It is practically impossible to make a precise and accurate statement about any current mathematical research that is intelligible to a non-mathematician. With this in mind, here is an oversimplification that to the best of my ability should give the general area of the thesis, and should make sense to most anyone with an interest. Technically speaking, it is false. A technical description will follow.
The thesis deals with a generalization of a technique known as Grobner bases that was developed in the 1960s for solving systems of polynomial equations and is especially useful in computer algebra systems such as Mathematica and Maple.
The generalization continues work that has been done by others recently to apply the Grobner basis technique to more complicated types of equations such as differential equations.
The starting point for the thesis is the theory of Grobner bases over k-algebras where k is a field. To date, the most general definition for Grobner bases requires that the generating set is a semigroup with a well order that is strictly compatible with multiplication. An Algebra with Term Order is an algebra with a well ordered basis possessing a sort of compatibility which at least relaxes the assumption that the generating set is a semigroup. All standard theorems concerning Grobner bases can be proved in this setting including the Buchberger algorithm and determination and lifting of syzygies.
Known examples of ATOs are polynomial rings, free algebras, and 'Solvable Polynomial Rings' (of H. Kredel). Solvable Polynomial Rings include rings of differential operators and quantum rings. Theorems on lifting of syzygies in Solvable Polynomial Rings had not previously been stated or proved; achieving this was the original motivation for the work.
One application given provides some examples of ATOs whose generating sets are not semigroups. These are quotient rings that are fortunate (and rare) to inherit an ATO structure from the parent ring.
One last topic gives a technique for describing any total order on a free monoid or free group. For those that are decidable, an algorithm is described that can decide any given inequality. I am currently working on a more concise description of this technique that will be applicable to relations in general, and I hope will be suitable for publication.