A survey of geometric structure in geometric analysis

Abstract

The subject of geometric analysis evolves according to our understanding of geometry and analysis. However, one should say that ideas of algebraic geometry and representation theory have been extremely powerful in both global and local geometry.In fact, the spectacular idea of using geometry to understand Diophantine problem has already widened our concept of space. The desire to find suitable geometry to accommodate unified field theory in physics would certainly drastically change the scope of geometry in the near future.In the following lectures, we shall focus on an important branch of geometric analysis: the construction of geometric structures over a given topological space.