Abstract Nonsense

Current Deal

Ok, so here’s what’s going to be happening (hopefully) on this blog in the upcoming months.

To make a long story short, my life has been somewhat hectic over the last few weeks/months–both personally and professionally (as a student). Consequently, I have not been able to blog nearly as much as I had hoped.

I definitively want to get back into blogging, but somewhat of an annoyance is part of this. I had originally intended (thinking that I would be able to blog more than I could!) to do complex analysis and the basics of differential topology before what I started talking about what really interested me–Riemann surfaces. Unfortunately, this is impossible at this point. I really want to start talking about Riemann surfaces but don’t have the time to catch up (in a time frame acceptable to me) the amount of background that is necessary.

Consequently, I will start blogging about Riemann surfaces assuming that the reader has a background knowledge in complex analysis and (to a far lesser degree) differential topology.

September 29, 2012 Posted by Alex Youcis | Uncategorized | Leave a Comment

Meromorphic Functions on the Riemann Sphere (Pt. II)

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

$\text{ }$

October 7, 2012 Posted by Alex Youcis | Complex Analysis, Riemann Surfaces | Classification, Complex Analysis, Intuition, Meromorphic Functions, Riemann Sphere, Riemann Surface, Riemann Surfaces | Leave a Comment

Meromorphic Functions on the Riemann Sphere (Pt. I)

Point of Post: In this post we classify the meromorphic functions on the Riemann sphere $\mathbb{C}_\infty$ .

$\text{ }$

Motivation

$\text{ }$

If a random kid off the street asked you “what are the continuous functions $(0,1)\to(0,1)$ ?” or “what are all the smooth maps $S^4\to\mathbb{R}$ ?” you would probably replay with a definitive “Ehrm…well…they’re just…” This is because such a description (besides “they’re just the continuous functions!”) is beyond comprehension in those cases! For example, it would take quite a bit of ingenuity to come up with something like the Blancmange function–in fact, such crazy continuous everywhere, differentiable nowhere functions are, in a sense dense in the space of continuous functions.

$\text{ }$

Thus, it should come as somewhat of a surprise that after reading this post you will be able to answer a kid asking “what are all the meromorphic functions on the Riemann sphere?” with a “Ha, that’s simple. They’re just…”. To be precise, we have already proven that $\mathbb{C}(z)\subseteq\mathcal{M}(\mathbb{C}_\infty)$ , and we shall now show that the reverse inclusion is true! Now, more generally, we shall be able to give a satisfactory (algebraic!) of the meromorphic functions of any compact Riemann surface. While this won’t be quite as impressive as the explicit, simple characterization of $\mathcal{M}(\mathbb{C}_\infty)$ but still a far cry from our situation with trying to characterize $C([0,1],[0,1])$ , since we don’t really even have an algebraic (ring theoretic) description of this (in terms of more familiar objects). This should be, once again, another indication that the function theory of compact Riemann surfaces is very rigid–they admit meromorphic functions, but not so many that the computation (algebraically) of their meromorphic function field is untenable.

$\text{ }$

Ok, now that we have had some discussion about the philosophical implications of actually being able to describe $\mathcal{M}(\mathbb{C}_\infty)$ let’s discuss how we are actually going to prove $\mathcal{M}(\mathbb{C}_\infty)=\mathbb{C}(z)$ . The basic idea comes from the fact that we can actually find a function $r$ which has prescribed zeros $\lambda_1,\cdots,\lambda_n$ and poles $p_1,\cdots,p_m$ such that $\text{ord}_{\lambda_i}(r)=e_i$ and $\text{ord}_{p_i}(r)=-g_i$ for any $e_i,g_i\in\mathbb{N}$ –namely, the function $\displaystyle r(z)$ given by $(z-\lambda_1)^{e_1}\cdots(z-\lambda_n)^{e_n}(z-p_1)^{-g_1}\cdots(z-p_m)^{-g_m}$ . Thus, if $f$ is a meromorphic function on $\mathbb{C}_\infty$ with zeroes and poles described as in the last sentence we see that $\displaystyle \frac{f}{r}$ is a meromorphic function on $\mathbb{C}_\infty$ and which has no zeros or poles on $\mathbb{C}$ . In particular, $\displaystyle h=\frac{f}{r}$ is a function meromorphic on $\mathbb{C}_\infty$ but holomorphic on $\mathbb{C}$ , and with no zeros. Now, it’s a common fact from complex analysis that the only entire function with a pole at infinity (recall that the somewhat confusing definition of pole at infinity now makes a lot more sense!) is a polynomial. Thus, $h$ is a polynomial, but since $h$ has no zeros on $\mathbb{C}$ we know from the fundamental theorem of algebra that $h$ is constant. Thus, $f$ is really just a constant multiple of $r(z)$ ! Note that the key to this proof is that ability specify poles and zeros of a given multiplicity, except perhaps specifying a pole at infinity, and that the point infinity is well-behaved (in the sense that things that have poles there are pretty tame).

