# Resources for Intro-Level Graduate Courses

In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses. Now before we jump in, let me preface the list by saying

it's FAR from comprehensive!

Most of the items below are accessible at the advanced undergraduate/early graduate level, so if you're looking for more advanced resources, you might be disappointed. Also, if you read my ramble about qualifying exams you'll know that my quals were in algebra, real analysis, and topology. So naturally my list is a little biased. But, hey, I figured a biased list is better than no list at all! In particular, I have very few items under differential geometry. That's simply because I haven't taken the course. But I am taking it this year, so I'll update this post as the months go on. (Bear with me, friends!)

And I've undoubtedly omitted several great resources either because I forgot to include them or I'm simply unaware of their existence. For this reason, I strongly encourage you to share your recommendations in the comments below! What books/links/etc. do you think are gems? Let me know! I'd love to hear.

Lastly, you'll see that I've included 'my favorite' books below. These aren't necessarily the most popular or 'the best.' They are simply the ones I enjoyed reading because of their accessibility and intuitive explanations.

And now, without further ado, I present to you The List!

## Books

### My favorite

• Basic Abstract Algebra by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul (I especially found the discussion on modules over a PID, Smith normal form, and rational canonical form helpful. See chapters 20-21.)

• Shameless plug: I once wrote a non-technical introduction to Galois theory. It's one of the blog's more popular posts!
• Keith Conrad has a goldmine of expositions (on pretty much everything) on his website. The reading is fairly easy, and there are plenty of worked-out examples.
• Here is a helpful collection of videos by Matthew Salome on Galois theory (field extensions, minimal polynomials, Galois groups, field automorphisms, etc.)
• CUNY's own MathDoctorBob also has a nice collection of videos on everything from the definition of a group to Sylow's Theorems to polynomial rings to Galois theory!
• What's the big deal with conjugation in group theory? I really like the accepted answer on this StackExchange post.

## Books

### My favorite:

• Real Analysis by N. L. Carothers (This is really a fantastic book (see my review here)! I found the discussion on absolute continuity especially illuminating. See chapter 20.)

## Books

### My favorite

• The Shape of Space by Jeffrey R. Weeks. (Filled with pictures and easy-to-read explanations, this book is a great resource for those first learning algebraic topology. I've written a review about it here!)

## Books

### My favorite

• Visual Complex Analysis by Tristan Needham (The intended audience for this book is undergraduate students in math, physics, and engineering, but it's so wonderfully written that I'm compelled to recommend it. I've also written a review about it here.)

• Shameless plug: Computations got you in the blues? Here's a motivational post for why we care about the automorphisms of the unit disc (upper half plane), complex plane, and Riemann sphere.
• Here's a video lecture series on advanced complex analysis: part 1 (Rouche's theorem, the open mapping theorem, Schwarz's Lemma, the Riemann mapping theorem, etc.) and part 2 (the Casorati-Weierstrass theorem, Picard's theorems, the Arzelà-Ascoli theorem, Montel's theorem etc.)

• coming soon!

# A Ramble About Qualifying Exams

This is my typical study environment. (Because Who needs a desk when you've got the floor?)

Today I'm talking about about qualifying exams! But no, I won't be dishing out advice on preparing for these exams. Tons of excellent advice is readily available online, so I'm not sure I can contribute much that isn't already out there. However, it's that very web-search that has prompted me to write this post.

You see, before I started graduate school I had heard of these rites-of-passage called the qualifying exams.* And to be frank, I thought they sounded terrifying. A sequence of exams, the results of which determine whether or not I'm 'qualified' to stay in the program? From the information I found online and from the first-hand stories I heard from actual graduate students, I was under the miserable assumption that it's impossible to pass these exams unless

1. you're constantly depressed (and any accidental feelings of happiness should be followed by remorse)
2. your health has sufficiently deteriorated due to lack of sleep, forgetting to eat, drinking too much coffee** etc.
3. your only extracurricular activities consist of 1) crying and 2) re-evaluating your life goals

Sounds traumatic, right? Well, now that I'm on the other side of things, I'm happy to report that it was not that bad! Difficult? For sure. But impossible? No.

