A Problem of Pálfy and Saxl
Suppose the lattice shown below is a congruence lattice of an algebra.
Conjecture: If the three \(\alpha_i\)'s pairwise permute, then all pairs in the lattice permute.
2014¶
GroupSound
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.
IEProps
The IEProps GitHub repository contains a draft of the article Interval enforceable properties of finite groups, to appear in Communications in Algebra.
Isotopy
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.
Overalgebras
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.