Posts by Collection

Ordinal graphs and their \(C^*\)-algebras

Submitted to Mathematica Scandinavica, 2025

We consider Cuntz-Krieger algebras for a class of left-cancellative categories with a fibration into the ordinals.

Principal Actions on Topological Quivers and Associated Operator Dynamics

Submitted to Documenta Mathematica, 2025

We study topological quivers \(Q\) admitting a free and proper action by a locally compact group \(G\) together with their associated \(C^*\)-algebras.

Topological Quiver \(C^*\)-Algebras & Group Actions

Published: February 29, 2024

Topological quivers are a somewhat natural generalization of topological graphs, where we insist that edge space is additionally endowed with a family of Radon measures, indexed by the vertex space, satisfying some regularity conditions. In this series of talks, we aim to present the needed background material to understand what a topological quiver \(C^*\)-algebra is, as well as discuss some results we hope to generalize from Deaconu, Kumjian and Quigg in the topological graph setting. One of these main results being that a locally compact group acting freely and properly on a topological quiver induces a Morita equivalence between the reduced crossed product of the quiver \(C^*\)-algebra and the \(C^*\)-algebra of the corresponding ‘quotient quiver’.

How Quantum Mechanics Motivates \(C^*\)-algebras

Published: April 18, 2024

In this talk we introduce Quantum mechanics from first principles. We illustrate how Stone’s theorem and the Stone-von Neumann theorem bridge the gap between the canonical commutation relations for unbounded operators and the Weyl relations for unitary groups. Time permitting, we’ll recast the Stone-von Neumann theorem in terms of \(C^*\) crossed products and discuss how this motivates natural generalizations of the Stone-von Neumann theorem.

Topological Quiver \(C^*\)-Algebras & Group Actions

Published: June 06, 2024

Topological quivers are a somewhat natural generalization of topological graphs, where we insist that edge space is additionally endowed with a family of Radon measures, indexed by the vertex space, satisfying some regularity conditions. In this series of talks, we aim to present the needed background material to understand what a topological quiver \(C^*\)-algebra is, as well as discuss some results we hope to generalize from Deaconu, Kumjian and Quigg in the topological graph setting. One of these main results being that a locally compact group acting freely and properly on a topological quiver induces a Morita equivalence between the reduced crossed product of the quiver \(C^*\)-algebra and the \(C^*\)-algebra of the corresponding ‘quotient quiver’.

A Cuntz-Krieger Uniqueness Theorem for a Class of Left-Cancellative Categories

Published: August 29, 2024

We introduce a class of left-cancellative small categories for which there is a Cuntz-Krieger uniqueness theorem. The categories arise naturally as a generalization of directed graphs for which paths are allowed to have lengths which are ordinals. We will describe a collection of \(C^*\)-correspondences which generalize the usual correspondence for a directed graph. Under suitable conditions, these correspondences satisfy condition (S) of Eryüzlü and Tomforde, and we describe how a transfinite induction argument yields the uniqueness theorem.

Ordinal graphs and their \(C^*\)-algebras

Published: November 22, 2024

We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.

Ordinal graphs and their \(C^*\)-algebras

Published: December 04, 2024

We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.

Ordinal graphs and their \(C^*\)-algebras

Published: January 24, 2025

We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.

Ordinal graphs and their \(C^*\)-algebras

Published: April 12, 2025

We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.

Ordinal graphs and their \(C^*\)-algebras

Published: May 24, 2025

We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.

Ordinal graphs and their \(C^*\)-algebras

Published: May 26, 2025

We introduce a class of left-cancellative categories which generalizes the category of paths of a directed graph by allowing paths to have ordinal length. We use generators and relations to study the Cuntz-Krieger algebra defined by Spielberg. For each ordinal we construct an associated \(C^*\)-correspondence. Then we apply Eryüzlü and Tomforde’s condition (S) and obtain a Cuntz-Krieger uniqueness theorem for ordinal graphs.

Ideals, approximate units

Published: June 30, 2025

I lectured on ideals and approximate units in \(C^*\)-algebras for GOALS 2025, which was held virtually. Here is a recording. Unfortunately zoom decided to cover half of the screen with the participants.

Graph \(C^*\)-algebras

Published: July 10, 2025

I lectured on universal \(C^*\)-algebras and graph \(C^*\)-algebras. Here is a recording. I learned from last time not to trust zoom, and I used my own recording software this time.

\(C^*\)-correspondences for Ordinal Graphs

Published: August 28, 2025

There is a certain class of ordinal graphs whose \(C^*\)-algebras are always Cuntz-Pimsner algebras in a natural way. In these examples, homomorphisms from algebras of distinguished subcategories into the ordinal graph algebra are injective. We will discuss this construction and some of the interesting details of the proof that these algebras are Cuntz-Pimsner algebras.

MAT 266, Calculus II for Engineers

Undergraduate course, Arizona State University, SoMSS, 2021

This was my first experience as a TA. I ran weekly review sessions, during which students completed practice problems, and graded the students’ homework.