Generalized Geometries and their Applications

Geometry is one of the pillars of both ancient and modern mathematics. It also plays a vital role in many scientific applications, in particular in physics. In recent years it has become clear, by studying duality symmetries in String Theory, that there exist certain generalizations of geometry which are not only crucial in the construction of a manifestly duality invariant formulation of String Theory (also known as M-theory), but have other important applications as well. These include noncommutative geometry and Hitchin's generalized geometry. In this project we will further develop the mathematical aspects of these generalized geometries, both from a differential and algebraic point of view, as well as their applications.