Panorama of Mathematics - Peter Scholze

Explore a captivating conference talk on the intricate relationship between locally symmetric spaces and Galois representations. Delve into the world of mathematical sciences as Peter Scholze, speaking at the "Panorama of Mathematics" conference, unravels complex concepts such as symmetric spaces, hyperbolic 3-space, and the homology of Bianchi manifolds. Discover the connections between these topics and their relevance to Galois representations, Hecke operators, and non-abelian reciprocity laws. Follow Scholze's step-by-step construction and gain insights into cutting-edge mathematical trends, results, and challenges in this 50-minute lecture organized by the Hausdorff Center for Mathematics.

Intro
What is the relation between the following?
Symmetric spaces
Hyperbolic 3-space
The homology of Bianchi manifolds
The torsion homology of Bianchi manifolds
Existence of Galois representations
Relation to Galois representations
Hecke operators
Figueiredo's example
Hecke eigenvalues
Answer: A non-abelian reciprocity law
Where does K/K come from?
Where does K/K actually come from?
Steps of the construction
The key step: Sketch