$\text{ }$

October 7, 2012 Posted by Alex Youcis | Complex Analysis, Riemann Surfaces | Classification, Complex Analysis, Intuition, Meromorphic Functions, Riemann Sphere, Riemann Surface | 1 Comment

Meromorphic Functions on Riemann Surfaces (Pt. III)

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

$\text{ }$

October 7, 2012 Posted by Alex Youcis | Complex Analysis, Riemann Surfaces | Complex Analysis, Independent of Chart, Intuition, Maps to Riemann Sphere, Meromorphic Functions, Order, Poles, Riemann Surfaces | Leave a Comment

Meromorphic Functions on Riemann Surfaces (Pt. II)

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

$\text{ }$

October 7, 2012 Posted by Alex Youcis | Complex Analysis, Riemann Surfaces, Uncategorized | Complex Analysis, Independent of Chart, Intuition, Maps to Riemann Sphere, Meromorphic Functions, Order, Poles, Riemann Surfaces | 2 Comments

Meromorphic Functions on Riemann Surfaces (Pt. I)

Point of Post: In this post we define meromorphic functions on Riemann surfaces and discuss several properties of such maps, including the correspondence between holomorphic functions and holomorphic maps to the Riemann sphere.

$\text{ }$

Motivation

$\text{ }$

So, up until this we’ve point we’ve defined Riemann surfaces and the structure preserving maps between them (the holomorphic ones). We should feel accomplished, no? I mean, we at least have the bare basics to start trying to do complex analysis on surfaces/use complex analysis to analyze the surfaces, right? Well, anyone who has taken a serious course in complex analysis should be vehemently shaking their head (you are, right?). Indeed, looking only at holomorphic functions leaves out a huge part of complex analysis. In particular, we should be weary of any discussion of complex analytic concepts that ignore meromorphic functions. Indeed, without considering meromorphic functions we wouldn’t have things like Cauchy’s integral formula, the theorem that enriches all of our study of holomorphic functions.

$\text{ }$

Another very obvious reason that we would want to study meromorphic functions comes from our previous revelatio that compact Riemann surfaces admit no non-constant holomorphic functions! We saw, looking at the proof that, perhaps the issues with this is that for globally defined functions into $\mathbb{C}$ their is a boundedness is the issue. Thus, allowing functions with singularities (and consequently for functions to get unbounded near these singularities!) will perhaps allow us to create meaningful (non-constant) functions from compact Riemann surfaces into $\mathbb{C}$ , which should be useful in the study of the compact Riemann surfaces themselves. Of course, such functions would be (if we were doing regular complex analysis!) meromorphic functions.

$\text{ }$

So, now that we have figured out why we would want to study meromorphic functions on Riemann surfaces, it remains to figure out precisely how to define them. Well, unsurprisingly, we define them like we define all local concepts on Riemann surfaces–via charts. Namely, just like in the case of holomorphic functions, we shall say that a function $f$ on a Riemann surface $X$ is meromorphic at $p$ if when we pretend that locally around $p$ and $f(p)$ we are dealing with subsets of $\mathbb{C}$ , and when we pretend that $f$ is a mapping between these two subsets, that $f$ is meromorphic at $p$ . Of course, this is rigorously done by checking the meromorphicity of $f$ ‘s coordinate map associated to a chart. We then have to verify that this meromorphicity of coordinate map is independent of chart transformation, but this is clear since if one conjugate a meromorphic map by biholomorphisms one gets a meromorphic map.

