This is a continuation of Hui Tan's talk on joint work studying the structure and classification of q-Araki-Woods factors. I will focus on the proofs of the main results: the dichotomy for subalgebras in continuous cores underlying strong solidity, and the failure of biexactness for q-Araki-Woods factors with infinite-dimensional representations via norm estimates from Nou and Hiai.

Let $\mathrm{Mat}_{n\times m}(\mathbb{C})$ be the affine space of $n\times m$ complex matrices, and let $\mathcal{Z}_{n,m,r}$ (resp. $\mathcal{UZ}_{n,m,r}$) be the locus in $\mathrm{Mat}_{n\times m}(\mathbb{C})$ corresponding to rook placements with exactly (resp. at least) $r$ rooks. The orbit harmonics method yields two quotient rings $R(\mathcal{Z}_{n,m,r})$ and $R(\mathcal{UZ}_{n,m,r})$, where both rings have the additional structures of $\mathfrak{S}_n\times\mathfrak{S}_m$-modules. We find the generators of their defining ideals and compute their graded Frobenius image. Furthermore, we give a nontrivial generalization of Viennot's shadow line avatar of the Schensted correspondence to rook placements in $\mathcal{UZ}_{n,m,r}$. This generalization is used to determine the standard monomial basis of $R(\mathcal{UZ}_{n,m,r})$ with respect to a diagonal term order. Joint with Jasper (Moxuan) Liu.

Vaughan Jones introduced an index for inclusions of certain von Neumann algebra in the 1980's and proved that it is surpisingly rigid. This rigidity is due to a rich combinatorial structure that is inherent to the representation theory of a subfactor with finite index. Subfactor representations reveal interesting unitary tensor categories, or quantum symmetries, whose algebras of intertwiners always contain the Temperley-Lieb algebras and, if an intermediate subfactor is present, the Fuss-Catalan algebras of Jones and myself. The case of two intermediate subfactors is much more involved and not much progress had been made since the late 1990's.

I will discuss recent work with Junhwi Lim in which we determine the quantum symmetries of a subfactor when two intermediate subfactors occur, and the four algebras form a cocommuting square. These new symmetries turn out to be related to partition algebras and Bell numbers.

The Wasserstein metric over a metric space X is an optimal-transport based distance on the set of probability measures on X. Metric spaces for which the optimal transport problem is "easiest" to solve are trees,  in the sense that the Wasserstein metric on trees isometrically embeds into $L^1$. Property A is a coarse invariant of metric spaces introduced by Yu as an approach to solving the coarse Baum-Connes conjecture. We prove a new characterization of bounded degree graphs X with Property A as precisely those that are coarsely equivalent to another space Y whose Wasserstein metric admits a biLipschitz embedding into $L^1$. Applications to group actions on Banach spaces will be discussed. Based on joint work with Tianyi Zheng and Ignacio Vergara.

We will first recall some basic results on harmonic functions: the mean value property, the maximum principle, the Liouville theorem, the Harnack inequality, the Bocher theorem, the capacity and removable singularities. We will then present a number of more recent results on some conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville theorems, Harnack inequalities, and Bocher theorems.

In this talk, I will begin by introducing graphons, which are infinite-size limits of adjacency matrices of sequences of growing graphs. I will then define graph reaction-diffusion (RD) equations, which are systems of differential equations that are defined on the nodes of a graph. For a sequence of growing graphs that converges to a graphon, the solutions of the sequence of graph RD equations also converge. The limiting solution solves a nonlocal differential equation that we call a graphon RD equation. Furthermore, the graph RD equation is related to a stochastic birth-death process on graphs. I will show that this birth-death process converges to the graphon RD equation via a hydrodynamic limit.