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Explore advanced model theory through this mathematical lecture examining Ax-Kochen-Ershov (AKE) principles for valued fields across different characteristics and structures. Delve into the foundational 1960s work by Ax, Kochen, and Ershov demonstrating how first-order theories of complete non-Archimedean valued fields are determined by their residue field and value group theories. Master the transition from absolute values to valuations and from completeness to henselian conditions, understanding how these modifications enable powerful model-theoretic methods. Investigate embedding lemmas for henselian valued fields of equal characteristic zero, learning how embeddings between residue fields and value groups extend to valued field embeddings under saturation hypotheses. Discover how these embedding results yield AKE principles for existential theories and extend through back-and-forth arguments to other classical fragments including existential-universal sentences. Study the state-of-the-art theory of separably tame valued fields, examining contributions from Kuhlmann, Knaf, and Pal that establish comprehensive AKE principles. Examine three major extension families: AKE principles for expanded languages including difference fields 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.
