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Explore advanced concepts in algebraic geometry through this sixth lecture of a minicourse on Mumford–Tate groups in Hodge theory, delivered by Professor Andrey Soldatenkov from UNICAMP as part of IMPA's Summer School 2026. Delve into the intricate mathematical framework that connects algebraic cycles, motives, and representation theory through the lens of Hodge structures. Examine how Mumford–Tate groups serve as fundamental tools for understanding the arithmetic and geometric properties of algebraic varieties, particularly in the context of abelian varieties and their endomorphism algebras. Learn about the classification of Hodge structures through their associated Mumford–Tate groups and discover applications to problems in arithmetic geometry and transcendental number theory. Build upon previous lectures in this comprehensive series to deepen your understanding of this sophisticated area of mathematics that bridges algebraic geometry, representation theory, and number theory.
