Model theory of expansions of valued fields

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My area of research is model-theoretic algebra. This is a subfield of mathematical logic, in which techniques of model theory are applied to other parts of mathematics including number theory, algebraic geometry, representation theory and real analytic geometry. The last twenty years have seen some very exciting developments in this area, including Pila's applications of o-minimality to counting points on analytic varieties, Hrushovski's proof of the Manin-Mumford conjecture in characteristic p and the work of Cluckers, Hales and Loeser on motivic integration which they used to transfer Ngo's proof of the Fundamental Lemma of the Langlands program in characteristic p to also have the statement in characteristic 0. All of the above work either uses the theory of valued fields or has analogies in the theory of valued fields. My own work is in continuing to develop the foundations of the model theory of valued fields. The overall goal of my research program is to increase the range of possible applications of model theory to address more problems in other areas of mathematics which involve analytic functions on a valued field. This requires a deep understanding of the model-theoretic properties of these structures. In the next five years, I will pursue this program by addressing projects which fall into two broad categories. The first category is the study of imaginaries (that is, quotients by definable equivalence relations) in valued fields with restricted analytic functions. Based on earlier work of myself with collaborators, we now understand fairly well the imaginaries in the algebraic valued field theory. The collaboration continues in order to tackle the much richer structure of the valued field with restricted analytic functions. The second category of projects is the study of fields with both a valuation and a convex ordering. By analogy with the non-ordered case, I plan to develop the theory of residue field domination for both the algebraic and the analytic structures. This will have consequences for creating a theory of ordered Berkovich space.