This page gives an overview of my research. Other sources of information about my work are my
Open notebook science
Recently I started an open notebook at williamdemeo.github.io where I record progress on problems and projects, and gather notes about topics that are new to me. Consequently, the posts are often rough and occasionally misguided. It is a notebook for doing research, not a blog.
- Isotopic algebras with nonisomorphic congruence lattices, Algebra Univers. (accepted) 2014.
- Interval enforceable properties of finite groups, Communications in Algebra (accepted) 2014.
- Expansions of finite algebras and their congruence lattices, Algebra Univers. 69 2013.
- Topics in nonabelian harmonic analysis and DSP applications, Proceedings
of the International Symposium on Musical Acoustics, Nara, Japan, 2004 (best paper award).
- Characterizing musical signals with Wigner-Ville interferences,
Proceedings of the International Computer Music Conference, Goteborg, Sweden, 2002.
- Approximating eigenvalues of large stochastic matrices, Proceedings of
the 8th Copper Mt. Conference on Iterative Methods, Colorado, USA, 1998.
- On a problem of Palfy and Saxl (in preparation).
- Congruence lattices of intransitive G-sets, with Ralph Freese (in preparation).
- Small congruence lattices, with Ralph Freese and Peter Jipsen (in preparation).
My student Matthew Corley won a Magellan Scholar grant to work with me on the GroupSound project. For this application I recently developed a map reduce algorithm for convolution over finite groups, which is implemented in a Sage worksheet—group convolution in just four lines of Python.
Below is a list of some open questions that interest me.
- Is every finite lattice isomorphic to the congruence lattice of a finite algebra? (related work)
- Does every finite lattice appear as an interval in the subgroup lattice of a finite group? (related work)
- Does every algebra with a finite dimensional modular congruence lattice have a unique factorization? (related work)
- The Jósson-Park Conjecture: Does every finitely generated variety with a finite residual bound have a finite equational basis?
- Find a general (non-group-theoretic) proof of the theorem of Oates and Powell: The variety generated by a finite group is finitely based.
- The Černý Conjecture: If an n-state automaton has a reset word, then it has one of length at most (n–1)². (related work)
- A question of Péter Pálfy and Jan Saxl: If a congruence lattice is isomorphic to Mn and there is a set of three or more atoms that pairwise permute, do all the atoms pairwise permute? (related work)
I learned about finite basis problems from George McNulty, who has some very nice papers and notes about these problems linked to his webpage. Ross Willard has some excellent notes on this topic as well. His overview of modern universal algebra is also highly recommended. An nice exposition of the Oates-Powell Theorem can be found in Chapter 5 of Hanna Neumann’s book, Varieties of Groups.
- Notes on permuting equivalence relations and permuting subgroups, May 2013.
- Dedekind’s Transposition Principle for lattices of equivalence relations, Nov 2012.
Some recent talks
- Small congruence lattices, BLAST Conference, Chapman University, Orange CA 8/8/2013.
- Dedekind’s Transposition Principle, permuting subgroups and equivalence relations, Zassenhaus Group Theory Conference, Asheville NC 5/24/2013.
- Dedekind’s Transposition Principle and isotopic algebras with nonisomorphic congruence lattices — AMS Sectional Meeting, Boulder CO, 4/13/2013.
- Syncronizing Automata and the Černý Conjecture — Graduate Algebra Seminar, Boulder CO, 4/11/2013.