Complex geometry: Its brief history and its future Personal perspective

There are surprisingly rich properties of these holomorphic functions. The possibility of holomorphic continuation of holomorphic functions forces us to consider multi-valued holomorphic functions. The concept of Riemann Surfaces was introduced to understand such phenomena. The ideas of branch cuts and branch points immediately relate topology of these surfaces to complex variables. The possibility of two Riemann surfaces can be homeomorphic to each other without being equal was realized in nineteenth century where remarkable uniformization theorems were proved by Riemann for simply connected surfaces. Although it took Hilbert many years later to make Riemann’s work on variational principle to be rigorous, the Dirichlet principle of constructing harmonic functions and hence holomorphic functions has tremendous influence up to modern days.