This work involves a uniting of investigations in stability theory, model theory of algebra, and finite model theory. One underlying technique is the construction of homogeneous universal models with respect to a notion of strong submodel. On the one hand, this links stability theory with probability on finite models by providing a technique for not only proving 0-1 laws but obtaining model theoretic properties of the almost sure theory. These model theoretic properties of the almost sure theory can be applied to problems in finite model theory. On the other hand, this technique is exploited to attempt to construct a bad field. Finally, investigations of embedded finite models solves problems of database database theory using techniques from stability theory. One theme in this work is model theory, a branch of mathematical logic. The general aim is to understand `ordinary mathematics' at a higher level of abstraction. This has previously resulted in applications in the area of algebra, specifically group theory. The present project will involve an attempt to find some specific new structures which expand the complex numbers. Some of these same insights will be used to explore the asymptotic properties of finite graphs. A further application of this abstract framework has been in the development of `embedded finite model theory,' where the techniques of model theory are employed to find limits on the expressibility of database queries.