Don't get me wrong though! I'm not saying quals are a walk in the park. Far from it. And of course the experience is different for every student. Even so, I'd like to ramble on a bit about my trauma-free experience, not because I think it's interesting (it's not, truly), but just in case a prospective or first-year graduate student will Google, "how to prepare for math quals," stumble upon this post, and walk away feeling like all is not lost. (I personally savored 'qual success stories' before entering graduate school because they always gave me a glimmer of hope.)

the best part of taking a topology course is that one gets to use colored pencils on a regular basis.

Now lest you think quals are manageable only for "extraordinarily gifted" students, let me be quick to say that I do not consider myself as possessing exceptional (or even ordinary) mathematical talents. What do I mean? To put it simply:

Math has never come easy to me.

I just happen to love it.

A lot.

Interestingly enough, I didn't even like math until my sophomore year of college! And while I'm divulging all my secrets, I might as well tell you*** that I also did not earn a master's degree before entering my PhD program. (Some students may find their quals doable since they've seen the material previously in a master's program.) In fact I had never taken a graduate-level course before! This of course made the transition to grad school challenging. For instance, my algebra professor covered everything - everything! - I knew from college within the first three weeks of my first semester. Three weeks, people! THREE WEEKS. I felt as if I had had the mathematical-wind knocked out of me.

Yes, i created flash cards for my algebra qual. you may think that's silly, but see that card in the upper left-hand corner? i had written incorrect information on the back of it, and one of my classmates pointed out the mistake in a group study session A week BEFORE THE exam. Embarrassed, i went home that night and carefully shored up my misunderstandings. And as it turned out, one of the problems on the qual the following week was based on the information on that very card. And you bet i got it right that time! (not surprisingly, it also became a blog post.)

Fortunately things have gotten easier as the semesters go on. Not the material, mind you, but the speed at which I can comprehend and analyze new material. It's very much like lifting weights. Your muscle fibers tear during the process and then they rebuild. Even though it's painful, your strength - your ability to handle greater loads - increases.

But metaphors aside, I was well aware of my background, and so I started preparing for the quals from "day one." This just means that my one and only goal was to pass these exams. So I solved my homework problems, took lecture notes, took notes on my lecture notes, and studied for my midterm/final exams all the while knowing that the material could appear on the quals. I deliberately chose not to attend many seminars, or take special-topics courses or independent studies, or attempt to read articles/papers on the side. Now it goes without saying that this path is not necessary (or even helpful) for every student! It's just the path that worked best for me.

So here's the deal. At my school, we are required to pass three exams - offered each September and May - chosen from six subjects: real analysis, abstract algebra, topology, logic, complex analysis, and differential geometry. Some students take (and pass!) all three of their quals at once. But I chose to take one qual at a time. Here's what my timeline looked like, beginning with August 2014 when I first entered graduate school:

As you can see, I started studying for the algebra and topology quals five months in advance. Here, "studying" equates to working on problems from old quals, solving and resolving homework exercises, reviewing major theorems/proofs, asking lots of questions, and even starting a blog! Also, my classmates and I formed study groups, and this was a tremendous help.

I mean, t-r-e-m-e-n-d-o-u-s.

Admittedly, I'm not much of a group-work kind of person. I usually prefer to study alone. But I'm certain that I would have failed all of my exams if not for my classmates - who are amazing, by the way! - and their help. (If you're one of them and are reading this now, thank you!!)

some scratch notes i made the night before my topology qual. Pretty ain't it?

Well, I think that's about it! So you see? There was nothing too exciting to report. Really, a qual is just one big final exam. And like any exam, you (presumably) devote 100% of your efforts from the start of the semester to avoid cramming/panic attacks/etc. So in that sense, the quals weren't such a big deal. But again, everyone's experience is unique. If you ask any other graduate student, they'll likely have a very different story to tell!

And as promised, I haven't (intentionally) given any advice on preparing for the quals. But if you are looking for some, this link might be a good place to start. Asking professors and other graduate students in your program for tips is a good idea as well. Of course, if you have any specific questions/comments, feel free to leave them below!

So to all those preparing - mentally or mathematically - for their qualifying exams, I wish you all the best!

