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Explore the Ax-Kochen-Ershov (AKE) theory and its applications in this mathematical lecture that examines how first-order theories of valued fields are determined by their residue fields and value groups. Delve into the foundational work from sixty years ago showing that complete fields with non-Archimedean absolute values of characteristic 0 have theories determined by their component structures. Learn about the transition from absolute values to valuations and from completeness to henselian conditions to facilitate model-theoretic methods. Understand the embedding lemma for henselian valued fields of equal characteristic zero, which demonstrates how embeddings between residue fields and value groups extend to valued field embeddings under saturation hypotheses. Examine how this lemma yields AKE principles for existential theories and extends through back-and-forth arguments to other classical fragments including existential-universal sentences. Discover the state-of-the-art setting for AKE principles in the theory of separably tame valued fields, building on work by Kuhlmann and collaborators including Knaf and Pal. Follow a uniform presentation beginning with classical equal characteristic zero settings and extending to finitely ramified valued fields in mixed characteristic and separably tame fields. Investigate three families of extensions and applications: AKE principles for expanded languages including difference fields and differential fields, analysis of existential theories of henselian valued fields in both equal and mixed characteristics, and groundwork for the Taming Theorem of Jahnke and Kartas that provides AKE results for valued fields admitting finite extensions with nontrivial defect.
