WJD Open Notebook

This post includes some links to notes about various topics related (sometimes tangentially) to programming languages. These are very rough, often unfinished, and not really meant for public consumption. However, I don't care if others look at them. There is no personal or confidential information on these pages.

The GroupSound project is about harmonic analysis on finite groups. Classical dsp filtering algorithms can be implemented as operations involving functions (e.g., audio signals) defined on a finite group. That is, the group serves as the domain, or "index set," of the functions. In this project, we explore the idea of using the finite group as an adjustable parameter of a digital audio filter.

The IEPost GitHub repository contains a draft of the article Interval enforceable properties of finite groups, to appear in Communications in Algebra.

🔗 Equivalence of algebraic structures

Consider two finite algebras that are the same except for a relabeling of the operations. Our intuition tells us that these are simply different names for the same mathematical structure. Formally, however, not only are these distinct mathematical objects, but also they are nonisomorphic.

Isotopy is another well known notion of "sameness" that is broader than isomorphism. However, I recently proved a result that shows why isotopy also fails to fully capture algebraic equivalence.