I’m very interested in modern approaches to mathematical texts. I’ve been using preTeXt, an xml derived typesetting language, to build textbooks and course notes. Most of these projects are works in progress, but I’m happy to share.

Supplements and code

Dynamics of low-degree rational skew-products on \(\mathbb{T}^2\)

This is a collection of pictures and code exploring the dynamics of a simple case of rational inner functions iterating the 2-torus. You can find a library of examples and the code used to produce the pictures here.

Open source texts

Hilbert Spaces: A sequel to linear algebra

This text is inspired by and partly based on the outstanding “Introduction to Hilbert Spaces” by N. J. Young, which is sadly out of print. This book fills a niche that is somewhat unique: a course on Hilbert space theory that is rigorous but follows from a strong undergraduate curriculum, rather than a more typical second year graduate course in functional analysis. The long-term goal is for this text to serve as a follow-up to Sheldon Axler’s Linear Algebra Done Right, which lays a lot of the linear algebraic groundwork for operator theory.

This is very much a work in progress. Current topics include: inner products, norms and metric spaces, closed linear subspaces, limits in Hilbert space, \(\ell^2\), \(L^2\), and a bit of Hilbert space geometry. Future topics will include Fourier series and an introduction to operators.

The text can be found here, and the source, which is compiled from preTeXt, is here.

Mathematical Cryptography

These are course notes from a class I taught to a group of computer science students that asked for a more rigorous introduction to the theory of cryptography. The notes loosely follow the structure of “An Introduction to Mathematical Cryptography”, by Hoffstein, Pipher, and Silverman. The book is an excellent text, but the mathematics were a bit proof heavy for the course I ran. I tried to split the difference between mathematical rigor and implementation. I taught this course in python, and the text has interactive code demonstrations that execute in the text using SageMath’s CoCalc server.

The text can be found here and the source is here.

Numerical Analysis

These are course notes I prepared for an upper division course in numerical analysis. The audience was largely engineers, so I focused on implementation of various ideas. The language I used in this course was Octave, as I didn’t have a site license for Matlab, which was my preferred language. On the plus side, Octave can be rendered in executable blocks in preTeXt, so choosing Octave was a net positive. Topics include root-finding, polynomial approximation, interpolation, numerical integration, and an introduction to Fourier series.

The text can be found here and the source is here.

Linear analysis

These are in progress course notes for a second course on linear analysis. Topics include the Laplace and Fourier transforms, series solutions to differential equations, Fourier, Legendre, and Frobenius series, and the linear algebraic view of functions in special spaces.

These notes follow Annin and Goode’s Differential Equations and Linear Algebra and Osgoode’s Lecture Notes on the Fourier Transform, with various interjections from functional analysis.

The text is here.

Differential equations

This is an online text that I wrote with Phanuel Mariano of Union College, based on a set of lecture notes that Phanuel wrote over several semesters. The text includes extensive exercises and covers a typical semester of a first course in differential equations, following the general path of Boyce and DiPrima. The text source is available for adapatation. The text is located here.

Notes on Complex Analysis

This is a collection of notes I wrote to support a year-long course in undergraduate complex analysis. It is very much a work in progress. The most complete feature is the final section, which is a proof of the prime number theorem, synthesized from a variety of sources stemming from Newman’s short, complex analytic proof, and providing an explication accessiblw to an undergraduate. I could not resist a brief foray into basic measure theory to pick up the dominated convergence theorem. The text is located here.

Operator Theory Community

OPT-IN

The Operator Theory Information Network (OPT-IN) is an in-development community hub for mathematicians and grad students interested in operator theory. It was inspired by the Operator Algebra Searchable Information Site (OASIS) and the Internet Analysis Seminar. I designed the website and the directory. The initiative is loosely led by me, Kelly Bickel, and J. E. Pascoe.

OTTER

Operator Theory Talks for Early Researchers (OTTER) is a related project aimed specifically at mathematical discourse and professional guidance for graduate students and postdocs. I’m involved in the project as a mentor and speaker.