Abstract:
I will discuss a way to study von Neumann algebras via quantifying ''how many'' finite-dimensional approximations they have. This line of study goes back in the 90's with work of Voiculescu, who used it to solve many long-standing open problems. My first lecture will start with a brief overview of this area of operator algebras. I will then transition to some recent developments, highlighting connections to random matrices as well as group theory.

Abstract:
Abstract of the 1st lecture:
Recently, Davidson and I introduced a new framework for noncommutative convexity and noncommutative function theory, along with a corresponding noncommutative Choquet theory that generalizes much of classical Choquet theory. A key point is that the category of compact noncommutative sets is dual to the category of operator systems. In recent work with Kim and Manor, we extended this duality to the category of (potentially) non-unital operator systems in the sense of Werner and Connes-van Suijlekom. I will motivate and discuss these developments, as well as some applications to C*-algebras and noncommutative dynamics.

Abstract of the 2nd lecture:
I will discuss new descriptions of some universal flows associated to a discrete group, obtained using what we view as a kind of ''topological Furstenberg correspondence.''The descriptions are algebraic and relatively concrete, involving subsets of the group satisfying a higher order notion of syndeticity. We utilize them to establish new necessary and sufficient conditions for strong amenability and amenability. Throughout, I will discuss connections to operator algebras. This is joint work with Sven Raum and Guy Salomon.

Abstract:
We will present applications of boundary actions to the investigation of the ideal and tracial structure of C*-algebras generated by unitary representations. In particular, for quasi-regular representations associated to stabilizers of boundary actions we have a complete understanding of the trace space of such C*-algebras and when they are simple. We will also present some examples coming from Thompson's groups and topological full groups which do not fit yet in the general theory.

Abstract:
We define an invariant for an inclusion of operator systems based on Pimsner and Popa's formulation for the Jones index of a subfactor. We will discuss how this index is connected with certain invariants studied in quantum information theory. This is joint work with Roy Araiza and Colton Griffin.

Abstract:
A well-popularized problem in C*-algebra theory is the classification of C*-algebras up to isomorphism. Through an intertwining argument, this is generally achieved through understanding and classifying *-homomorphisms between C*-algebras - a problem that is also natural in its own right. In joint work with J. Carrión, J. Gabe, C. Schafhauser, and S. White, we have classified *-homomorphisms using minimal (and abstract) hypotheses on the domain A and codomain B: A can be any separable exact C*-algebra satisfying the UCT, and B can be any separably Z-stable C*-algebra with compact tracial state space and strict comparison with strict comparison with respect to traces. I will discuss this result in the context of classification of C*-algebras, and some of the key ideas in our approach.