## Culminating Workshop

Saturday, July 24th

09:00-09:30 am | Welcome and Coffee
09:30-10:20 am | Jack Spielberg: Discrete dynamical systems spontaneously generated by a semigroup (Notes)

A traditional discrete dynamical system is given by a discrete group, along with a (locally) compact Hausdorff space on which the group acts by homeomorphisms. Remarkably, if we start with a semigroup instead of a group, a space X appears out of thin air. The semigroup then defines a dynamical system in the form of a(n étale) groupoid with unit space X. I will describe this process of obtaining a space from a semigroup, and will illustrate it with several examples. Groupoids will be peddled lightly.

10:30-11:00 am | Coffee Break
11:00-11:50 am | Ben Hayes: Applications of random matrices and free probability to the structure of von Neumann algebras: 1-bounded entropy (Notes)

Building off Hartglass' talks, I will discuss the notion of 1-bounded entropy, an invariant for von Neumann algebras which arises out of an attempt to "count" how many matricial approximations a von Neumann algebras. I will discuss applications to structural results of von Neumann algebras, particularly for free group factors. Time permitting, I may discuss my recent work with Jekel-Kunnawalkam Elayavalli on matricial approximations of Property (T) algebras. No prior knowledge outside of the expository talks will be assumed (particularly of Property (T)).

12:00-02:00 pm | Lunch
02:00-02:50 pm | Sarah Browne: The Baum-Connes conjecture

In this talk, we will see the Baum-Connes conjecture without coefficients. The conjecture states whether, for a group $G$, a particular map is an isomorphism, which will be defined during the talk. We will give an overview of the many known results for which the conjecture holds. We will discuss some joint work with Sara Azzali, Maria Paula Gomez Aparicio, Lauren Ruth and Hang Wang in which we make computations to show explicitly the conjecture holds for the pure braid groups. Dependent on time, this may be limited to the right hand side of the map, which is defined using the reduced group C*-algebra of $G$.

03:00-03:30 pm | Coffee Break
03:30-04:20 pm | Emily Peters: Category theory in Operator Algebras: an example (Notes)

Category theory is a growing monster, which eats up other fields of mathematics but is also being changed by the fields it has eaten. In this talk I will touch on the questions of (1) what is category theory? (2) What can categories do for operator algebras? and (3) what can operator algebras do for categories? All these questions will be investigated through the example of bimodule categories for von Neumann algebras.

04:30-05:20 pm | Adam Fuller: C*-envelopes of operator algebras (Notes)

Let $A$ be a nonselfadjoint algebra of operators. Different copies of $A$ can lead to $A$ generating different C$^*$-algebras, depending on where $A$ is represented. There is, however, a unique smallest C$^*$-algebra which $A$ generates: the C$^*$-envelope. In this talk we will give an overview of C$^*$-envelopes. We will begin with the functional analytic roots of the topic in Choquet theory; look at Arveson's seminal work in noncommutative Choquet theory; and Hamana's approach using injective envelopes. Throughout we will provide several examples relating to familiar C$^*$-algebras, e.g. abelian C$^*$-algebras; the Cuntz algebra; and crossed product C$^*$-algebras.

06:30-08:30 pm | Conference Dinner

Sunday, July 25th

09:00-09:50 am | Sarah Plosker: Operator Algebra Techniques in Quantum Information Theory (Notes)

In quantum information theory (QIT), we study positive, trace-one trace-class operators (called quantum states) and linear transformations between quantum states (completely positive, trace-preserving linear maps, called quantum channels). In this lecture, I will introduce these and other fundamental objects and show how a number of important problems in QIT can be approached through an operator algebra framework, highlighting some familiar tools and techniques, as well as my and others' results in the area.

10:00-10:30 am | Coffee Break

Abstract

10:30-11:20 am | Samantha Pilgrim: Coarse Geometry and Operator Algebras (Notes)

Coarse geometry is the study of metric spaces up to an equivalence which preserves only 'large-scale' structure. This talk will introduce the basic theory of coarse geometry and coarse properties (e.g. asymptotic dimension, property A, coarse embeddings in Hilbert space), discuss some relations between coarse geometry and C*-algebras, and describe some of the presenter’s work on dynamic asymptotic dimension. Assumed prerequisites are minimal, though there may be quite a few new concepts introduced.

11:30-12:20 pm | David Jekel: The metric space of non-commutative laws is not separable (Notes)

This talk connects non-commutative probability theory with the study of group C* and von Neumann algebras. We explain how the non-commutative law (probability distribution) of a $d$-tuple $(U_1,\dots,U_d)$ of unitaries in a tracial von Neumann algebra is described by a character on the free group $\mathbb{F}_d$, and this serves as a non-commutative analog of probability measures on the torus $\mathbb{T}^d$. Following Biane and Voiculescu, we define the Wasserstein distance between two characters $\mu$ and $\nu$ as the infimum of the $L^2$ distance of $(U_1,\dots,U_d)$ and $(V_1,\dots,V_d)$, where $\mu$ is the distribution of $U$ and $\nu$ is the distribution of $V$. In stark contrast to classical probability theory, we show that the space of characters on $\mathbb{F}_d$ is not separable with respect to the Wasserstein distance, which is one indication of how hard it is to classify von Neumann algebras. This is a consequence of a group-theoretic theorem due to Olshanskii (inspired by Gromov and adapted by Ozawa) that there is a Property (T) group $G$ that has uncountably many non-isomorphic quotient groups $(G_\alpha)_{\alpha \in I}$. We show that these distinct quotients produce uncountably many non-commutative laws that are all separated in Wasserstein distance by some fixed $\epsilon > 0$. Based on joint work with Wilfrid Gangbo, Kyeongsik Nam, and Dimitri Shlyakhtenko.