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Explore the mathematical theory of o-minimality and its applications to counting rational points in definable sets through this Members' Colloquium lecture. Learn the fundamental concepts of o-minimality before discovering how this framework enables proving asymptotic estimates for rational points within definable mathematical structures. Examine how diverse mathematical problems from various fields can be reframed as questions about rational points in appropriately chosen definable sets. Understand the powerful implications that o-minimality theory provides for solving these reformulated problems across different areas of mathematics. Gain insights into this sophisticated mathematical tool that bridges abstract theory with concrete applications in number theory and algebraic geometry.
