The classification of NIP fields is a major open problem in model theory. This talk will be an overview of an ongoing attempt to classify NIP fields of finite dp-rank. Let $K$ be an NIP field that is neither finite nor separably closed. Conjecturally, $K$ admits exactly one definable, valuation-type field topology (V-topology). By work of Anscombe, Halevi, Hasson, Jahnke, and others, this conjecture implies a full classification of NIP fields. We will sketch how this technique was used to classify fields of dp-rank 1, and what goes wrong in higher ranks. At present, there are two main results generalizing the rank 1 case. First, if $K$ is an NIP field of positive characteristic (and any rank), then $K$ admits at most one definable V-topology. Second, if $K$ is an unstable NIP field of finite dp-rank (and any characteristic), then $K$ admits at least one definable V-topology. These statements combine to yield the classification of positive characteristic fields of finite dp-rank. In characteristic 0, things go awry in a surprising way, and it becomes necessary to study a new class of "finite rank" field topologies, generalizing V-topologies. The talk will include background information on V-topologies, NIP fields, and dp-rank.
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- Logic Seminar