In this paper we discuss the integrable chiral Potts model, as it clearly relates to how we got befriended with Vaughan Jones, whose birthday we celebrated at the Qinhuangdao meeting. Remarkably we can also celebrate the birthday of the model, as it has been introduced about 30 years ago as the first solution of the star-triangle equations parametrized in terms of higher genus functions. After introducing the most general checkerboard Yang–Baxter equation, we specialize to the star-triangle equation, also discussing its relation with knot theory. Then we show how the integrable chiral Potts model leads to special identities for basic hypergeometric series in the $q$ a root-of-unity limit. Many of the well-known summation formulae for basic hypergeometric series do not work in this case. However, if we require the summand to be periodic, then there are many summable series. For example, the integrability condition, namely, the star-triangle equation, is a summation formula for a well-balanced $_4 \Phi _3$ series. We finish with a few remarks about the relation with quantum groups at roots of unity.

We give a precise definition for when a subfactor arises from a conformal net which can be motivated by classification of defects. We show that a subfactor $N \subset M$ arises from a conformal net if there is a conformal net whose representation category is the quantum double of $N \subset M$.

Growing out of the initial connections between subfactors and knot theory that gave rise to the Jones polynomial, Jones’ axiomatization of the standard invariant of an extremal finite index II$_1$ subfactor as a spherical C$^∗$-planar algebra, presented in [16], is the most elegant and powerful description available. We make the natural extension of this axiomatization to the case of finite index subfactors of arbitrary type. We also provide the first steps toward a limited planar structure in the infinite index case. The central role of rotations, which provide the main non-trivial part of the planar structure, is a recurring theme throughout this work.

In the finite index case the axioms of a C$^∗$-planar algebra need to be weakened to disallow rotation of internal discs, giving rise to the notion of a rigid C$^∗$-planar algebra. We show that the standard invariant of any finite index subfactor has a rigid C$^∗$-planar algebra structure. We then show that rotations can be re-introduced with associated correction terms entirely controlled by the Radon-Nikodym derivative of the two canonical states on the first relative commutant, $N′ \cap M$.

By deforming a rigid C$^∗$-planar algebra to obtain a spherical C$^∗$-planar algebra and lifting the inverse construction to the subfactor level we show that any rigid C$^∗$-planar algebra arises as the standard invariant of a finite index II$_1$ subfactor equipped with a conditional expectation, which in general is not trace preserving. Jones’ results thus extend completely to the general finite index case. We conclude by applying our machinery to the II$_1$ case, shedding new light on the rotations studied by Huang [11] and touching briefly on the work of Popa [29].

In the case of infinite index subfactors there are obstructions to having a full planar algebra theory. We constructing a periodic rotation operator on the L$^2$-spaces of the standard invariant of an approximately extremal, infinite index II$_1$ subfactor. In the finite index case we recover the usual rotation. We also show that the assumption of approximate extremality is necessary and sufficient for rotations to exist on these L$^2$-spaces.

The potential complexity of the standard invariant of an infinite index subfactor is illustrated by the construction of a II$_1$ subfactor with a type III central summand in the second relative commutant, $N′ \cap M_1$. The restriction to L$^2$-spaces does not see this part of the standard invariant and Izumi, Longo and Popa’s [13] examples of subfactors that are not approximately extremal provide a further challenge to move beyond the L$^2$-spaces in the construction of rotation operators. The present construction is simply an initial step on the road to a planar structure on the standard invariant of an infinite index subfactor.

We study 2-cabled analogs of Voiculescu’s trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable definition, finiteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the standard invariant of the associated inclusion is exactly the original planar algebra. We also give conditions that imply that the associated von Neumann algebras are non-$\Gamma$ non-$L^2$ rigid factors.

We give two different proofs of the existence of the $AH + 2$ subfactor, which is a 3-supertransitive self-dual subfactor with index $\frac{9+\sqrt{17}}{2}$. The first proof is a direct construction using connections on graphs and intertwiner calculus for bimodule categories. The second proof is indirect, and deduces the existence of $AH + 2$ from a recent alternative construction of the Asaeda-Haagerup subfactor and fusion combinatorics of the Brauer-Picard groupoid.

