This area covers research and education of modern geometry and topology. Nowadays, geometric or topological ideas can be found in many areas of mathematics and mathematical sciences. The main objects we study in modern geometry are manifolds and their related spaces. The aim of the subject is to clarify the global structures of those spaces by applying various topological and/or differentio-geometric methods. Our special emphases are on the following subjects: topology of three and four-dimensional manifolds, topological study of various dynamical systems on manifolds, homotopy theory of spaces, arithmetic geometry.