What does a nonabelian group sound like?
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 IEProps GitHub repository contains a draft of the article Interval enforceable properties of finite groups, to appear in Communications in Algebra.
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.
In the Overalgebras GitHub repo resides the article Expansions of finite algebras and their congruence lattices, Algebra Universalis, 69, 2013. Also included is the GAP software I wrote for constructing new finite algebras by extending and expanding transitive G-sets, as described in the paper.
I learned about Diaconescu's Theorem from Andrej Bauer's lecture.
This post describes the proof as it was presented by Bauer. These notes are rough and intended for my own reference. Please see François Dorais' blog post for a nice discussion of this topic.