Quantization PDEs and Geometry : The Interplay of Analysis and Mathematical Physics This book presents four survey articles on different topics in mathematical analysis that are closely linked to concepts and applications in physics. Specifically it discusses global aspects of elliptic PDEs Berezin Toeplitz quantization the stability of solitary waves and sub Riemannian geometry. The contributions are based on lectures given by distinguished experts at a summer school in Göttingen. The authors explain fundamental concepts and ideas and present them clearly. Starting from basic notions these course notes take the reader to the point of current research highlighting new challenges and addressing unsolved problems at the interface between mathematics and physics. All contributions are of interest to researchers in the respective fields but they are also accessible to graduate students. Introduction Todor Gramchev: Gelfand Shilov Spaces: Structural Properties and Applications to Pseudodifferential Operators in \\R^n Miroslav Englis: An Excursion into Berezin Toeplitz Quantization and Related Topics Andrew Comech: Global Attraction to Solitary Waves Irina Markina: Geodesics in Geometry with Constraints and Applications.