3}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1p1duals1v1v1x2v1}\end{array}$}
}child {node [fill=red!30]{$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\end{array}$}
}}child {node {$\begin{array}{c}\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1p0x1duals1v1v1x2v1x3x2x4x5}\\\bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1p0x1duals1v1v1x2v1x2x3}\end{array}$}
};
\end{tikzpicture}
}

\end{frame}

\begin{frame}
\frametitle{The classification up to index 5}
\begin{thm}[Morrison-Snyder, part I, \arxiv{1007.1730}]
Every (finite depth) $II_1$ subfactor with index less than $5$ sits inside one of 54 families of vines (see below), or 5 families of weeds:
\begin{align*}
\cC &= \left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0p0x0x1duals1v1v1x2v2x1x3},\bigraph{bwd1v1v1v1p1v1x0p1x0v0x1v1p1duals1v1v1x2v1}\right), \\
	\cF &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0p0x0x0x1v1x0x0p1x0x0p0x1x0p0x0x1v1x0x0x0p0x0x1x0p0x0x0x1duals1v1v1x2v1x3x2x4v1x4x3x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0p0x1v1x0x0p0x1x0v0x1p1x0p0x1duals1v1v1x2v1x2v2x1}\right), \\
	\cB &=\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right), \\
	\cQ  &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v2x1x3} \right), \\
	\cQ' &= \left(\bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v0x0x1duals1v1v1x2x3} \right).
\end{align*}
\end{thm}

\begin{thm}[Morrison-Penneys-P-Snyder, part II, \arxiv{1007.2240}]
Using quadratic tangles techniques, there are no subfactors in the families
$\cC$ or $\cF$.
\end{thm}
\end{frame}

\begin{frame}
\begin{thm}[Calegari-Morrison-Snyder, \arxiv{1004.0665}]
In any family of vines, there are at most finitely many subfactors, and there is an effective bound.
\end{thm}

\begin{cor}[Penneys-Tener, part IV, \arxiv{1010.3797}]
There are only four possible principal graphs of subfactors coming from the 54 families
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1v1v1x2v2x1}, \bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p1x0duals1v1v1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1v1x2v1}\right)$
\item $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}\right).$
\end{itemize}
\end{cor}
\end{frame}

\begin{frame}
\frametitle{Recent results}
\begin{thm}[Morrison-Penneys-Peters-Snyder, part V, work in progress]
There are no subfactors coming from the weed $\cB =\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x0p0x1p0x1v0x1x0x0p0x0x1x0v1x0p0x1v1x0p0x1duals1v1v1x2v1x3x2x4v1x2}, \bigraph{bwd1v1v1v1p1v1x0p1x0v1x0p0x1v0x1p1x0v1x0p0x1v0x1p1x0duals1v1v1x2v1x2v2x1}\right )$
\end{thm}
\begin{proof}
A connection on the principal graph only exists at a certain index (one for each supertransitivity), but the only graphs with exactly that index are certain infinite graphs which are easily ruled out.
\end{proof}

\begin{block}{Izumi, work in progress}
Also by a connection argument, the only subfactor coming from the weeds $\cQ$ or $\cQ'$ is $3311$. $\left(\bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}\right)$
\end{block}
\end{frame}

\begin{frame}
We're thus very close to completing the classification up to index 5:
\begin{conj}
There are exactly ten subfactors other than Temperley-Lieb with index between $4$ and $5$.
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1x2v2x1} \bigraph{bwd1v1v1v1p1v1x0p1x0duals1v1v1x2}\right)$,
\item $\left(\bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p0x1v1x0p0x1duals1v1v1v1v1x2v2x1}, \bigraph{bwd1v1v1v1v1v1v1v1p1v1x0p1x0duals1v1v1v1v1x2}\right)$,
\item $\left(\bigraph{bwd1v1v1v1v1v1p1v1x0p0x1v1x0p0x1p0x1v1x0x0v1duals1v1v1v1x2v2x1x3v1}, \bigraph{bwd1v1v1v1v1v1p1v0x1p0x1v0x1v1duals1v1v1v1x2v1}\right)$,
\item The 3311 GHJ subfactor (MR999799), with index $3+\sqrt{3}$   $\left(\bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0v1duals1v1v1x2x3v1}\right)$,
\item Izumi's self-dual 2221 subfactor (MR1832764), with index $\frac{5+\sqrt{21}}{2}$ $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0duals1v1v2x1}\right)$
\end{itemize}
along with the non-isomorphic duals of the first four, and the non-isomorphic complex conjugate of the last.
\end{conj}
\end{frame}

