# Abstract Nonsense

## An Update

Hey everyone. It’s been a fair bit of time since I last posted here. Since then, a lot of things have happened. Most relevant for this blog, I graduated UMD and started attending the University of California, Berkeley.

I recently decided that I wanted to start blogging again. Instead of posting on Abstract Nonsense again, I thought I would start a new blog: Hard Arithmetic. If you are interested, you should come and take a look.

Because of the popularity of this blog though, I have decided to leave it up–it makes me so happy it’s helped so many people since I’ve left it. This decision is despite the fact that reading some of my older posts is quite embarrassment inducing. That said, if you’d like to contact me, it would probably more effective to either contact me at my new blog, or shoot me an email at ayoucis@berkeley.edu

## Meromorphic Functions on Riemann Surfaces (Pt. II)

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

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October 7, 2012

## Holomorphic Maps and Functions (Pt. I)

Point of Post: In this post we define holomorphic maps between surfaces and prove various properties that such maps possess. We then specialize this to looking at holomorphic functions.

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Motivation

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Now that we have defined Riemann surfaces and given ample examples we can define the arrows in our category–in other words, the structure preserving maps between Riemann surfaces. Of course, these should just be the maps between Riemann surfaces that are holomorphic locally, where this makes sense since locally Riemann surfaces are just open subsets of $\mathbb{C}$.

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How do we make this rigorous? Precisely the way we did with smooth maps between smooth manifolds. Namely, if we have a map $f:X\to Y$ between two Riemann surfaces, and a point $p\in X$ then $f$ should be holomorphic at $p$ if, when we pretend that $X$ is just an open subset of $\mathbb{C}$ locally around $p$, and that $Y$ is just an open subset of $\mathbb{C}$ around $f(p)$, and then pretend that $f$ is just a map between these open subsets of $\mathbb{C}$, that we get something holomorphic. Of course, we need to unravel this to make it slightly more rigorous. The first rigorization (new word?) we would like to perform is to make explicit what we mean by “pretending” things look, locally, like an open subset of $\mathbb{C}$. In particular, choosing charts $(U,\varphi)$ and $(V,\psi)$ at $p$ and $f(p)$ allows us to pretend that $U$ and $V$ are just the open subsets $\varphi(U)$ and $\psi(V)$ of $\mathbb{C}$ (we need to, for set theoretic technicalities, assume that $f(U)\subseteq V$–but this is just technical, and should [intuition wise!] just be ignored). Fine, but, how do we “pretend” that $f$ is a map $\varphi(U)\to\psi(V)$? Well, it’s fairly obvious that we would want the method of “pretending” to be consistent with the method of “pretend” we performed on the identifications $U\leftrightarrow\varphi(U)$ and $V\leftrightarrow\psi(V)$. In particular, we should pull back $f$ to a map on a subset of $\mathbb{C}$ the same way we pulled $U$ back–using $\varphi$. Similarly, we should use $\psi$ to pull $f$ to a map on a subset of $\mathbb{C}$. Putting this all together we see that the map $\varphi(U)\to\psi(V)$ we should be considering is $\psi\circ f\circ\varphi^{-1}$. Thus, this is the map we want to be holomorphic at $\varphi(p)$ (the “pretend” $p$).

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Of course, just as in the case of smooth manifolds, the somewhat unsettling aspect of this definition is the idea that, a priori, this definition depends on our method of “pretend”. In particular, why can’t we pick different charts and get a function which is not holomorphic? Well, this is precisely why Riemann surfaces aren’t defined as just topological manifolds of one complex dimension. Namely, the requirement that our complex structure be internally (holomorphically) compatible is precisely so that this definition is independent of chart choice.

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Ok, now that we have a notion of holomorphic mappings between Riemann surface we can start to ask how many of the theorems from complex analysis transfer over to this context? Does the Open Mapping Theorem hold? Does the Identity Theorem Hold? It turns out that the meta principle that (almost!) any mapping property that holds for true for holomorphic mappings between domains in $\mathbb{C}$ holds true for mappings between Riemann surfaces. This should be intuitively true since holomorphic mapping properties are most often local theorems, and locally Riemann surfaces and the maps between them, are just holomorphic mappings between domains!

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Of course, this is a meta principle, and as my parenthetical disclaimer indicates, things don’t always hold true for Riemann surfaces in general. What are some of the issues that would prevent us from carrying over a true statement about holomorphic maps between domains to a true statement about holomorphic mappings between Riemann surfaces? Probably the most egregious issue, and one of the simplest conceptually, is that perhaps there isn’t even an obvious analog for the theorem! For example, the Maximum Modulus Principle involves the notion of $|f|$. Now, if we have a mapping $f:X\to Y$ for abstract Riemann surfaces $X$ and $Y$, what does $|f|$ even mean? That said, most of the issues are ones where we really only require $Y$ to be not-so-abstract. Thus, we are naturally led (via trying to generalize theorems in complex analysis) to consider the special case of holomorphic mappings $X\to\mathbb{C}$. Such holomorphic mappings shall be called (for historical reasons) holomorphic functions (i.e. the word function is reserved [instead of mapping] for when the codomain is $\mathbb{C}$).

