# Some Notes on Taking Notes

These are My favorite notebooks, along with my favorite pencil. I write in notebook number "zero" during seminars and meetings. And my more carefully written day-to-day notes are kept in notebooks "one," "two" and "three."

A quick browse through my Instagram account and you might guess that I take notes. Lots of notes. And you'd be spot on! For this reason, I suppose, I am often asked the question, "How do you do it?!" Now while I don't think my note-taking strategy is particularly special, I am happy to share! I'll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to "stick." So while some people think aloud (or quietly),

I think on paper.

My study habits, then, are built on this fact. Of course not everyone learns in this way, so this post is not intended to be a how-to guide. It's just a here's-what-I-do guide.

With that said, below is a step-by-step process I tried to follow during my final years of undergrad and first two years of grad school.**

## Step 1

### Read the appropriate chapter/section in the book before class

I am an "active reader," so my books have tons of scribbles, underlines, questions, and "aha" moments written on the pages. I like to write while I read because it gives me time to pause and think about the material. For me, reading a mathematical text is not like reading a novel. It often takes me a long time just to understand a single paragraph! Or a single sentence. I also like to mark things that I don't understand so I'll know what to look for in the upcoming lecture.

Category Theory in Context by Emily Riehl. Illegible pencil markings by me.

I own several PDF texts too, though I prefer physical books. (It's hard to snuggle up with your computer at night.) But I like to mark up those PDFs as well. I use the notation tools available in Apple's Preview app for this.

Algebra + Homotopy = Operad By Bruno Vallette. Colorful Annotations courtesy of Apple.

## STEP 2

### Attend lecture and take notes

This step is pretty self-explanatory, but I will mention this: I write down much more than what is written on the chalkboard (or whiteboard). In fact, a good portion of my in-class notes consists of what the professor has said but hasn't written. And perhaps now is a good time to tell you...

• I used to take lecture notes with a Lamy fountain pen, then I moved on to this gel pen and also this one. Now I use a mechanical pencil (see next bullet). But I always have one red and one green pen with me. I use green for questions ("What did the prof just say?" "What does this mean?" "How is this even possible?") and red for things I really want to remember. ("Oh that's what she meant." "Here's the key!" "Of course it must be possible!") I use this green/red scheme in my books and PDFs, too.

• While taking notes at home (step 3 below) I use a Rotring 600 drafting pencil. And I love dark, silky-smooth fine lines, so I use 0.5mm 3B grade lead. To date, my favorite is this one.

• For years I took notes on lined paper, but Jeremy Kun convinced me otherwise. Nowadays lined paper makes me cringe. Instead, I use blank white printing paper for my in-class and scratch notes (I always have a clip-board with me) and these spiral-bound, unruled Maruman Mnemosyne notebooks for all other notes*** (pictured above; also see step 3 below).

• I don't use LaTeX for my everyday note-taking. (But I used to type up my homework solutions in LaTeX.) While I do enjoy typesetting math, it can be time-consuming, especially when including pictures and diagrams. Plus, I like the feel of pencil-to-paper and, admittedly, the look of my own handwriting!

My arsenal

## STEP 3

### Rewrite lecture notes at home

My in-class notes are often an incomprehensible mess of frantically-scribbled hieroglyphs. So when I go home, I like to rewrite everything in a more organized fashion. This gives the information time to simmer and marinate in my brain. I'm able to ponder each statement at my own pace, fill in any gaps, and/or work through any exercises the professor might have suggested. I'll also refer back to the textbook as needed.

Sometimes while rewriting these notes, I'll copy things word-for-word (either from the lecture, the textbook, or both), especially if the material is very new or very dense. Although this can be redundant, it helps me slow down and lets me think about what the ideas really mean. Other times I'll just rewrite things in my own words in a way that makes sense to me.

A semester's worth of notes!

As for the content itself, my notes usually follow a "definition then theorem then proof" outline, simply because that's how material is often presented in the lecture. But sometimes it's hard to see the forest for the trees (i.e. it's easy to get lost in the details), so I'll occasionally write "PAUSE!" or "KEY IDEA!" in the middle of the page. I'll then take the time to write a mini exposition that summarizes the main idea of the previous pages. I've found this to be especially helpful when looking back at my notes after several months (or years) have gone by. I may not have time to read all the details/calculations, so it's nice to glance at a summary for a quick refresher.

And every now and then, I'll rewrite my rewritten notes in the form of a blog post! Many of my earlier posts here at Math3ma were "aha" moments that are now engrained in my brain because I took the time to blog about them.

## STEP 4

### Do homework problems

Once upon a time, I used to think the following:

How can I do problems if I haven't spent a bajillion hours learning the theory first?

But now I believe there's something to be said for the converse:

How can I understand the theory if I haven't done a bajillion examples first?

In other words, taking good notes and understanding theory is one thing, but putting that theory into practice is a completely different beast. As a wise person once said, "The only way to learn math is to DO math." So although I've listed "do homework problems" as the last step, I think it's really first in terms of priority.

