Associate Professorepeters3 (at) luc (dot) edu
My research interests are in, broadly, quantum symmetry, and more narrowly, subfactors/fusion categories. I use planar algebras and "proof by pictures" when possible, and also study knots and their invariants.
I am co-organizing an MSRI program on Quantum Symmetries, January-May 2020.
I organized the 14th East Coast Operator Algebras Symposium at Loyola, October 1 and 2, 2016.
My advisor was Vaughan Jones, and my dissertation was on "A planar algebra construction of the Haagerup subfactor."
Papers and PreprintsMy papers on the arXiv.
My papers on MathSciNet
I assembled some notes from "Operator Algebras and Conformal Field Theories," August 16-21 2010, University of Oregon. These notes are the result of lots of people giving great talks and other people's great liveTeXing of those talks.
ResearchMy focus is on constructing planar algebras, and exotic subfactors. Planar algebras are a tool for doing certain computations by drawing pictures; more technically, a planar algebra is a structure in which certain diagrams in the plane are interpreted as morphisms. Subfactors of von Neumann algebras give an example of a planar algebra.
The simplest example of a planar algebra is the Temperley-Lieb algebra, which first arose in statistical mechanics. The planar algebra perspective on Temperley-Lieb helps explain why this algebra has close ties to both subfactors and knot theory (specifically, the Jones polynomial).
I'm particularly interested in using planar algebras to construct exotic subfactors. Exotic subfactors are those which don't come from previously studied algebraic objects, like groups, quantum groups, or rational conformal field theories. In my dissertation, I used planar algebras to construct the Haagerup subfactor, and also to find a non-standard embedding (I use this term loosely) of the Haagerup planar algebra in a graph planar algebra.
My current interests are in developing the Atlas of Subfactors, and classifying principal graphs up to small index limits (Joint with Vaughan Jones, Scott Morrison, Dave Penneys, Noah Snyder, and others), and also learning more about tensor categories. If you'd like more details, check out my NSF project proposal , read some blog posts about planar algebras, or email me!
Handouts and SlidesPlanar algebras and evaluation algorithms. Slides from my talk at UW Madison, April 1, 2013.
Classifying Subfactors: beyond index 5. Slides from my talk at NCGOA 2012, Vanderbilt University, May 9, 2012.
Non-commutative galois theory and the classification of small index subfactors. Slides from my colloquium at UMass Boston, November 30 2011.
The classification of subfactors up to index 5. Slides from my talk at the IHP conferences "II1 factors: rigidity, symmetries and classification", May 27, 2011.
Knots, the Four-color Theorem, and von Neumann Algebras. Slides from my talk in the D.W. Weeks seminar at MIT, April 20, 2011.
Classification of Subfactors: Slides from ECOAS 2010 at Dartmouth College.
Constructing the extended Haagerup subfactor: Slides from WCOAS 2009 in Reno, NV.
Planar Algebras and Knots: A poster from the AWM poster session at the 2009 Joint Math Meetings in DC.
Generators and Relations for the Haagerup planar algebra: Handout from my talk at the University of Toronto, December 12, 2008.
Planar algebras and the Haagerup subfactor, at the University of Tokyo operator algebras seminar, October 23, 2008. Beamer slides
Exceptional Tensor Categories in Subfactor Theory, at the MFO (Mathematisches Forschungsinstitut Oberwolfach) Arbeitsgemeinschaft on Algebraic structures in conformal field theories, April 4, 2007. Extended Abstract.
TeachingFun! Watch a video of me talking about the platonic solids. This was produced by Girls' Angle, a math club for middle school girls, as part of their Women In Math video series.
I'm a regular ringer at MathILy, a summer math program for mathematically talented high school students. My past week of chaos classes have included non-Euclidean geometry, voting methods, Shor's algorithm, and how to play poker.
Many years ago, I taught the high school students' section of the Boston Math Circle. We had classes on knot theory, math and politics, auction theory, coding theory, and polytopes.
At the University of New Hampshire, I taught calculus 1 and multivariable calculus, and organized a graduate seminar on knot theory.
Here are my job application materials from AY 2011-12: Research Statement, CV, Teaching Statement.