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\author[Emily Peters, Noah Snyder]{Emily Peters \\ \url{http://euclid.unh.edu/~eep} \\ Noah Snyder \\ \url{http://math.columbia.edu/~nsnyder} \\ joint work with Calegari, Jones, Morrison, Penneys}
\title{Classifying subfactors up to index 5}
\date{Shanks Workshop on Subfactors and Tensor Categories \\ \url{http://euclid.uhn.edu/~eep/Shanks2010.pdf}}

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\begin{document}

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       \frametitle{Outline}
       \tableofcontents
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       \frametitle{Outline}
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\section{Haagerup's classification up to index $3+\sqrt{3}$}
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\frametitle{principal graphs up to $3+\sqrt{3}$}
\begin{theorem}[Haagerup, Asaeda-Haagerup, Bisch, Asaeda, Asaeda-Yasuda, Bigelow-Morrison-Peters-Snyder]
There are only three principal graphs with index in the range $(4,3+\sqrt{3})$:    
$\mathfig{0.15}{graphs/Haagerup}, \mathfig{0.225}{graphs/EH}, \mathfig{0.3}{graphs/HA}$
\end{theorem}
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\begin{frame}
\frametitle{Our goal:  principal graphs up to $\sqrt{5}$}
\begin{conj}[Goodman-de la Harpe-Jones, Xu, work in progress of Calegari, Jones, Morrison, Penneys, Peters, Syder]
There are only two principal graphs with index in the range $[3+\sqrt{3}, \sqrt{5})$:    
$\bigraph{gbg1v1v1p1p1v1x0x0p0x1x0}$,
$\bigraph{gbg1v1v1v1p1p1v0x0x1v1}$
\end{conj}
\end{frame}







\begin{frame}
Graphs with duals and Ocneanu-style 4-partite graphs.  (I.e. we
explain what the individual pictures mean.)
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\begin{frame}
Running the odometer.  (Talk about extending one graph vs.
extending both graphs.  Index bound means this is finite.)

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\end{frame}






\begin{frame}
How to actually get small lists: triple point obstruction, square
test.  (Don't go into much detail on each, but emphasize that they all
apply independent of supertransitivity.)
\end{frame}









\begin{frame}
The quadratic tangles attack.  (Again not too much detail.  Vaughan
will have already talked about it.  Emphasize: doesn't work when w=-1,
but can work for some "weeds" not just for "vines.")
\end{frame}









\begin{frame}
The number theory attack.  (Emphasize: works for all vines.)
\end{frame}









\begin{frame}
The cases we're stuck on.  (Quadruple point with odd
supertransitivity and only one leg continuing.  "The Bad Seed."
Emphasize that we're hopeful that QT will eventually work for the
former, but we're genuinely stuck on the latter.)
\end{frame}









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Relationship to monoidal categories/fusion categories.
(Implications of this for small objects in unitary categories.  Fusion
case.  Better number theory argument for fusion case.)
\end{frame}









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