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Explore the mathematical concept of Tate locus through this advanced lecture examining conjectures and current results in algebraic geometry. Delve into the etale counterpart of the Hodge locus of a variation of Hodge structures (VHS), focusing on geometrically connected varieties over fields. Learn about the degeneracy locus of l-adic local systems and understand how it relates to recent advances in Hodge theory, particularly those achieved through o-minimality techniques. Examine the main conjectures surrounding the Tate locus when the base field is a number field, and discover what can currently be proven in this area of research. The presentation covers joint research work with Jakob Stix and Akio Tamagawa, providing insights into cutting-edge developments in arithmetic geometry. Gain understanding of the applications and implications of these theoretical results, while exploring the connections between different areas of modern algebraic geometry and number theory.
