A lot is known about irreducible hyperfinite subfactors with small index and finite depth, the same cannot be said about the infinite depth case. We will discuss a new approach to classifying infinite depth subfactors coming from commuting squares. We will also use this approach to construct a new irreducible hyperfinite infinite depth subfactor with index $\frac{5+\sqrt{17}}{2}$.

K-theory is the generalized cohomology theory that has proven incredibly effective at classifying C*- algebras. In this talk, we discuss its dual homology theory, K-homology. K-homology and K-theory admit a bilinear pairing to the integers, called the index pairing, which allows one to learn about K-theory using K-homology and vice versa. In general, this pairing is very difficult to compute; however, if a C*algebra admits a dense subalgebra on which its K-homology is finitely Summable, Connes has equipped us with a more computible formula. Given a smooth manifold, $M$, of dimension $n$, the K-homology of $C_0(M)$ is finitely summable on $C_0^\infty(M)$ for "$p>n$," and so the degree summability also sees the dimension of the manifold. However, beyond the case of a manifold, few classes of C* algebrss are known to have finitely summable K-homology, and far fewer are known not too. In this talk, we will define K-homology, discuss summability, and progress in answering this question more generally.

The Effros-Shen algebra corresponding to an irrational number $\theta$ can be described by an inductive sequence of finite-dimensional subalgebras, where the connecting maps encode the terms of the continued fraction expansion of $\theta$. In recent work, Mitscher and Spielberg present the Effros-Shen algebra as the $C^*$-algebra of a category of paths determined by the continued fraction expansion of $\theta$. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. Drawing on a construction by Christensen and Ivan, we use this inductive limit structure to define a spectral triple. This is joint work with Konrad Aguilar and Jack Spielberg.

Let $G\stackrel{\alpha}{\curvearrowright}(M,\tau)$ be a trace-preserving action of a finite group $G$ on a tracial von Neumann algebra. Suppose that $A \subset M$ is a finitely generated unital $*$-subalgebra which is globally invariant under $\alpha$. We give a formula relating the von Neumann dimension of the space of derivations on $A$ valued on its coarse bimodule to the von Neumann dimension of the space of derivations on $A \rtimes_\alpha G$ valued on its coarse bimodule, which is reminiscent of Schreier's formula for finite index subgroups of free groups. This formula induces a formula for the free Stein dimension (defined by Charlesworth and Nelson), $\dim \text{Der}_c(A,\tau)$ (defined by Shlyakhtenko), and $\Delta$ (defined by Connes and Shlyakhtenko). For the latter, we assume that $G$ is abelian group. Using the formula for $\Delta$, we recover recent results of Shlyakhtenko on the microstates free entropy dimension.

Matui's HK-conjecture proposes an in-principle computation of the K-theory of the reduced C*-algebra of a (nice enough) groupoid in terms of the homology of the groupoid. While there are a number of positive results, the first counterexample is due to Scarparo in the case where the groupoid is essentially principal. Deeley gave the first principal counterexample using an action groupoid built from a flat manifold and an expansive self-cover. In this case, the invariants in the HK-conjecture are computed from the corresponding invariants of the flat manifold, and HK is a direct analogue of the following phenomenon in algebraic topology. For a CW-complex of dimension at most 3, the Atiyah–Hirzebruch spectral sequence computes K-theory explicitly as a direct sum of cohomology groups. However, this explicit computation fails in general for spaces of dimension 4 and greater. While Deeley's original counterexample uses a flat manifold of dimension at least 9, I will present a counterexample coming from a flat manifold of dimension 4. As a corollary, this shows that HK fails for groupoids with dynamic asymptotic dimension at least 4.

Topologically ordered quantum spin systems have become an area of great interest, in part because the ground state space for these systems is a quantum error correcting code. This was reflected in the axiomatization of topological order given by Bravyi, Hasting, and Michalakis. In this talk, we will describe new local topological order axioms given in recent joint work with Corey Jones, Pieter Naaijkens, and David Penneys. These axioms strengthen those of Bravyi, Hastings, and Michalakis, and they give rise to a 1-dimensional net of boundary algebras. We then provide an example satisfying these axioms, namely Kitaev’s quantum double model. We compute the boundary algebras for this model and show that they give nets of algebras either corresponding to Hilb(G) or Rep(G) depending on whether the boundary is rough or smooth. In either case, we have a canonical state on the boundary algebra which is tracial. This is joint work with Chian Yeong Chuah, David Penneys, and Shuqi Wei.