My Research

J.D. Phillips

 

 

Problems worthy

of attack

prove their worth

by fighting back

Piet Hein

 

 

For a list of my publications, please see my curriculum vitae. Most of my recent papers can be downloaded at the Front for the Mathematics ArXiv. Here are the Math Reviews of my papers on MathSciNet (note: a subscription is needed to access this page). For a link to some supplementary items (mostly Prover9 and Mace4 files) to a few of my papers, click here.

Most of my research is in quasigroup and loop theory. Quasigroups are "nonassociative groups." More precisely, a quasigroup is a set  together with three binary operations *, \, and /satisfying the following identities: x \ (x * y) = x * (x \ y) = y, and (x / y) * y = (x * y) / y  = x. A loop is a quasigroup with a 2-sided neutral element: x * 1 = 1 * x = x.

The theories, then, of quasigroups and loops are generalizations of the theory of groups. This generalization takes two main forms. First, since all groups can also be viewed as quasigroups, the theory of quasigroups includes the theory of groups as an appealing subtheory. This forces the theory into a leaner elegance and greatly simplifies certain notions from group theory. Applying techniques from quasigroup theory to groups viewed as quasigroups allows the mathematician to address questions whose answers are usually hidden under the additional assumptions of group theory. Second, the techniques involved in studying quasigroups and loops closely resemble the techniques of group theory. In fact, the theory of quasigroups is intimately related to the theory of certain groups associated with quasigroups. And so a knowledge of traditional group theory is required to study quasigroups and loops. This explains, for instance, the rich history of involvement by some of the world's most eminent group theorists in developing the theory of quasigroups and loops.
  
From another perspective, quasigroups and loops can be viewed as universal algebras, as above. As such, quasigroups lend themselves naturally to the full spectrum of techniques available to the universal algebraist, especially the powerful tools and techniques of computational mathematics. In fact, these techniques have proven to be much more effective in dealing with quasigroups than in dealing with groups, particularly in the rich area of representation theory. And so by virtue of their intimate connection with groups, quasigroups are natural vehicles for the group theorist wishing to use universal algebra and computational mathematics in his or her investigations.

My work exploits the powerful computational tools—for instance, automated theorem provers like Prover9, as well as finite model builders like Mace4—that are now widely available to algebraists and that are transforming some areas of algebra, especially those involving inquiries about structures that can be defined equationally, for instance, quasigroups and loops, as above. Ultimately, I enjoy working on problems involving any algebras that can be defined equationally, e.g., medial groupoids, digroups, alternative rings, etc.

One of the things I enjoy most about mathematics research is collaborating with interesting people. Here are links to the websites of some of the mathematicians I've been lucky enough to work with:

Michael Aschbacher.

 

Orin Chein.


 

Aleš Drápal.

 

Tuval Foguel.


 

Edgar Goodaire.

 

Tomáš Kepka.

 

Michael Kinyon.

 

Aleksandar Krapež.

 

Ken Kunen.


 

Bill McCune.

 

Gábor Nagy.


 

Markku Niemenmaa.


 

Jonathan Smith.

David Stanovský.

 

Joseph Urban.

 

Bob Veroff.

 

Petr Vojtěchovský.