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Explore advanced topics in valued field theory through this mathematical lecture that examines Ax-Kochen-Ershov (AKE) principles and their extensions to positive residue characteristic cases. Delve into the foundational work from sixty years ago showing how first-order theories of henselian valued fields in equal characteristic zero are determined by their residue field and value group theories. Learn about the embedding lemma methodology that facilitates model-theoretic approaches by replacing completeness hypotheses with henselian conditions. Discover how these classical results extend to finitely ramified valued fields in mixed characteristic and separably tame valued fields, building on contributions from Kuhlmann, Knaf, and Pal. Examine three key application areas: AKE principles for expanded languages including difference and differential fields, analysis of existential theories in henselian valued fields across different characteristics, and groundwork for the Taming Theorem of Jahnke and Kartas addressing valued fields with finite extensions having nontrivial defect. Gain insight into Kaplansky's hypothesis regarding finite ramification and understand how modern developments in tame valued field theory provide uniform presentations of these fundamental mathematical principles.
