Review of geometry and analysis

In this article, we shall discuss what the author considers to be important in geometry and related subjects.Since the time of the Greek mathematicians, geometry has always been in the center of science.Scientists cannot resist explaining natural phenomena in terms of the language of geometry.Indeed, it is reasonable to consider geometric objects as parts of nature.Practically all elegant theorems in geometry have found applications in classical or modern physics.In order to understand the future of geometry, it is perhaps useful to review what was known in the past.Clearly what I consider to be important may not be viewed to be so by others.Also, we should always keep in mind that what is fashionable now may not be so tomorrow.A theory can be judged to be successful only if its consequences help us understand the basic structure or the beauty of geometry.While we shall divide the subject into several categories, the division is artificial, as the development of each section depends on other sections heavily.