In this short lecture, axioms for scale covariant and for conformal field theory in two dimensions are presented, some of their consequences are explored, and the connection to $\mathbb{Z}$-graded vertex algebras and to vertex operator algebras is made.

We review the definition of hypergroups by Sunder, and we associate a hypergroup to a type III subfactor $N \subset M$ of finite index, whose canonical endomorphism $\gamma \in \mathrm{End}(M)$ is multiplicity-free. It is realized by positive maps of $M$ that have $N$ as fixed points. If the depth is $ \gt 2$, this hypergroup is different from the hypergroup associated with the fusion algebra of $M$-$M$ bimodules that was Sunder's original motivation to introduce hypergroups.

We explain how the present hypergroup, associated with a suitable subfactor, controls the composition of transparent boundary conditions between two isomorphic quantum field theories, and that this generalizes to a hypergroupoid of boundary conditions between different quantum field theories sharing a common subtheory.

This note is an announcement of the paper [BC16]. We derive a Laurent series expansion in $d$ for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra $\mathrm{TL}_k(d)$, converging for all complex loop parameters $d$ with $|d| \gt 2\cos\big(\frac{\pi}{k+1}\big)$. The coefficients appearing in each Laurent expansion are shown to have a natural combinatorial interpretation and their sign is explicitly understood. As an application, we solve a series of questions raised by Jones and improve substantially our understanding of the Jones Wenzl projection.

In their study of almost group representations, Manuilov and Mishchenko introduced and investigated the notion of asymptotic stability of a finitely presented discrete group. In this paper we establish connections between connectivity of amenable groups and asymptotic stability and exhibit new classes of asymptotically stable groups. In particular, we show that if $G$ is an amenable and connective discrete group whose classifying space $BG$ is homotopic to a finite simplicial complex, then $G$ is asymptotically stable.

It is known that the category of representations of a rational vertex operator algebra (RVOA) is a modular tensor category. An interesting and important question is the converse, reconstruction: given a modular tensor category $C$, is there a RVOA whose category of representations is $C$? In this paper we explain how, given $C$, to recover the affine algebra and lattice vertex operator subalgebras possible inside a RVOA realizing $C$. We apply this to the most celebrated 'exotic' modular tensor category: that associated to the Haagerup subfactor. Whether this can be done consistently is a highly nontrivial test of the existence of the still-hypothetical Haagerup VOA.

We show that any type $\mathrm{III}_1$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $\ast$-subalgebra that is the range of a normal conditional expectation. We also investigate stability properties of CBP under finite index extensions/restrictions of type $\mathrm{III}_1$ factors.

We give an algorithm to compute the $K$-groups of the crossed product by the flip automorphism for a nuclear C$^*$-algebra satisfying the UCT.

We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product multiplicative unitary have the same Hilbert space representations. We also show that the multiplicative unitaries associated to two regular objects for the same tensor category are equivalent and hence generate isomorphic $\mathrm{C}^*$-quantum groups. In particular, a $\mathrm{C}^*$-quantum group is determined uniquely by its tensor category of representations on Hilbert space, and any functor between representation categories that does not change the underlying Hilbert spaces comes from a morphism of $\mathrm{C}^*$-quantum groups.

Recently, the classification problem of gapped Hamiltonians attracts a lot of attentions. We consider this problem for a class of Hamiltonians on quantum spin chains. We review the motivation of the problem and the background, and explain our recent result, about bulk-classification of frustration-free Hamiltonians.

We study actions of countable discrete amenable groups on unital separable simple nuclear $\mathcal{Z}$-absorbing $\mathrm{C}^*$-algebras. Under a certain assumption on tracial states, which is automatically satisfied in the case of a unique tracial state, the crossed product is shown to absorb the Jiang-Su algebra $\mathcal{Z}$ tensorially.

Some recent results on KMS states on chiral components of two-dimensional conformal quantum field theories are reviewed. A chiral component is realized as a conformal net of von Neumann algebras on a circle, and there are two natural choices of dynamics: rotations and translations.

For rotations, the natural choice of the algebra is the universal $C^*$-algebra. We classify KMS states on a large class of conformal nets by their superselection sectors. They can be decomposed into Gibbs states with respect to the conformal Hamiltonian.

For translations, one can consider the quasilocal $C^*$-algebra and we construct a distinguished geometric KMS state on it, which results from diffeomorphism covariance. We prove that this geometric KMS state is the only KMS state on a completely rational net. For some non-rational nets, we present various different KMS states.