From Holomorphic Functions to Holomorphic Sections

It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer. To be more precise, I am a complex differential geometer, although I am equally interested in real differential geometry. To many people, geometry is a kind of mathematics that is related to length, area, volume, etc. For the Euclidean geometry, this is indeed the case. Differential geometry is a kind of geometry on curved space. The basic geometric objects like length, area and volume are also very important in differential geometry. However, I prefer to think differential geometry is a kind of calculus that takes the underlying curved space into account. What we mean here is that in calculus, we study very complicated functions on relatively simple spaces: 1-d or n-d Euclidean spaces; on the other hand, in topology, we study sophisticated spaces but with relative simple function theory on it. In this sense, differential geometry is a balanced mathematics that pays equal attention to the function theory (calculus) and the topology of the space where the functions are defined. Thus it is not surprising that there is an interaction between calculus and topology in differential geometry. Or in other words, the deep connections between analysis and topology are expected to be found in differential geometry. One of the perfect example of this kind is called the Atiyah-Singer index theorem. But today, I will not discuss this topic, as it is too lengthy. The main point I will make today is that even one is a purely analyst, working on the function theory on simple space like the complex plane, then to some level, topology is naturally introduced, and it will interacts with the analysis. I begin with the simplest complex analysis everybody knows. Since I am a complex geometer, I will begin with complex function of one variable. There are three different ways to define a holomorphic function. Namely the definition of Cauchy-Riemann, the definition of Cauchy and the definition of Weierstrass. Let’s briefly go over the three definitions. 1. The point of view of Cauchy-Riemann. A holomorphic function f on a domain Ω is a C1 complex valued function whose real and imaginary part u and v satisfy the so-called Cauchy-Riemann equations { ∂u ∂x = ∂v ∂y ∂v ∂x = − ∂y .