\begin{frame}
\frametitle{Index exactly $5$}
There are 5 principal graphs that come from group-subgroup subfactors, and these are known to be unique.
\begin{itemize}
\item  $\left(\bigraph{bwd1v1p1p1p1duals1v4x3x2x1}, \bigraph{bwd1v1p1p1p1duals1v4x3x2x1}\right)$ \tiny $1 \subset \Integer/5\Integer$
\item $\left(\bigraph{bwd1v1p1v1x1v1duals1v1x2v1}, \bigraph{bwd1v1p1v1x1v1duals1v1x2v1}\right)$ \tiny $\Integer/2\Integer \subset D_{10}$
\item $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v2x1x3}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v2x1x3}\right)$ \tiny $\Integer/4\Integer \subset \Integer/5\Integer \rtimes \operatorname{Aut}(\Integer/5\Integer)$ 
\item $\left(\bigraph{bwd1v1v1v1p1p1v0x0x1p0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1v1p1p1v0x1x0p0x0x1v1x0p0x1duals1v1v1x2x3v2x1}\right)$  \tiny $A_4 \subset A_5$
\item $\left(\bigraph{bwd1v1v1v1p1v1x0p0x1v1x0p1x1v1x0v1duals1v1v1x2v1x2v1}, \bigraph{bwd1v1v1v1p1v0x1p0x1v1x0p1x0p0x1v0x1x0v1duals1v1v1x2v1x2x3v1}\right)$  \tiny $S_4\subset S_5$
\end{itemize}
We still have a few other possibilities to rule out
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1p1p1v0x1x0p0x0x1v1x0p0x1duals1v1v1x3x2v2x1}, \bigraph{bwd1v1v1v1p1p1v0x1x0p0x0x1v1x0p0x1duals1v1v1x3x2v2x1}\right)$
\item $\left(\bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1p1p1v1x0x0p0x1x0p0x0x1duals1v1v1x2x3}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1v1p1p1v1x0x0p0x1x0v1x0v1v1duals1v1v1v1x2x3v1v1}, \bigraph{bwd1v1v1v1v1v1p1p1v1x0x0p0x1x0v1x0v1v1duals1v1v1v1x2x3v1v1}\right)$
\end{itemize}
\end{frame}





\section{Beyond 5:  wilderness}



\begin{frame}
\frametitle{Index beyond $5$}


Somewhere between index $5$ and index $6$, things get wild:

\begin{thm}[Bisch-Nicoara-Popa]
At index $6$, there is an infinite one-parameter family of irreducible, hyperfinite subfactors having isomorphic standard invariants.  
\end{thm}

and

\begin{thm}[Bisch-Jones]
$A_2 * A_3$ is an infinite depth subfactor at index $2\tau^2 \sim 5.23607$.

\begin{tikzpicture}[scale=.6]
	\filldraw (0,0) circle (.5mm) node[above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\filldraw (1,1) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\filldraw (2,1) circle (.5mm);
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,0) circle (.5mm);
	\filldraw (4,1) circle (.5mm);
	\filldraw (5,0) circle (.5mm);
	\filldraw (6,0) circle (.5mm);
	\filldraw (6,1) circle (.5mm);
%	\filldraw (7,0) circle (.5mm);
	\node at (7,0) {$\cdots$};
	
	\draw (0,0)--(6.5,0);
	\draw (1,0)--(1,1);
	\draw (2,0)--(2,1);
	\draw (4,0)--(4,1);
	\draw (6,0)--(6,1);
\end{tikzpicture},
%
\begin{tikzpicture}[scale=.6]
	\filldraw (0,0) circle (.5mm) node[above] {$*$};
	\filldraw (1,0) circle (.5mm);
	\filldraw (2,0) circle (.5mm);
	\filldraw (3,1) circle (.5mm);
	\filldraw (3,0) circle (.5mm);
	\filldraw (4,0) circle (.5mm);
	\filldraw (5,1) circle (.5mm);
	\filldraw (5,0) circle (.5mm);
	\filldraw (6,0) circle (.5mm);
	\filldraw (7,1) circle (.5mm);
	\filldraw (7,0) circle (.5mm);

