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One of my major research interests is in fusion categories and subfactors.

With Masaki Izumi, Vaughan Jones, David Penneys, Emily Peters, Noah Snyder and James Tener, I have recently completed the classification of subfactors with index less than 5. This classification is described in our survey paper
The classification of subfactors of index at most 5
Joint with Vaughan F. R. Jones and Noah Snyder
pdf · arxiv
and the detailed proofs appear in a series of four papers:
Subfactors of index less than 5, part 1: the principal graph odometer
Joint with Noah Snyder, accepted at Communications in Mathematical Physics, June 28 2011.
pdf · arXiv · journal
Subfactors of index less than 5, part 2: triple points
Joint with David Penneys, Emily Peters and Noah Snyder, International Journal of Mathematics Vol. 23, No. 3 (2012) 1250016 (33 pages).
pdf · arXiv · journal
Subfactors of index less than 5, part 3: quadruple points
Joint with Masaki Izumi, Vaughan Jones and Noah Snyder, accepted at Communications in Mathematical Physics, October 8 2011.
pdf · arXiv
Subfactors of index less than 5, part 4: vines
Written David Penneys and James Tener, International Journal of Mathematics, Vol. 23, No. 3 (2012) 1250017 (18 pages).
pdf · arXiv · journal
Noah wrote a short blog post outlining the main results, at the Secret Blogging Seminar. Since then, we have a number of papers giving further results on the classification of small index subfactors.
An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem
pdf · arxiv
Constructing spoke subfactors using the jellyfish algorithm
Joint with David Penneys, to appear Transactions of the American Mathematical Society.
pdf · arXiv · project summary
The little desert? Some subfactors with index in the interval $(5,3+\sqrt{5})$
Joint with Emily Peters.
pdf · arXiv
You can see slides from the following talks about this project, by myself and collaborators. An important tool in the classification program was a theorem using arithmetic conditions to reduce certain infinite classes of potential subfactors to finitely many cases, described in a blog post and proved in my paper
Cyclotomic integers, fusion categories, and subfactors
Joint with Frank Calegari and Noah Snyder, Communications in Mathematical Physics Volume 303, Issue 3 (2011), pp. 845-896.
pdf · arXiv · journal · mathscinet
Prior to beginning the classification of subfactors with index less than 5, I completed the classification of subfactors with index less than $3+\sqrt{3}$, by constructing the extended Haagerup subfactor, in the paper
Constructing the extended Haagerup planar algebra
Joint with Stephen Bigelow, Emily Peters and Noah Snyder, in press at Acta Mathematica.
pdf · arXiv
This work is described in a blog post, and you can also see the slides or view the video of my talk Emily Peters also gave some talks on this construction Further, with Noah Snyder I used the exotic Haagerup and extended Haagerup subfactors to show that not all fusion categories can be defined over a cyclotomic field. This is described in a blog post at the Secret Blogging Seminar, and our paper:
Non-cyclotomic fusion categories
Joint with Noah Snyder, in press at Transactions of the American Mathematical Society.
pdf · arXiv