This thesis is about a generalized concept of distance in geometry and how it is captured by the modes of vibration of geometric objects. On a smooth surface, the distance between two points is determined by the length of a shortest path connecting them. In some settings, however, such as transport problems, it is necessary to generalize this notion of distance. In fact, a more relevant figure than the spatial distance between two sites may be the cost of transporting a unit of mass from one site to the other. Mathematicians still think of transportation cost as a kind of distance. In noncommutative geometry, the field of research of this thesis, such a generalized notion of distance is related to the modes of vibration of geometric objects. We study approximations of such distances arising from low-frequency modes.
Malte Leimbach studied Mathematics at Freie Universität Berlin, Germany, where he obtained his MSc in 2021. During his studies he spent one semester each at Universidad de Granada, Spain, and at Université Paris Didérot (Paris VII), France. He joined Radboud Universiteit Nijmegen as a PhD candidate in September 2021. His research project “Scale-dependent approach to noncommutative geometry: applications to quantum physics” was supervised by Walter van Suijlekom.