	\filldraw (2,-1) circle (.5mm);
	\filldraw (3,-1) circle (.5mm);
	\filldraw (4,-1) circle (.5mm);
	\filldraw (5,-1) circle (.5mm);
	\filldraw (5,-2) circle (.5mm);
	\filldraw (6,-1) circle (.5mm);
	\filldraw (7,-1) circle (.5mm);
	\filldraw (7,-2) circle (.5mm);

	\node at (8,0) {$\cdots$};
	\node at (8,-1) {$\cdots$};
	
	\draw (0,0)--(7.5,0);
	\draw (3,0)--(3,1);
	\draw (5,0)--(5,1);
	\draw (7,0)--(7,1);
	
	\draw (1,0)--(2,-1)--(7.5,-1);
	\draw (5,-1)--(5,-2);
	\draw (7,-1)--(7,-2);
\end{tikzpicture}



\end{thm}

\end{frame}








\begin{frame}
Classification above index 5 looks hard, but we can still fish for examples (only supertransitivity $> 1$)!  

Here are some graphs that we find.  (A few are previously known)
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3},
\bigraph{bwd1v1v1p1v1x0p0x1p0x1v0x1x0p1x0x1duals1v1v2x1x3}\right)$ \\ (from $SU_q(3)$ at a root of unity, index $\sim 5.04892$)
\end{itemize}
At index $2\tau^2 \sim 5.23607$
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1v1v1p1v1x1p0x1duals1v1v1v1x2}, \bigraph{bwd1v1v1v1v1p1v1x1p0x1duals1v1v1v1x2}\right)$
\item $\left(\bigraph{bwd1v1v1v1p1p1v0x1x0p0x1x0p0x0x1v0x0x1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1v1p1p1v1x0x0p0x1x0p0x0x1v1x0x0p0x1x0p0x0x1duals1v1v1x2x3v2x1x3}\right)$
\item $\left(\bigraph{bwd1v1v1v1v1p1p1v1x0x0p0x1x0p0x0x1v1x0x0p0x1x0v1x0p0x1duals1v1v1v2x1x3v2x1}, \bigraph{bwd1v1v1v1v1p1p1v1x0x0p0x1x0p0x0x1v1x0x0p0x1x0v1x0p0x1duals1v1v1v2x1x3v2x1}\right)$
\item $\left(\bigraph{bwd1v1v1p1v1x0p1x0p0x1p0x1v0x1x0x1duals1v1v1x4x3x2}, \bigraph{bwd1v1v1p1v1x1v1v1duals1v1v1v1}\right)$
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1p1v1x1p0x1v0x1duals1v1v1x2}, \bigraph{bwd1v1v1p1v1x1p0x1v0x1duals1v1v1x2}\right)$ \\ (``Haagerup $+1$'' at index $\frac{7+\sqrt{13}}{2} \sim 5.30278$)
\item $\left(\bigraph{bwd1v1v1p1p1v0x1x1p0x0x1duals1v1v1x2}, \bigraph{bwd1v1v1p1p1v0x1x1p0x0x1duals1v1v1x2}\right)$ at $\frac{1}{2} \left(4+\sqrt{5}+\sqrt{15+6 \sqrt{5}}\right) \sim 5.78339$
\item $\left(\bigraph{bwd1v1v1v1v1p1p1v0x1x1p0x0x1duals1v1v1v1x2}, \bigraph{bwd1v1v1v1v1p1p1v0x1x1p0x0x1duals1v1v1v1x2}\right)$ at $3+2\sqrt{2} \sim 5.82843$
\end{itemize}
And at index $6$
\begin{itemize}
\item $\left(\bigraph{bwd1v1v1p1v1x0p1x1p0x1v1x0x1duals1v1v1x2x3}, \bigraph{bwd1v1v1p1v1x0p1x1p0x1v0x1x0v1duals1v1v1x2x3v1}\right)$
\item $\left(\bigraph{bwd1v1v1p1v1x0p1x1p0x1v0x1x0v1duals1v1v1x2x3v1}, \bigraph{bwd1v1v1p1v1x0p1x1p0x1v0x1x0v1duals1v1v1x2x3v1}\right)$
\end{itemize}
and several more!
\end{frame}

\begin{frame}
\begin{center}
 The End!
\end{center}
\end{frame}





\end{document}
% ----------------------------------------------------------------
