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.