The alt-comm repository contains a note giving an alternate decription of the (non-modular) commutator that yields a polynomial time algorithm for computing it. This is inspired by the alternate description of the commutator given by Kearnes in [1].

open notebook

William DeMeo

William DeMeo

I am a postdoc in the Math Department at University of Hawaii (2016–2017). In fall 2017 I will move to University of Colorado, Boulder.

I have worked at Iowa State University (2014–2016) and University of South Carolina (2012–2014).

My primary research area is universal algebra; current projects focus on lattice theory, computational complexity, and universal algebraic approaches to constraint satisfaction problems. Other research interests include logic, computability theory, type theory, category theory, functional programming, dependent types, and proof-carrying code.

Most of my research papers are posted on my Math arXiv page or my CS arXiv page. A more comprehensive collection of my work resides in my Github repositories.

The alt-comm repository contains a note giving an alternate decription of the (non-modular) commutator that yields a polynomial time algorithm for computing it. This is inspired by the alternate description of the commutator given by Kearnes in [1].

I started a GitHub repository called UniversalAlgebra/Conferences in an effort to keep up with the vast number of meetings in these areas and help make summer travel plans accordingly.

The UniversalAlgebra/Conferences repository is simply a list of links to the conferences listed.

The repository par-alg-rep contains some notes describing a few results that we worked out in the fall of 2016, while I was a postdoc at University of Hawaii, although the main result—a straight-forward proof of the fact that every finite lattice is the congruence lattice of a finite partial algebra—was discovered during a visit to Chapman University in October 2016.

It is not hard to see that 3-SAT
reduces to the problem of deciding whether all coatoms in a certain partition
lattice are contained in the union of a collection of certain principal filters.
Therefore, the latter problem, which we will call the
*covered coatoms problem* (CCP), is NP-complete.
In this post we simply define CCP. Later we check that 3-SAT reduces to CCP,
and then develop some ideas about constructing a feasible algorithm to solve CCP.

The questions below appeared on an online test administered by Jane Street Capital Management to assess whether a person is worthy of a phone interview.

This is a markdown version of the tutorial Learn You an Agda and Achieve Enlightenment! by Liam O’Connor-Davis.

I made this version for my own reference, while working through the tutorial, making some revisions and additions, and a few corrections. You may prefer the original.

*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.

Suppose the lattice shown below is a congruence lattice of an algebra.

**Claim:**
If the three $\alpha_i$’s pairwise permute, then all pairs in the lattice permute.

I started a GitHub repository called TypeFunc in an effort to get my head around the massive amount of online resources for learning about type theory, functional programming, category theory, $\lambda$-calculus, and connections between topology and computing.

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.