$\text{ }$

Ok, fine, so this is the obvious, not new definition of meromorphicity. But, as we have said before, not only does using abstract Riemann surfaces allow us to study such objects (you need to have Riemann surfaces and their pursuant concepts to study Riemann surfaces!) but it allows us to illuminate some old concepts. In particular, suppose that we wanted to try to extend a meromorphic function $f:\mathbb{C}\to\mathbb{C}$ to a holomorphic function? To put this on less confusing terms, we know that there is a discrete set $P$ of poles of $f$ such that $f:\mathbb{C}-P\to\mathbb{C}$ is holomorphic, and we want to extend this to a holomorphic map $\widetilde{f}:\mathbb{C}\to\mathbb{C}$ . Well, naively, if this $\widetilde{f}$ is to be holomorphic, it better be continuous. Thus, we have no choice but to define $\displaystyle \widetilde{f}(p)=\lim_{z\to p}f(z)$ for $p\in P$ . Now, there is an obvious issue with this since (by definition!) $\displaystyle \lim_{z\to p}|f(z)|=\infty$ , and so (with this definition) $\widetilde{f}(p)$ is not a well-defined element of $\mathbb{C}$ . Ah, but therein lies the rub. Namely, $\widetilde{f}(p)$ is not a well-defined element of the complex numbers. This doesn’t stop us from thinking about $\mathbb{C}$ as sitting inside a bigger Riemann surface $X$ and trying to extend $f:\mathbb{C}-P\to X$ (given by the composition $\mathbb{C}-P\xrightarrow{f}\mathbb{C}\hookrightarrow X$ ) where, perhaps, $\displaystyle \lim_{z\to p}f(z)$ may exist!

$\text{ }$

Thus, we are left looking for a Riemann surface $X$ with an embedding $\mathbb{C}\hookrightarrow X$ and a notion of an “infinite complex number”. Hopefully you aren’t thinking too hard. Namely, the extended complex plane $\mathbb{C}_\infty$ is exactly such a Riemann surface! Not only does $\mathbb{C}$ sit comfily inside of $\mathbb{C}_\infty$ , but the topology on $\mathbb{C}_\infty$ allows us to think of $\infty$ as an element for which defining $\displaystyle \widetilde{f}(p)$ makes sense! In particular, the knowledge that $|f(z)|$ gets arbitrarily large tells us that as $z\to p$ we have that $|f(z)|$ lies in discs $|z|>R$ for arbitrarily large values of $R$ . Now, if one starts looking at what the discs $|z|>R$ on the Riemann sphere one sees that the only point in all such discs is $\infty$ ! Thus, it actually makes topological sense to define $\widetilde{f}(p)=\infty$ . Thus, we see that meromorphic mappings $\mathbb{C}\to\mathbb{C}$ really correspond to maps $\mathbb{C}\to\mathbb{C}_\infty$ , and if there is any justice in the world, these maps will be holomorphic. Luckily for us, this is in fact true (the just world part might be pushing it though), and we shall see that meromorphic mappings $X\to\mathbb{C}$ (where $X$ is any Riemann surface!) correspond exactly to holomorphic maps $X\to\mathbb{C}_\infty$ . Well, almost, the only thing we need to watch out for is the condition that poles are discrete. But, since preimages of holomorphic non-constant maps are discrete we see the only trouble map is the map $X\to\{\infty\}\subseteq\mathbb{C}_\infty$ , which we shall just ignore.

$\text{ }$

Well, once we have the notion of meromorphic functions, the most basic thing we’d want to do with them is inspect how “bad” of a singularity they have at a given poin, or more specifically, how bad of a singularity either they or their inverse has at a given point. This leads us naturally to the notion of the order of a meromorphic function at a point. If we were doing plain old complex analysis on subsets of $\mathbb{C}$ this would be as simple as expanding our function as a Laurent series and finding the smallest non-zero coefficient (the integer attached to this coefficient would be the order). Simpleminded as we are, we would like to do the same thing here. Of course, the issue with this is the one that we have encountered at least fourteen times now–the fact that a Laurent series doesn’t make sense for a map $X\to\mathbb{C}$ when $X$ is an abstract Riemann surface. And, like the fourteen other times, we fix this by thinking locally. We will define the Laurent series of $f$ at $p$ , relative some chart $(U,\varphi)$ , to merely by the Laurent series of $f\circ\varphi^{-1}$ at $\varphi(p)$ . Now, in a perfect world of sunshine and lollipops, the Laurent series of $f$ would be the same with respect to any chart. Sadly, this is not true, for if it were true then one would have that the Laurent series of a function $f:\mathbb{C}\to\mathbb{C}$ at $p$ is the same as the Laurent series of $f\circ\psi$ at $\psi^{-1}(p)$ for any biholomorphism $\psi$ (on a neighborhood of $p$ )–you can find counterexamples to this. But, the key is that while the Laurent series is extremely chart dependent, the integer at which its lowest non-zero coefficient appears, does not. Thus, we get the well-defined notion of order that shall play such a pivotal role in our later studies.

