Theory & Research

A Pile of Shifts II: Structure and K-Theory
With S. Hebert, S. Klimek, and M. McBride. We study the structure of C*-algebras generated by weighted shifts on s-adic trees and compute their K-theory.

A Higher Order Local Mesh Method for Approximating 1-Laplacians on Unknown Manifolds
With J. Harlim. We introduce a higher-order numerical method for approximating differential operators on vector fields, given only point-cloud data sampled from an unknown manifold.

Noncommutative Geometry of Hensel–Steinitz Algebras
With S. Hebert, S. Klimek, and M. McBride. We analyze derivations and K-theory of smooth subalgebras of Hensel–Steinitz algebras, crossed products arising from multiplication maps on p-adic integers.

Crossed Product C*-Algebras Associated with p-adic Multiplication
With S. Hebert, S. Klimek, and M. McBride. We investigate C*-algebras associated with multiplication in the p-adic integers and compute their K-theory.

Radial Basis Approximation of Tensor Fields on Manifolds: From Operator Estimation to Manifold Learning
With J. Harlim and S. W. Jiang. We use radial basis functions and local tangent space approximation to learn many different operators associated to an unknown manifold from point-cloud data.

A Note on Quantum Odometers
With S. Klimek and M. McBride. We explore aspects of noncommutative geometry of smooth subalgebras within Bunce–Deddens–Toeplitz algebras.

Noncommutative Geometry of the Quantum Disk
With S. Klimek and M. McBride. We examine derivations in a smooth subalgebra of the Toeplitz algebra, viewed as the noncommutative analog of the unit disk.

Aspects of Noncommutative Geometry of Bunce–Deddens Algebras
With S. Klimek and M. McBride. We define smooth subalgebras of Bunce–Deddens C*-algebras and study their derivations, K-theory, and K-homology.

Spectral Convergence of Symmetrized Graph Laplacian on Manifolds with Boundary
With J. Harlim. We prove spectral convergence rates of graph Laplacians toward the Laplace–Beltrami operator on manifolds with boundary, generalizing the theory behind Diffusion Maps.

A Note on Spectral Triples on the Quantum Disk
With S. Klimek and M. McBride. We construct spectral triples on the quantum disk using covariant derivations.

Talk Slides from SIAM Conference on Mathematics of Data Science
In the Fall of 2022, I gave a talk on my research at the SIAM Conference on Mathematics of Data Science (MDS22). Attached are the slides I prepared for my talk.

Poster from FoCM 2023
I was an invited poster presenter at the ninth conference of the Foundations of Computational Mathematics (FoCM) Society in Paris during the Summer of 2023. Attached is a .pdf of my poster.

Solutions to Elements of Statistical Learning
A key part of my Data Science journey has been solving exercises from the seminal textbook ESL. Here are my solutions to the exercises from this excellent book.