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Now anyone familiar with smooth manifold theory will be saying to themselves “of course, holomorphic functions will undoubtedly be extremely important in our studies!” This intuition comes from the fact that in real manifold theory, this is very much true. Studying the smooth functions $M\to\mathbb{R}$ can tell us a surprisingly, fantastically huge amount of information about $M$ (pronounced Morse theory). This is, in fact, not entirely true. For example, we shall see that the only holomorphic functions on compact Riemann surfaces shall be constant maps (intuitively, this makes sense because any function shall have to assume it’s maximum from where [using our intuition given by the Maximum Modulus Principle on domains] we should guess our map is contant). This shall be the first indication of a very fundamental fact of Riemann surfaces. Namely, there are two types of Riemann surfaces: the compact and the non-compact. To be less cryptic, we shall see that the theorems/techniques used in the study of compact Riemann surfaces shall vary greatly from those in the study of non-compact Riemann surfaces. A good rule of thumb is that the study of compact Riemann surfaces feels algebraic/algebraic geometric (this has very precise, rigorous categorical statement) that the study of non-compact Riemann surfaces feels much more analysis/ geometric analysis like.

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October 4, 2012

## Riemann Surfaces (Pt. II)

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

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October 2, 2012

## 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.

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Motivation

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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.

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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 $\mathbb{R}$). 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.

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September 3, 2012

## Current Schedule:Summer REU and Blogging

I thought that I would give any (if there exists any) regular readers of my blog a heads-up as to what is coming up in the following months in terms of blogging.

This summer I will be attending the SMALL program at Williams College in Williamstown, MA. Specifically I shall be working in the Algebraic and Geometric Combinatorics project with Elizabeth Beazley.

## Complex Differentiable and Holmorphic Functions (Pt. III)

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

May 1, 2012

## Complex Differentiable and Holmorphic Functions (Pt. II)

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

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May 1, 2012

## The Degree of an Extension (Pt. II)

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

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March 8, 2012

## Extension of Scalars (Pt. I)

Point of Post: In this post we discuss some of the fundamental ideas concerning extension of scalars.

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Motivation

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We finally can now get around to discussing a use of tensor products that I touted in my original introduction: extension of scalars. Namely, the idea is that we are handed some $R$-module $M$ and some superring $S\supseteq R$ and would like to see to what extent we can consider $M$ to be an $S$-module. In other words, are are considering the problem which is dual to taking an $S$-module and considering it as an $R$-module by just “restricting” scalars. Unfortunately, this is often impossible to do. For example, $\mathbb{Z}$ can’t be made into a $\mathbb{Q}$-space since, if it could, whatever $\frac{1}{2}\cdot 1$ would be (and note, it has to be an INTEGER since the multiplication is a map $\mathbb{Q}\times\mathbb{Z}\to\mathbb{Z}$) it would satisfy $2(\frac{1}{2}\cdot1)=1$ which is impossible! So, the next best thing one could hope to do is perhaps extend the module $M$ in some “minimal” way so that it can be naturally imbued with an $S$-module structure. In other words, we want to find some $S$-module $N$ for which $M$ embeds into $N$ as an $R$-module, and doing this in some minimal sort of way.  For example, while $\mathbb{Z}$ cannot be given the structure of an $\mathbb{Q}$-space it can surely be $\mathbb{Z}$-embedded into such a space, namely $\mathbb{Q}$ itself. Once again, this may not always be possible, for example if $A$ is a finite abelian group (e.g. $\mathbb{Z}$-module) then $A$ can never be $\mathbb{Z}$-embedded into a $\mathbb{Q}$-space since (as can be easily proven!) every element of a $\mathbb{Q}$-space has infinite order. Thus, we can really only hope to ask for a “best case scenario”. What $S$-module maximizes both the ability to faithfully (to some degree) embed $M$ and is minimal in some sense. In the case of our abelian group $A$ it’s clear that we’re going to have to take $0$ to be our $\mathbb{Q}$-space since this is the only such space in which $A$ can be “embedded” (albeit very unfaithfully). This is what we mean by extension of scalars, such an $S$-module $N$.

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If the obvious intellectual curiosity isn’t enough to motivate this problem I can mention that it has many uses. For example, I have in the past discussed the notion of induced representations which can be seen as extension of scalars problem. Namely, we suppose that we have some group $G$ and some subgroup $H\leqslant G$. Roughly then what we wish to do is pass from an $H$-representation to a $G$-representation, which can be thought of as extending an $\mathcal{A}(H)$-module (where $\mathcal{A}(H)$ is the group algebra) to an $\mathcal{A}(G)$-module.

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So, why might we expect that the tensor product is the correct route for such an extension of scalars? There is actually a quite natural way one might realize this. The first is the naive attempt that one might actually try to make a given $R$-module $M$ into an $S$-module in the most brutish way. Namely, let’s define a “formal multiplication” of $S$ and $M$ elements. Namely, given $s\in S$ and $m\in M$ let $s\star m$ just be  formal symbol, our “multiplication”. We then see that if this “multiplication” is to create a valid $S$-module structure extending that of $M$‘s preexisting $R$-module structure, we’re going to need certain identities to hold. For example, by mere definition of a module we are going to need that $\star$ is linear in each entry (this is because we should have that $(s+s')\star m=s\star m+s'\star m$, etc.). Moreover, since we want $r\star m=rm$ (since we are extending the $R$-module structure) and $(ss')\star m=s\star(s'\star m)$ we see that we are going to have $(sr)\star m=s\star(r\star m)=s\star(rm)$ for all $s\in S$, $r\in R$, and $m\in M$. Thus, we see that $\star$ is an $R$-biadditive map $S\times M\to S\star M$. Therefore if we’d like to consider a “universal” way to define an $S$-module structure on $M$ it seems that we should be looking for a “universal” $R$-biadditive map $S\times M\to S\star M$ and so really we want $S\star M$ to just be $S\otimes_R M$ and that this should be our extension of scalars.

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January 24, 2012