Typically, then, I'll make a short to-do list (which includes homework assignments along with other study-related duties) each morning. And I'll give myself a time limit for each task. For example, something like "geometry HW, 3 hours" might appear on my list, whereas "do geometry today" will not. Setting a time gives me a goal to reach for which helps me stay focused. And I may be tricking my brain here, but a specific, three-hour assignment sounds much less daunting than an unspecified, all-day task. (Of course, my lists always contain multiple items that take several hours each, but as the old adage goes, "How do you eat an elephant? One bite at a time.")

By the way, in my first two years of grad school I often worked with my classmates on homework problems. I didn't do this in college, but in grad school I've found it tricky to digest all the material alone - there's just too much of it! So typically I'd first attempt exercises on my own, then meet up with a classmate or two to discuss our ideas and solutions and perhaps attend office hours with any questions.

As far as storage goes, I have a huge binder that contains all of my rewritten notes*** from my first and second year classes. (I use sheet protectors to keep them organized according to subject.) On the other hard, I use a paper tray like this one to store my lecture notes while the semester is in progress. Once classes are over, I'll scan and save them to an external hard drive. I've also scanned and saved all my homework assignments.

Well, I think that's about it! As I mentioned earlier, these steps were only my ideal plan. I often couldn't apply them to every class -- there's just not enough time! -- so I'd only do it for my more difficult courses. And even then, there might not be enough time for steps 1 and 3, and I'd have to start working on homework right after a lecture.

But as my advisor recently told me, "It's okay to not know everything." Indeed, I think the main thing is to just do something. Anything. As much as you can. And as time goes on, you realize you really are learning something, even if it doesn't feel like it at the time.

Alright, friends, I think that's all I have to share. I hope it was somewhat informative. If you have any questions, don't hesitate to leave it in a comment below!

*Truly. I've kept a journal since I was seven years old!

**Typically, the first one or two years of grad school are much different than the latter years. (I've just finished my third year.) Once the written qualifying exams are over, less time is spent in the classroom and more time is spent on research. For that reason, my current note-taking strategy doesn't follow this four-step scheme. See next footnote.

***A word of clarification: Up until several months ago, I would rewrite my lecture notes on loose-leaf paper. But these days I don't rewrite my lecture notes because I don't take many classes! (That's just the nature of graduate school.) I do, however, learn things on my own (guided by my advisor) and therefore still take lots of notes. And that is what I use those black, spiral-bound Maruman notebooks are for.

# "One-Line" Proof: Fundamental Group of the Circle

Once upon a time I wrote a six-part blog series on why the fundamental group of the circle is isomorphic to the integers. (You can read it here, though you may want to grab a cup of coffee first.) Last week, I shared a proof* of the same result. In one line. On Twitter. I also included a fewer-than-140-characters explanation. But the ideas are so cool that I'd like to elaborate a little more.

As you might guess, the tools are more sophisticated than those in the original proof, but they make frequent appearances in both topology and category theory, so I think it's worth a blog post. (Or six. Heh.) To keep the discussion at a reasonable length, I'll have to assume the reader has some familiarity with algebraic topology and basic category theory. But even if some of the words sound foreign, I encourage you to read as much as you can! My hope is that this post will whet your appetite to study further.

So without further ado, I present

### Proof:

Let's take a closer look at each of the three isomorphisms.

There are two important functors in topology called based loop $\Omega$ and reduced suspension $\Sigma$:
The loop functor $\Omega$ assigns to each pointed space $X$ (that is, a space with a designated basepoint) the space $\Omega X$ of based loops in $X$, i.e. loops that start and end at the basepoint of $X$. On the other hand, $\Sigma$ assigns to each $X$ the (reduced) suspension $\Sigma X$ of $X$. This space is the smash product of $X$ with $S^1$. In general it might not be easy to draw a picture of $\Sigma X$, but when an $n$-dimensional sphere, it turns out that $\Sigma S^n$ is homeomorphic to $S^{n+1}$ for $n\geq 0$. So for $n=1$ the picture is

The loop-suspension adjunction is a handy, categorical result which says that $\Omega$ and $\Sigma$ interact very nicely with each other: up to homotopy, maps out of suspension spaces are the same as maps in to loop spaces. More precisely, for all pointed topological spaces $X$ and $Y$ there is a natural isomorphism
Here I'm using the notation $[A,B]$ to indicate the set of homotopy classes of basepoint-preserving maps from $A\to B$. (Two based maps are homotopic if there is a basepoint-preserving homotopy between them. This is an equivalence relation, and the equivalence classes are given the name homotopy classes.) This, together with the observation that the $n$th homotopy group $\pi_n(X)$ is by definition $[S^n,X]$, yields the following:
And that's the first isomorphism above!**

Remark: The $\Omega-\Sigma$ adjunction is just one example of a general categorial construction. Two functors are said to form an adjunction if they are - very loosely speaking - dual to each other. I had planned to blog about adjunctions after our series on natural transformations but ran out of time! In the mean time, I recommend looking at chapter 4 of Emily Riehl's Category Theory in Context for a nice discussion.

