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Explore a deep dive into the Galois group of the category of mixed Hodge-Tate structures in this advanced mathematics lecture. Delve into the joint work of Alexander Goncharov and Guangyu Zhu, examining the canonical equivalence between rational mixed Hodge-Tate structures and finite-dimensional graded comodules over a graded commutative Hopf algebra. Discover the natural explicit construction of the Hopf algebra and its generalization to a Hopf dg-algebra for variations of Hodge-Tate structures on complex manifolds. Learn about the application of refined periods in weight n variations of mixed Hodge-Tate structures and their single-valued nature. Conclude with an exploration of the p-adic variant of the construction, connecting to Fontaine's crystalline and semi-stable period rings in p-adic Hodge theory.
