| Résume | Shelah's conjecture predicts that any infinite NIP field is
either separably closed, real closed or admits a non-trivial henselian
valuation. Recently, Johnson proved that Shelah's conjecture holds for
fields of finite dp-rank, also known as dp-finite fields. The aim of these two talks is to give an introduction to dp-rank in some algebraic structures and an overview of Johnson's work.
In the first talk, we define dp-rank (which is a notion of rank in NIP theories) and give examples of dp-finite structures. In particular, we discuss the dp-rank of ordered abelian groups and use them to construct multitude of examples of dp-finite fields. We also prove that every dp-finite field is perfect and sketch a proof that any valued field of dp-rank 1 is henselian.
In the second talk, we give an overview of Johnson's proof that every
infinite dp-finite field is either algebraically closed, real closed or
admits a non-trivial henselian valuation. Crucially, this relies on the notion of a W-topology, a natural generalization of topologies arising from valuations, and the construction of a definable W-topology on a
sufficiently saturated unstable dp-finite field. |