* For those not in the know, the qualifying exams (a.k.a preliminary exams) are a series of two to four - depending on the university - written exams that graduate students must pass within their first year or two of study. Students who do not pass the exams are not able to remain in the program. (At my school, if we happen to fail an exam, we have exactly one chance to retake it.)

In some programs, such as mine, students must also pass an oral examination. In this exam, administered after the quals are passed, the student gives an oral presentation to a committee of faculty members on a specialized topic. Passing the oral exam is next on my agenda.

** Ha! Who am I kidding? There's no such thing as "too much coffee."

***I suppose my reason for sharing this is to let you know that one does not need to be a genius to pass one's quals. (Of course, it doesn't hurt either!) One does, however, need to work hard.

# Automorphisms of the Riemann Sphere

Welcome to the final post in our summer series on automorphisms of four (though, for all practical purposes, it's really three) different Riemann surfaces: the unit disc, the upper half plane, the complex plane, and the Riemann sphere. Last time, we proved that the automorphisms of the complex plane take on a certain form. Today, we'll close the series by proving a similar result about automorphisms of the Riemann sphere.

If you missed the introductory/motivational post for this series, be sure to check it out here!

Also in this series:

## Automorphisms of the Riemann Sphere

Theorem: Every automorphism $f$ of the Riemann sphere $\hat{\mathbb{C}}$ is of the form $f(z)=\frac{az+b}{cz+d}$ where $a,b,c,d\in\mathbb{C}$ such that $ad-bc\neq 0$.

Proof. If $f(z)=\frac{az+b}{cz+d}$ for $a,b,c,d\in\mathbb{C}$ with $ad-bc\neq 0$, then $f$ is a linear fractional transformation and is thus an automorphism of $\hat{\mathbb{C}}$. The converse direction is a corollary of the following proposition.

Proposition: A function $f$ is meromorphic* in $\hat{\mathbb{C}}$ if and only if it is a rational map.

Proof. First suppose $f$ is a meromorphic function. Then $f$ has only a finite number of poles in $\hat{\mathbb{C}}$. This follows since the poles of $f$ form a discrete, closed subset of $\hat{\mathbb{C}}$. (The set is closed because its complement, the collection of points where $f$ is holomorphic, is open. To see this, note that for any point $z$ at which $f$ is holomorphic, we can find a small enough neighborhood $N$ about $z$ such that $f$ is also holomorphic on $N$.) Hence it must be finite.

So suppose the poles of $f$ are $a_1, a_2,\ldots, a_n$ and consider the Laurent expansion of $f$ at $a_1$: $$f(z)=\frac{A_{-m}}{(z-a_1)^m}+\cdots+\frac{A_{-1}}{z-a_1}+A_0 +A_1(z-a_1) + \cdots.$$ Let $p_1(z)=\frac{A_{-m}}{(z-a_1)^m}+\cdots+\frac{A_{-1}}{z-a_1}$ be the principal part of $f$ and let $f_1(z)=A_0 +A_1(z-a_1) + \cdots$ so that $f(z)=p_1(z)+f_1(z)$. Observe that $p_1$ is holomorphic for all $z\neq a_1$ and has a pole at $z=a_1$. Moreover, $f_1$ has the same number of poles as $f$ except at $z=a_1$.

Repeating this argument for each $i=1,2,\ldots, n$, we obtain the collection of principal parts $p_i(z)$ of $f$ at the pole $a_i$ . So consider the function $$f_n(z)= f(z)-p_1(z)-\cdots - p_n(z)$$ and observe that $f_n$ is holomorphic on $\mathbb{C}$ and has a pole at $\infty$. This implies that $f_n$ must be a polynomial and so $f(z)=f_n(z)+p_1(z)+\cdots p_n(z)$ is a rational function, i.e. there are polynomials $P$ and $Q$ with coefficients in $\mathbb{C}$ for which $f(z)=\frac{P(z)}{Q(z)}$.

