## Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. III)

**Point of Post: **This is a continuation of this post.

## Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. II)

**Point of Post: **This is a continuation of this post.

## Loci of Holomorphic Functions and the Inverse Function Theorem (Pt. I)

**Point of Post: **In this post we discuss a new way of getting Riemann surfaces by taking loci of holomorphic functions on . We necessarily then discuss the (holomorphic) inverse function theorem.

**Motivation**

Just like when studying smooth manifolds, perhaps one of the richest examples of Riemann surfaces come from looking at loci of holomorphic functions. In particular, for us, we shall care most about the zero sets of of holomorphic functions with a domain in . Of course, there are two issues with this. Firstly, what does it mean for a function to be holomorphic? Secondly, why is it clear that such a locus is actually a Riemann surface?

The first is one for which I can only give an unsatisfactory answer for now. For us, a holomorphic function of two variables will be one which is holomorphic in each variable separately. The fact that this definition of holomorphic yields what we expect holomorphic functions to be (functions which are locally power series!) is a relatively deep fact known as (or at least coming from) Hartog’s theorem. This is a theorem whose place is rightfully in the study of Several Complex Variables (SCV), and is something that I can’t do justice to right now. Moreover, it is perhaps the “other branch” (complex analysis joke fully intended) one could take when they want to go to a second graduate course in complex analysis. So, for anyone interested in WHY this is the correct notion of holmorphicity in higher dimensions I recommend taking a look at SCV in general. To make up for my lack of actually explaining these concepts, let me be at least helpful in the narrowing down the length list of SCV books to, what is in my humble opinion, the best four books. The classic books on SCV are [6] and [7]. They are both fantastic books, not only exposition, but in thoroughness–pretty much anything you would want to know is contained in these two books. That said, if one is looking for a compactified, straight to the point, version of these books , I would suggest [8]. Ohsawa definitely is terse, but will give you the necessary facts and theorems that one may want to know–I will warn you that it is definitely an analyst’s book. Lastly, if one is looking for a broader, more geometric approach to SCV I would suggest [9].

Anyways, coming back from that rant. The second issue about why such a locus is actually a Riemann surface is answered, much like in the case of smooth manifolds, by the implicit function theorem IFT. The intuition for the inverse function theorem is much like the case of the usual IFT Of course, we shall need an adaptation of the usual IFT so as that we get not just a smooth inverse but a holomorphic one. This is where the necessary groan should be emitted by anyone familiar with the proof of the smooth IFT (or at least the equivalent, inverse function theorem). Now, before you decide to skip the proof let me assure you–it’s not bad. In fact, following the general differences between real and complex analysis, not only is the holomorphic IFT much simpler than the real analog, but is much prettier. The basic idea is that by using a slight generalization of the Argument Principle we are able to explicitly right down a function which gives the roots of an equation within a given region. Moreover, we are able to write this function as an integral in holomorphic functions, and thus it will be a holomorphic function itself. See, not so bad?

## Compact Riemann Surfaces are Topologically g-holed Tori

**Point of Post: **In this post we prove that topologically compact Riemann surfaces are just -holed tori.

**Motivation**

A very natural question when one starts studying Riemann surfaces is “what topological spaces admit a complex structure?” It turns out that while difficult in general, for compact Riemann surfaces these are precisely the -holed tori. While this sounds pretty deep, to anyone who is familiar with the classification of compact surfaces, we can mask most of the difficulty by just proving that the Riemann surface is orientable. This somewhat surprising fact follows immediately from the Cauchy-Riemann equations. The somewhat surprising part is that the converse is true. Namely, if is any smooth orientable -manifold, then admits a complex structure. This is somewhat of a deep fact, one that takes some serious (unavailable) machinery to prove and so we’ll just state it.

## Riemann Surfaces (Pt. I)

**Point of Post: **In this post we formally introduce the notion of Riemann surfaces and discuss some important examples.

**Motivation**

We now begin what is, in my very humble and uninformed opinion, one of the most beautiful subjects in the entirety of basic graduate mathematics–Riemann surfaces. Such a bold statement begs two immediate questions: what are Riemann surfaces, and why are they so pretty?

The first question is one which has a simple, albeit somewhat esoteric, response–a Riemann surface is merely a one-dimensional connected complex manifold. What is such an object? Well, anyone who is likely to get a lot out of these posts probably is familiar with the concept of a smooth manifold, which is merely topological spaces with a well-defined notion of how to ‘do calculus’ on them. From this, it’s not hard to guess what a complex manifold is, it’s a topological space that has well-defined way of doing complex analysis on them. So, Riemann Surfaces are nothing more than (connected!) topological spaces which locally look like open subsets of , and that this local notion pieces together nicely enough to give a global notion of what a holomorphic mapping between two such Riemann surfaces looks like.

Now that we have a very rough idea of what a Riemann surface should be, we can at least try to explain why the theory of Riemann surfaces is so beautiful. Everyone who has taken complex analysis at an advanced undegraduate/graduate level is aware of the fact that complex analysis is much more intimately (or at least more immediately!) related to algebra and topology than real analysis is. For example, for a domain one has that the simple connectedness of is equivalent to every harmonic function admitting a holomorphic function such that , which is equivalent to every holomorphic function admitting a primitive on (these allow us to define cohomology via complex analysis!).

This pervasive feeling of deep algebraic and geometric connections will continue when we discuss Riemann surfaces. We shall prove some truly deep, and truly beautiful theorems in this vein. For example, we shall prove that, in a very precise sense, doing work with compact Riemann surfaces is the same thing as working with projective plane curves–in particular, we shall see that every algebraic function field (algebraic extension of ) is just the meromorphic functions on some compact Riemann surface. While I could go on and on about how interesting and amazing this subject is, I think that it would be better that I attempt to inject my paltry insight as we go along, and let you see for yourself why this subject makes me so excited.

## Smooth Maps and the Category of Smooth Manifolds (Pt. III)

**Point of Post: **This is a continuation of this post.

## Smooth Maps and the Category of Smooth Manifolds (Pt. II)

**Point of Post: **This is a continuation of this post.

## Smooth Maps and the Category of Smooth Manifolds (Pt. I)

**Point of Post:** In this post we define what it means for a map between two manifolds to be smooth.

**Motivation**

We literally defined smooth manifolds to be the topological spaces where we will have a relatively sound meaning of what a “smooth map” is. Thus, it would seem that the first order of business is to fully define and explore this notion of smooth map. The basic idea though is precisely what we have said before. A map between smooth manifold will be smooth if it is smooth locally around each point and its image–when we think about the space locally as Euclidean space.

The interesting part is that once we define smooth map we will then be able to define the category of (finite dimensional) smooth manifolds. We will then be able to discuss the functor which takes a smooth manifold to it’s algebra of smooth functions (for us, function will mean a map into ). We will then be able to make sense of the following statement: the smooth structure of a manifold is largely encoded in its algebra of smooth functions.