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 and
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 , accepted at Communications in Mathematical Physics, June 28 2011.
pdf · arXiv · journal
Subfactors of index less than 5, part 2: triple points
Joint with , and , 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 , and , accepted at Communications in Mathematical Physics, October 8 2011.
pdf · arXiv
Subfactors of index less than 5, part 4: vines
Written and , 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 , 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
pdf · arXiv
• November 30 2011, Non-commutative Galois theory and the classification of small-index subfactors, by , University of Massachussetts, Boston. (slides)
• October 29 2011, Connections and planar algebras, Von Neumann algebras and Conformal Field Theory at Vanderbilt. (abstract)
• October 19 2011, Small index subfactors, Toronto colloquium. (abstract, prezi)
• October 8, Classifying fusion categories and subfactors, SIAM Mini-symposium on Algebraic Aspects of Quantum Computing, Raleigh. (slides)
• September 24 2011, The classification of small-index subfactors, by , Wabash extramural modern analysis miniconference. (slides)
• May 27 2011, The classification of subfactors up to index 5, by , IHP conference on "$II_1$ factors: rigidity, symmetries and classification". (slides)
• May 20 2011, Classifying small index subfactors, Great Plains Operator Theory Symposium. (slides, prezi)
• Match 24 2011, Finite quantum groups, by , Vanderbilt Colloquium. (slides)
• January 13 2011, Fusion categories and subfactors, UCSD Colloquium. (slides, prezi)
• January 8 2011, Eliminating weeds and vines in the classification of subfactors to index 5, by , AMS Joint Meetings in New Orleans. (slides)
• November 10 2010, Fusion categories and subfactors, Caltech Colloquium. (slides, prezi)
• October 29 2010, Classifying subfactors up to index 5, Kyoto Operator Algebra seminar. (slides)
• October 24 2010, Classification of subfactors, by , ECOAS 2010 (slides)
• October 8 2010, Subfactors at index 5 and beyond, UCLA/DARPA subfactors meeting. (slides)
• September 2 2010, Classification of subfactors up to index 5. Quantum groups, Clermont-Ferrand. (prezi)
• August 11 2010, Classification of subfactors up to index 5. Operator algebras satellite conference, Chennai (prezi)
• June 4 2010, Classifying fusion categories. Miller Institute Symposium. (poster)
• May 11 2010, Classification of subfactors up to index 5., Non-commutative geometry and operator algebras, Vanderbilt. (prezi)
• October 18 2009, Fusion categories and small index subfactors, part I, by , special session on Fusion Categories at AMS Waco meeting. (slides)
• October 18 2009, Fusion categories and small index subfactors, part II, special session on Fusion Categories at AMS Waco meeting. (slides)
• September 3 2009, Fusion categories, UC Berkeley Colloquium. (blackboards)
• June 3 2009, Classifying subfactor planar algebras, UC Riverside colloquium.
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 and , 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 , and , 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
also gave some talks on this construction
• April 20 2011, Knots, the Four-color Theorem, and von Neumann Algebras, D.W. Weeks seminar at MIT. (slides)
• October 17 2009, Constructing the extended Haagerup subfactor, WCOAS 2009, Reno. (slides)
Further, with 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 , in press at Transactions of the American Mathematical Society.
pdf · arXiv