Topics in model theory

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My proposed research program lies in model theory, a branch of mathematical logic, and more specifically in stability and simplicity theory. I am currently working on three broadly defined projects:Fine structure for classifiable theories: The following question arises when considering models found as leaves on decomposition trees in a classifiable theory. Suppose that p is a regular type of depth zero over a countable model M and M(p) is a model dominated by a realization of p over M. If p is strongly regular than M(p) is prime over M and the realization of p; if p is locally modular then in joint work with Bouscaren, Hrushovski and Laskowski, M(p) is well-understood.  If p is trivial the structure of M(p) is currently not understood. This extends work in "The uncountable spectra of countable theories".Open problems in the foundations of simple theories: Three problems of interest that remain open regarding simple theories are1. elimination of imaginaries in a general simple theory,2. the stable forking conjecture3. general group construction and group existence theoremsThese problems are related in a way to the following topic:First-order continuous logic: Ben-Ya'acov has recently introduced the notion of first-order continuous logic and has begun studying various stability related notions.  One of the advantages of this new setting is that the corresponding notion of hyperimaginary can be thought of as a "real" element of the structure, unlike the classical first order case.  There are a variety of projects here that are interesting including understanding the outcome of classical problems in this setting, determining if some of the classical problems have solutions in the continuous context and finally seeing to what degree these results pull back to the standard setting.