Conversely let's assume $f=\frac{P(z)}{Q(z)}=\frac{a_nz^n+\cdots+a_0}{b_mz^m+\cdots+b_0}$ is a rational function expressed in simplest terms so that $P$ and $Q$ share no common roots. Suppose $z_0$ is a zero of $Q(z)$ of order $k$ so that $Q(z)=(z-z_0)^kQ_1(z)$ where $Q_1$ is a polynomial of degree $m-k$ satisfying $Q_1(z_0)\neq 0$. Then $$f(z)=\frac{P(z)}{(z-z_0)^kQ_1(z)}=\frac{\varphi(z)}{(z-z_0)^k}$$ where $\varphi(z)=P(z)/Q_1(z)$ is holomorphic and $\varphi(z_0)\neq 0$. It follows that $z_0$ is a pole of $f$. Hence the zeros of $Q$ are precisely the poles of $f$ (and are of the same order). Thus $f$ is meromorphic.

$\square$

Corollary: If $f$ is an automorphism of $\hat{\mathbb{C}}$ then $f$ is of the form $f(z)=\frac{az+b}{cz+d}$ where $a,b,c,d\in\mathbb{C}$ satisfy $ad-bc\neq 0$.

Proof. Suppose $f$ is an automorphism of $\hat{\mathbb{C}}$. Then $f$ must be meromorphic as $f$ must map one point to $\infty$ and thus have a pole. By the previous proposition, $f$ must be of the form $f(z)=\frac{P(z)}{Q(z)}$ for polynomials $P$ and $Q$. But by assumption $f$ is injective and so $P$ and $Q$ must be linear in $z$. In other words $f(z)=\frac{az+b}{cz+d}$ for $a,b,c,d\in\mathbb{C}$. (And a quick check shows that the condition $ad-bc\neq 0$ is needed to guarantee that $f^{-1}$ exists.) This completes the proof of the Theorem above.

$\square$

*that is, $f$ is holomorphic on all of $\hat{\mathbb{C}}$ except for a set of isolated points, namely the poles of $f$.
Comment

# Automorphisms of the Complex Plane

This is part three of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the complex plane.

# Automorphisms of the Upper Half Plane

This is part two of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the upper half plane.

# Automorphisms of the Unit Disc

This is part one of a four-part series in which we prove that the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere each take on a different form. Today our focus is on the unit disc.

# Three Important Riemann Surfaces

In this post we ramble on about Riemann surfaces, the uniformization theorem, universal covers, and two secret (or not-so-secret!) techniques that mathematicians use to study a given space. Our intent is to provide motivation for an upcoming mini-series on the automorphisms of the unit disc, upper half plane, complex plane, and Riemann sphere.

# Good Reads: The Princeton Companion to Mathematics

Next up on Good Reads: The Princeton Companion to Mathematics, edited by Fields medalist Timothy GowersThis book is an exceptional resource! With over 1,000 pages of mathematics explained by the experts for the layperson, it's like an encyclopedia for math, but so much more. Have you heard about category theory but aren't sure what it is? There's a chapter for that! Seen the recent headlines about the abc conjecture but don't know what it's about? There's a chapter for that! Need a crash course in general relativity and Einstein's equations, or the P vs. NP conjecture, or C*-algebras, or the Riemann zeta function, or Calabi-Yau manifolds? There are chapters for all of those and more

# Clever Homotopy Equivalences

You know the routine. You come across a topological space X and you need to find its fundamental group. Unfortunately, X is an unfamiliar space and it's too difficult to look at explicit loops and relations. So what do you do? You look for another space Y that is homotopy equivalent to X and whose fundamental group is much easier to compute. And voila! Since X and Y are homotopy equivalent, you know that the fundamental group of X is isomorphic to the fundamental group of Y. Mission accomplished.

Below is a list of some homotopy equivalences which I think are pretty clever and useful to keep in your back pocket for, say, a qualifying exam or some other pressing topological occasion.

# Snippets of Mathematical Candor

A while ago I wrote a post in response to a great Slate article reminding us that math - like writing - isn't something that anyone is good at without (at least a little!) effort. As the article's author put it, "no one is born knowing the axiom of completeness." Since then, I've come across a few other snippets of mathematical candor that I found both helpful and encouraging. And since final/qualifying exam season is right around the corner, I've decided to share them here on the blog for a little morale-boosting.

# (Co)homology: A Poem

I was recently (avoiding doing my homology homework by) reading through some old poems by Shel Silverstein, author of The Giving TreeA Light in the Attic, and Falling Up to name a few. Feeling inspired, I continued to procrastinate by writing a little poem of my own - about homology, naturally!