Teichmüller Theory of the Torus

A lot is known about the topological structure of surfaces - one can classify all surfaces up to homeomorphism. However, one can have infinitely many non-isometric complete geometric structures on a surface. Informally, the moduli space is the space of all geometric structures on a surface up to isometry, the Teichmüller space is its universal cover, the mapping class group is its fundamental group, and the curve graph is its 1-skeleton. In this talk, we shall study these objects on a torus, highlight connections between these objects, and comment about generalizing these ideas for higher surfaces. This talk will assume zero prereqs from geometry/topology, and should be accessible to all.