There has been much interest in the area of model-based reasoning within the Artificial Intelligence community, particularly in its application to diagnosis and troubleshooting. The core issue in this thesis, simply put, is, model-based reasoning is fine, but whence the model? Where do the models come from? How do we know we have the right models? What does the right model mean anyway? Our work has three major components. The first component deals with how we determine whether a piece of information is relevant to solving a problem. We have three ways of determining relevance: derivational, situational and an order-of-magnitude reasoning process. The second component deals with the defining and building of models for solving problems. We identify these models, determine what we need to know about them, and importantly, determine when they are appropriate. Currently, the system has a collection of four basic models and two hybrid models. This collection of models has been successfully tested on a set of fifteen simple kinematics problems. The third major component of our work deals with how the models are selected.

One of the central themes of modern Number Theory is to study properties of Galois and automorphic representations and connections between them. In our dissertation, we describe two different projects that study properties of these objects. In our first project, which is analytic in nature, we consider Artin representations of Q of dimension 3 that are self-dual. We show that these occur with density 0 when counted using the conductor. This provides evidence that self-dual representations should be rare in all dimensions. Our second project, which is more algebraic in nature, is related to automorphic representations. We show the existence of canonical models for certain unitary Shimura varieties. This should help us in computing certain cohomology groups of these varieties, in which regular algebraic automorphic representations having useful properties should be found.