Motivated by our previous results, we investigate structural properties of probability measurepreserving actions of sofic groups relative to their Pinsker factor. We also consider the same properties relative to the Outer Pinsker factor, which is another generalization of the Pinsker factor in the nonamenable case. The Outer Pinsker factor is motivated by entropy in the presence, which fixes some of the “pathological” behavior of sofic entropy: namely increase of entropy under factor maps. We show that an arbitrary probability measure-preserving action of a sofic group is mixing relative to its Pinsker and Outer Pinsker factors and, if the group is nonamenable, it has spectral gap relative to its Pinsker and Outer Pinsker factors. Our methods are similar to those we developed in “Polish models and sofic entropy” and based on representation-theoretic techniques. One crucial difference is that instead of considering unitary representations of a group $\Gamma$, we must consider $\ast$-representations of algebraic crossed products of $L^\infty$ spaces by $/Gamma$.

We classify $C^\ast$ near-group categories by using Vaughan Jones theory of subfactors and the Cuntz algebra endomorphisms. Our results show that there is a sharp contrast between two essentially different cases, integral and irrational cases. When the dimension of the unique non-invertible object is an integer, we obtain a complete classification list, and it turns out that such categories are always group theoretical. When it is irrational, we obtain explicit polynomial equations whose solutions completely classify the $C^\ast$ near-group categories in this class.

We state a conjecture for the formulas of the depth 4 low-weight rotational eigenvectors and their corresponding eigenvalues for the $3^G$ subfactor planar algebras. We prove the conjecture in the case when $|G|$ is odd. To do so, we find an action of $G$ on the reduced subfactor planar algebra at $f^(2)$, which is obtained from shading the planar algebra of the even half. We also show that this reduced subfactor planar algebra is a Yang-Baxter planar algebra.

I present a method of calculating the coefficients appearing in the Jones-Wenzl projections in the Temperley-Lieb algebras. It essentially repeats the approach of Frenkel and Khovanov in [4] published in 1997. I wrote this note mid-2002, not knowing about their work, but then set it aside upon discovering their article.

Recently I decided to dust it off and place it on the arXiv — hoping the self-contained and detailed proof I give here may be useful.

The proof is based upon a simplification of the Wenzl recurrence relation. I give an example calculation, and compare this method to the formula announced by Ocneanu [13] and partially proved by Reznikoff [15]. I also describe certain moves on diagrams which modify their coefficients in a simple way.

In an attempt to find analogues, for higher relative commutants, of the Bose-Mesner algebra structure on the second relative commutant of a spin-model subfactor, we find that there do indeed exist other algebra structures on the higher relative commutants of any subfactor planar algebra which are induced by the action of tangles; in fact, we show they can only arise in the obvious fashion.

The Bianchi I metric describes a homogeneous, but anisotropic universe and is commonly used to fit cosmological data. A fit to the angular distribution of 740 supernovae of type Ia with measured redshift and apparent luminosity is presented. It contains an intriguing, yet non-significant signal of a preferred direction in the sky. The Large Synoptic Survey Telescope being built in Chile should measure some 500 000 supernovae within the next 20 years and verify or falsify this signal.

Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for $# \mathrm{Hom}(\pi_1(S),G)$ in terms of the Euler characteristic of $S$ and the dimensions of the irreducible representations of $G$. A similar formula in the nonorientable case was proved by Frobenius and Schur [FS06]. Both of these proofs use character theory and an explicit presentation for $\pi_1$. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory.

There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to Legendrian knots, cluster algebras, and planar algebras.

We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don’t cross, until you eventually get a linear combination of links with no crossings, which you turn into a polynomial. The other uses quantum groups: you construct a functor from a topological category to some category of representations in such a way that (directed framed) links get sent to endomorphisms of the trivial representation, which are just rational functions. Certain instances of these two constructions give rise to essentially the same invariants, but when one carefully matches them there is a minus sign that seems out of place. We discuss exactly how the constructions match up in the case of the Jones polynomial, and where the minus sign comes from. On the quantum group side, one is led to use a non-standard ribbon element, which then allows one to consider a larger topological category.