Teichmüller Space and the Segal Moduli Space - Talk 1, Part 1

Explore the fundamental concepts of Teichmüller theory in this 52-minute conference talk that introduces the Teichmüller space of a Riemann surface as a space of deformations of its complex structure. Learn how this Banach manifold possesses complex structures with many non-trivially equivalent models, most of which were established by the 1960s. Discover the connection between Teichmüller space and the Segal moduli space, which consists of equivalence classes of Riemann surfaces with boundary parametrizations and, despite being invented decades later, turns out to be nearly the same space up to a completion and discrete modular group action. Gain insight into the basic ideas of Teichmüller theory through an outline of fundamental principles, followed by a heuristic explanation of its relationship to the Segal moduli space and the resulting consequences, all presented with minimal technical details to make the material accessible.