$\text{ }$

Holomorphic Maps and Riemann Surfaces (Pt. IV)

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

$\text{ }$

October 4, 2012 Posted by Alex Youcis | Complex Analysis, Riemann Surfaces | Compact vs. Non-Compact, Complex Analysis, Fundamental Theorem of Algebra, Holomorphic Function, Holomorphic Map, Identity Theorem, Intuition, Riemann Surfaces, Weierstrass Product Theorem | 1 Comment

Holomorphic Maps and Functions (Pt. III)

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

$\text{ }$

Holomorphic Maps and Functions (Pt. II)

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

$\text{ }$

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.

$\text{ }$

Motivation

$\text{ }$

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 $\mathbb{C}$ .

$\text{ }$

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” $p$ ).

$\text{ }$

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.

$\text{ }$

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!

$\text{ }$

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 $\mathbb{C}$ ).

$\text{ }$

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.

$\text{ }$

October 4, 2012 Posted by Alex Youcis | Complex Analysis, Riemann Surfaces, Uncategorized | Compact vs. Non-Compact, Complex Analysis, Fundamental Theorem of Algebra, Holomorphic Function, Holomorphic Map, Identity Theorem, Intuition, Riemann Surfaces, Weierstrass Product Theorem | 1 Comment

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

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

$\text{ }$

October 3, 2012 Posted by Alex Youcis | Complex Analysis, Riemann Surfaces | Complex Analysis, Holomorphic, Holomorphic Implicit Function Theorem, Irreducible Algebraic Variety, Locus, Riemann Surface | Leave a Comment

« Previous Entries

About me

My name is Alex Youcis. I am currently a senior at the University of Maryland, College park getting my bachelor’s in mathematics. Before I came to UMD I was a math major at Drexel University in Philadelphia, Pennsylvania. I was born and raised in Lititz, Pennsylvania, a small town near Lancaster.

As is probably clear, I really enjoy doing math. It’s a passion for me, and I don’t means this in the weekend-hobby sense, I can’t imagine doing anything else other than mathematics. That said, I’m not one of those space cadets one often sees wandering around math departments, trying to solve others problems, or perpetrating some other equally creepy shenanigans. No, while math is a gargantuan aspect of my life, there are many, many other things that interest me. Perhaps the most consuming of these is my love of music. I find it difficult to go even one day without my iPhone, or grooveshark.com If you want to say something to me in the car, you have about eight seconds before my iPhone is plugged in and we’ve set sail on a musical adventure. I have hundreds, upon hundreds of bands that I intensely love, but at the time of writing I can emphatically say that Regina Spektor is my favorite artist–the woman is a musical Gauss. I also love dogs, friends, conversations about things, craft beer and the occasional online comic.

If you’d like to know anything about this blog see the pertinent page in all its splendorous glory.

If you have any questions regarding myself, math, music, or anything else feel free to e-mail me at alex.youcis@gmail.com

Current Course List:

Advanced Microeconomics

Numerical Analysis–uses Suli and Meyers

Riemann Surfaces–Uses Varolin. This is available, in near entirety, for free here.

Algebraic Number Theory–Using various course notes, including those of Milne. The course website is here.

Representation Theory of $p$ -adic Groups–This shall use various notes, the most intimidating of which is this manuscript of Bernstein, the least intimidating of which are these notes of Ban.

Search for:
Email Subscription

Enter your email address to subscribe to this blog and receive notifications of new posts by email.

Join 73 other followers
Tags

Algebra Algebraic Combinatorics Algebraic Topology Analysis Answers Category Theory Character Theory Complex Analysis Connectedness Differential Geometry Direct Limits Examples Field Theory Finite Dimensional Vector Spaces Full Solutions Functor Geometry Group Group Actions Group Algebra Groups Group Theory Halmos Homotopy Induced Representation Intuition Irreducible Characters Isomorphism Linear Algebra Linear Transformations Module Modules Module Theory Multivariable Analysis Munkres Representation Theory Review of Group Theory Riemann Surfaces Ring Rings Ring Theory Solutions Tensor Product Topology Total Derivative

Abstract Nonsense

Crushing one theorem at a time

Current Deal

Meromorphic Functions on the Riemann Sphere (Pt. II)

Meromorphic Functions on the Riemann Sphere (Pt. I)

Meromorphic Functions on Riemann Surfaces (Pt. III)

Meromorphic Functions on Riemann Surfaces (Pt. II)

Meromorphic Functions on Riemann Surfaces (Pt. I)

Holomorphic Maps and Riemann Surfaces (Pt. IV)

Holomorphic Maps and Functions (Pt. III)

Holomorphic Maps and Functions (Pt. II)

Holomorphic Maps and Functions (Pt. I)

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

About me

About Me

Blogroll

Lecture Notes

Categories

Top Posts

Email Subscription

Tags

Site info

Follow “Abstract Nonsense”