# T-duality, part I

T-duality is a geometric generalisation of the Fourier transform, where the circle is replaced by a family of circles (or more precisely a circle bundle) and bounded periodic functions are replaced by more sophisticated geometric structures. T-duality originated in string theory and is by the SYZ conjecture intimately connected to mirror symmetry and thus also the geometric Langlands duality.

In these lectures I will review the history of the topic, explain the mathematical setup and the action of T-duality on geometric objects such as twisted cohomology, twisted K-theory, Courant algebroids and certain vertex algebras. Time permitting, I will also explain how to extend the framework to include string structures. These are higher analogues of spin structures and are closely related to the existence of a Dirac-Ramond operator on loop spaces.

Only basic knowledge of differential geometry will be assumed.