Explore the second lecture in a series on class field theory and its generalizations, delivered by Professor Ivan Fesenko from the University of Nottingham at Kyoto University. Delve into the foundational mathematical concepts that A. Weil deemed as essential as Galois theory for all mathematicians. Learn about T. Takagi's groundbreaking 1920 contribution to existence theory in class field theory and examine how Sh. Mochizuki's Inter-Universal Teichmüller (IUT) theory has prompted a fresh examination of the field. Gain insights into the modern perspective of class field theory and its connections to the Langlands program, higher class field theory, and anabelian geometry, while discovering how these concepts relate to cutting-edge developments in IUT theory and two-dimensional adelic analysis and geometry.