Nonstandard models of Peano Arithmetic. Recursively saturated models of arithmetic and of other theories. Applications of nonstandard models to other areas of mathematics, especially algebra and group theory. Combinatorial game theory and nonstandard models. Satisfaction classes and notions of truth over a nonstandard model of arithmetic.
Structural properties of models of Peano arithmetic, and in particular their initial segments.
Sadie Kaye is one of the main workers in the area of models of first-order arithmetic. There are a number of themes to this research, but most structural information about models of arithmetic relates to the order structure of the model. Sadie's work includes linking this order structure to the automorphism group of models of PA, and to looking at new families of initial segments, such as generic cuts. In many cases the structural properties are best understood through a language expanding that of PA by adding other predicates and functions - one representing the cut in question for example. This leads to new ways of looking at second order theories of arithmetic (utilising coding devices for example). The recent and on-going work with TL Wong, a PhD student of Sadie's at Birmingham illustrates these ideas very well.
Strengthenings of the notion of recursive saturation and resplendency, in particular arithmetical saturation and transplendency.
Recursive saturation is a very natural and useful property that many nonstandard models of arithmetic have. In some cases (e.g. when the model has a nonstandard truth definition) recursive saturation is available "for free". Recursive saturation is closely related to the idea of resplendence in second order model theory. However, recursive saturation alone is often not enough for some results. An older result by Kotlarski, Kossak and Kaye shows that a countable recursively saturated model of PA has an automorphism moving every nondefinable point if and only if the model satisfies the stronger property of being arithmetically saturated. In recent work Kaye and her PhD student Engstrom looked at a powerful extension of this to form expansions of the model simultaneously omitting a type. The resulting notion - transplendency - is very powerful and not as yet fully understood and is still the subject of current research.
Properties of nonstandard algebraic structures, in particular nonstandard finite symmetric groups, abelian groups and linear groups.
Algebra and logic combine very well. Kaye has instigated a study of finite algebraic objects inside nonstandard models. This leads to some very attractive algebraic objects, including nonstandard symmetric groups (studied for example by Kaye and his research student Allsup) and nonstandard finite linear groups. Even nonstandard cyclic groups have interesting structure which is being investigated currently by another PhD student, Reading. Results in these areas show that symmetric groups are closely connected with so-called sofic groups, and nonstandard groups often have natural quotients with analytic structure and often with interesting measures. The work results in interesting new examples of algebraic objects with new means of reasoning about them. Nonstandard methods of this type can be applied to other areas too. Another PhD student of Kaye's is currently investigating Conway-style Combinatorial Games and the sorts of nonstandard games that arise from model theoretic considerations applied to these.