My main research interest in nonassociative mathematics. It relies on techniques from algebra, combinatorics, low-dimensional topology, computational algebra and automated deduction. There are many open questions suitable for students at all levels and for postdocs.

I am always game for new projects, so do not hesitate to contact me if you are interested in joint research.

Quasigroups and loops

Quasigroups are magmas in which all left and right translations are permutations. Loops are quasigroups with an identity element. Loop theory combines techniques from group theory and universal algebra. Rumor has it that Drapal, Kinyon and myself are working on a loop theory textbook.

Solvability and nilpotence

There are two competing notions of solvability in loop theory, one generalizing the usual definition from group theory (cf. Bruck) and another specializing a general theory from congruence-modular varieties (cf. Freeze and McKenzie). In general, the two notions differ for loops. It is an open question whether they coincide in certain varieties of loops.

With Stanovsky we developed commutator theory for loops and hence congruence solvability. With Drapal we showed that the two notions of solvability coincide in Moufang loops of odd order adn we proved that Moufang loops of odd order are congruence solvable, strenghtening a result of Glauberman.

Supernilpotence is a strengthening of the notion of nilpotence. It is known that a group is k-nipotent if and only if it is k-supernilpotent. For loops, it is not even known if k-supernilpotence has a finite basis. With Stanovsky we recently found a short and natural basis for 3-supernilpotent loops.

Finite simple Moufang loops and Okubo quasigroups

The variety of loops is vast. Hence it is customary to focus on subvarieties of loops satisfying equations that approximate associativity, or that have other properties making them closer to groups.

A loop is Moufang if it satisfies the identity x*(y*(x*z)) = ((x*y)*x)*z. The classification of nonassociative finite simple Moufang loops has been completed by Liebeck - it depends on the classification of finite simple groups. For every finite field there is a unique nonassociative finite simple Moufang loop and there are no other nonassociative finite simple Moufang loops. The finite simple nonassociative Moufang loops are known as Paige loops.

I proved that every Paige loop is generated by 3 elements. Since Moufang loops are diassociative, the number of generators cannot be further reduced. With Nagy we also described automorphism groups of Paige loops. There is a well-known correspondence between Moufang loops and so-called groups with triality (cf. Hall).

Paige loops arise naturally as the multiplicative structure of Zorn algebras, also known as split octonion algebras. By tweaking the multiplication slighlty, the para-Zorn and Okubo algebras are obtained. With Smith we established a number of results for Okubo quasigroups.

Code loops and combinatorial polarization

There is a one-to-one correspondence between doubly even binary codes and a certain class of loops, called code loops introduced by Griess. It turns out that code loops are precisely Moufang loops with two squares (cf. Chein and Goodaire). The most famous code loop is the Parker loop which was used by Conway to construct the largest sporadic group, the Monster group. Here, the associated code is the extended binary Golay code. It is a difficult question how to associate a doubly even code of minimal length to a given code loop.

Combinatorial polarization plays an important role in code loops since it explains how the squaring, commutator and associator mappings are related. I generalized these notions to mappings of a higher combinatorial degree. With Drapal we characterized code loops (and odd code loops) in various equivalent ways. With O'Brien we enumerated code loops of order up to 512.

Loops of Bol-Moufang type

There is a well-defined notion of loops/quasigroups of Bol-Moufang type. These loops encompass Moufang loops, Bol loops, C-loops, and other varieties of loops. Among other results, with Phillips we carried out the classification of varieties of loops/quasigroups of Bol-Moufang type and investigated C-loops. With Kinyon and Nagy we classified Bol loops and Bruck loops of order pq.

Automorphic loops

A loop is automorphic if all its inner mappings are automorphism. This is a very interesting and highly structured variety of loops.

In a series of papers with various coauthors (de Barros, Jedlicka, Grishkov, Kinyon, Kunen, Nagy and Phillips), we proved the Odd Order Theorem for automorphic loops (generalizing but also using the Feit-Thompson theorem), showed that commutative automorphic loops of odd prime power order are nilpotent, studied automorphic loops of order p³, and so on.

