Applications of model theory to algebra and geometry (MAGIC004)
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This course is aimed at PhD students, not necessarily working in model theory, but working in areas potentially linked to model theory (e.g. other parts of logic, or parts of algebra, algebraic geometry, number theory, combinatorics). The first 5 lectures will introduce fundamental model-theoretic concepts. The second part of the course will explore various `tameness’ conditions on first order theories (e.g. concepts associated with model-theoretic stability theory and its extensions), with a focus on examples from algebra, especially fields (e.g. algebraically closed, real closed and p-adically closed fields, and pseudofinite fields). A goal will be to exhibit potentially applicable methods.
Spring 2020 (Monday, January 20 to Friday, March 27)
Some familiarity with first order logic would be helpful but not essential.
Lectures 1—5: BASICS OF MODEL THEORY AND STABILITY THEORY: First order languages, structures and theories, compactness, types, saturation and homogeneity, quantifier elimination. Lectures 6—10: TAME THEORIES, EXAMPLES, APPLICATIONS: uncountable categorical and strongly minimal theories; stable, o-minimal, simple, and NIP theories; model-theoretic notions of independence and dimension, and their interpretation in algebraically important structures (e.g. algebraically closed and real closed fields).
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