Explore the Galois group of the category of mixed Hodge Tate structures in this 59-minute lecture by Guangyu Zhu, presented as part of the Hausdorff Trimester Program on Periods in Number Theory, Algebraic Geometry and Physics. Delve into the concept of mixed Hodge-Tate structures over Q as a mixed Tate category of homological dimension one, and understand how Tannakian formalism equates it to the category of graded comodules of a commutative graded Hopf algebra. Learn about Zhu's recent joint work with A. Goncharov, which provides a canonical description A (C) of the Hopf algebra. Discover how this construction can be generalized to A (R) for any dg-algebra R with a Tate line, offering insights into advanced topics in algebraic geometry and number theory.