Scott Morrison - Fusion categories and subfactors

my publications on the arXiv
my public calendar

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.

November 30 2011, Non-commutative Galois theory and the classification of small-index subfactors, by Emily Peters, University of Massachussetts, Boston. (slides)
October 29 2011, Connections and planar algebras, Von Neumann algebras and Conformal Field Theory at Vanderbilt. (abstract)
Abstract: With Emily Peters, I've been exploring subfactors with index in the interval $(5, 3+\sqrt{5})$. We've recently obtained a classification of 1-supertransitive subfactors in this range, and performed an extensive computer search in higher supertransitivities. I'll describe the examples of subfactors we've found. We have two main new techniques. First, even when we only know a fragment of a principal graph, we can extract certain inequalities by considering the norms of the entries of a connection. This allows the new classification result. Second, we extend the theory of bi-unitary connections to the bi-invertible case, and find we can then work over a fixed number field. This allows effective use of the "hybrid method" of constructing subfactors: given a not-necessarily flat bi-invertible connection, we can efficiently solve the equations for a flat element in the graph planar algebra. This lets us completely analyse the examples.
October 19 2011, Small index subfactors, Toronto colloquium. (abstract, prezi)
Abstract: Fusion categories provide a simple model for a collection of particles which can split and fuse. Despite the simplicity of the model, it has proved useful in describing exotic materials in condensed matter physics. On the mathematical side, fusion categories can be thought of as 'quantum' finite groups. I'll describe recent joint work on the classification of small fusion categories. The new examples we've encountered are rather strange objects. Moreover, the classification is much sparser than we had expected, encouraging the hope of extending our current knowledge to larger and larger classes of fusion categories. Along the way we'll use analysis (von Neumann algebras and subfactors), number theory (the geometry of cyclotomic integers), graph combinatorics, two-dimensional topology, and some representation theory!
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 Emily Peters, Wabash extramural modern analysis miniconference. (slides)
May 27 2011, The classification of subfactors up to index 5, by Emily Peters, 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 Noah Snyder, 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 David Penneys, 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 Emily Peters, 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 Noah Snyder, 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 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

March 22, Extended Haagerup exists!, Tensor Categories in Bloomington, Indiana.

Emily Peters 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 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

Projects

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