Knot theory and Yang-Baxter equation

The areas of knot theory and Yang-Baxter equation are related due to the 3rd Reidemeister move, which can be interpreted algebraically in various ways, cf. Dehornoy. I am mostly interested in algebraic structures that give rise to knot invariants and in set-theoretic solutions of the Yang-Baxter equation that give rise to one-sided quasigroups.

Racks and quandles

A rack is a magma in which all right translations are automorphisms. Quandles are idempotent racks. The knot quandle of Joyce and Matveev is a complete invariant of oriented knots up to mirror image and orientation reversal. Quandles can be used to color knots.

With Yang we enumerated racks and quandles up to order 13. With Hulpke and Stanovsky we studied connected quandles, that is, quandles in which the group generated by all right translations acts transitively. With Lages and Lopes we investigated the question whether latin quandles can be characterized among connected quandles by means of the cycle structure of their right translations. I am interested in the related Hayashi conjecture for connected racks.

Set-theoretic solutions of the Yang-Baxter equation

There are many different kinds of set-theoretic solutions of the Yang-Baxter equation, including racks and braces.

With Bonatto, Kinyon and Stanovsky we studied involutive latin solutions of the Yang-Baxter equation, and with Stanovsky we described idempotent solutions.

Computational algebra

A substantial part of my research is concerned with computational results. With Nagy, we wrote a GAP package LOOPS and are working on a more general package RightQuasigroups. One of the strengths of the packages are the enclosed libraries.

Some of our computational results are mentioned above. We also enumerated Moufang loops of order 64 and 81. With Daly, we showed how to count nilpotent loops up to isomorphism.

Automated deduction

Automated deduction is an interdisciplinary field for proving theorems (automatically, of course) from given axioms. There have been substantial improvements to automated deduction (both to the software and to the methods) in recent years. I use the theorem prover Prover9 and the finite model builder Mace4.

Nonassociative algebra appears to be especially well suited for automated deduction. Many deep results in loop theory would not be possible without automated deduction. For instance, with Kinyon and Veroff, we used automated deduction to settle a question for loops in which all inner mappings commute.

Other topics

Here are some additional areas are worked in and that do not fit neatly into the above categories.

Distances of groups

Drapal showed that groups whose multiplication tables are close (in the sense that they agree in many cells) are isomorphic.

Given a group Q, what is the minimum number of cells in the multiplication table of Q that must be changed in order to obtain a group different from Q? For sufficiently large groups, the answer is given by Drapal. I showed that the answer is 6p-18 whenever Q is a group of prime order p > 7. With Wanless, we completely settled the problem by calculated minimal distances among small groups.

For a 2-group Q, it is necessary to modify at least one quarter of the multiplication table of Q in order to obtain a group not isomorphic to Q. This "quarter" modifications are fruitful in loop theory, too.

Rainbow cycles

Let G be a complete, edge-colored graph. A cycle in G is rainbow if no two of its edges have the same color. When G is infinite, we define
S(G) = {n > 1 | there is no rainbow cycle of length n in G}.

The subset S(G) has very interesting properties. The class of graphs with 3 in S(G) (no rainbow triangles) is well understood, cf. Gallai, but nobody knows how to characterize the graphs with 4 in S(G). With Ball and Pultr we showed that S(G) - 2 is a submonoid of natural numbers, and we described all submonoids of natural numbers containing 2 that arise as S(G) - 2. It was shown by Alexeev that not every submonoid of natural numbers is of the form S(G) - 2 for some G. In fact, if S(G) contains an odd integer then S(G) contains all sufficiently large integers. I showed that every S(G) has either period one (that is, it contains all sufficiently large integers), or 2, or 4. There are some not very good estimates as to when the period in S(G) will kick in.

Displacement of permutations

For a permutation π on n points, let d(π), the displacement of π, be the average of i - π(i), where i ranges from 1 to n. With Daly we have proved many interesting facts about the displacement. For instance, the expected value of d(π)/n tends to 1/3 as n approaches infinity. We have also studied permutations that stretch intervals [i,i+1] as much as possible, which is related to the design of interleavers for turbo codes.

With Albow, Edgington and Lopez, we described all permutations of a 2-dimensional grid that maximally separate neighbors.