## The Homotopy Equivalence

Next, let's say a word about why $\Omega S^1$ and $\mathbb{Z}$ are homotopy equivalent. This equivalence will immediately imply the second isomorphism above since $\pi_0$ (and in fact each $\pi_n$) is a functor, and functors preserve isomorphisms. (That is, $\pi_0$ sends homotopy equivalent spaces to isomorphic sets.***) Now, why are $\Omega S^1$ and $\mathbb{Z}$ homotopy equivalent? It's a consequence of the

Claim: A homotopy equivalence between fibrations induces a homotopy equivalence between fibers.

Eh, that was a mouthful, I know. Let's unwind it. Roughly speaking, a map $p:E\to B$ of topological spaces is called a fibration over $B$ if you can always lift a homotopy in $B$ to a homotopy in $E$, provided the initial "slice" of the homotopy in $B$ has a lift. And the preimage $p^{-1}(b)\subset E$ of a point in $b$ is called the fiber of $b$. So the claim is that if $p:E\to B$ and $p':E'\to B$ are two fibrations over $B$, and if there is a map between them that is a homotopy equivalence (We'd need to properly define what this entails, but it can be done.) then there is a homotopy equivalence between fibers $p^{-1}(b)$ and $p'^{-1}(b)$.

### Example #1

The familiar map $\mathbb{R}\to S^1$ that winds $\mathbb{R}$ around the circle by $x\mapsto e^{2\pi ix}$ is a fibration, and the fiber above the basepoint $1\in S^1$ is $\mathbb{Z}$. Incidentally, we proved this in the original six-part series that I mentioned earlier!

### Example #2

The based path space $\mathscr{P}S^1$ of the circle gives another example. This is the space of all paths in $S^1$ that start at the basepoint $1\in S^1$. The map $\mathscr{P}S^1\to S^1$ which sends a path to its end point is a fibration. What's the fiber above $1$? By definition, a path is in the fiber if and only if it starts and ends at 1. But that's precisely a loop in $S^1$! So the fiber above 1 is $\Omega S^1$.
These examples give us two fibrations over the circle: $\mathbb{R}\to S^1$ and $\mathscr{P} S^1\to S^1$. And it gets even better. Both $\mathbb{R}$ and $\mathscr{P}S^1$ are contractible and therefore homotopy equivalent! By the claim above, $\Omega S^1$ and $\mathbb{Z}$ must be homotopy equivalent, too. This gives us the second isomorphism above.

Pretty neat, right? If you're interested in the details of the claim and ideas used here, take a look at J. P. May's A Concise Course in Algebraic Topology, chapter 7.5. By the way, there is a dual notion to fibrations called cofibrations. (Roughly: a map is a cofibration if you can extend - rather than lift - homotopies.) And both of these topological maps have abstract, categorical counterparts -- also called (co)fibrations -- which play a central role in model categories.

## The Integers are Discrete

The third isomorphism is relatively simple: we just have to think about what $\pi_0(X)$ really is. Recall that $\pi_0(X)=[S^0,X]$ consists of homotopy classes of basepoint-preserving maps $S^0\to X$. But $S^0$ is just two points, say $-1$ and $+1$, and one of them, say $-1$, must map to the basepoint of $X$. So a basepoint-preserving map $S^0\to X$ is really just a choice of a point in $X$. And any two such maps are homotopic when there's a path between the corresponding points! So $\pi_0(X)$ is the set of path components of $X$.

It follows that $\pi_0(\mathbb{Z})\cong\mathbb{Z}$ since there are $\mathbb{Z}$-many path-components in $\mathbb{Z}$. And that's precisely the third isomorphism above!

And with that, we conclude

## QED

Okay, okay, I suppose with all the background and justification, this isn't an honest-to-goodness one-line proof. But I still think it's pretty cool! Especially since it calls on some nice constructions in topology and category theory.

Well, as promised in my previous post I'm (supposed to be) taking a small break from blogging to prepare for my oral exam. But I had to come out of hiding to share this with you - I thought it was too good not to!

Until next time!

* I first learned of this proof from my advisor while taking a course in K-theory last semester. I've been meaning to blog about it ever since!

** You might worry that $\pi_0(\Omega S^1)$ is just a set with no extra structure. But it's actually a group! To see this, note that there is a multiplication on $\Omega S^1$ given by loop concatenation. It's not associative, but it is up to homotopy. (So loops spaces are not groups. They are, however, $A_\infty$ spaces.) So in general, sets of the form $[X,\Omega Y]$ are groups. For more, see May's book section 8.2.

*** Yes, sets. Not groups. In general, $\pi_0(X)$ is merely a set (unlike $\pi_n(X)$ for $n\geq 1$ which is always a group). But we're guaranteed that $\pi_0(\mathbb{Z})$ is a group since it's isomorphic to $\pi_0(\Omega S^1)$ (and see the second footnote).
Comment

# crumbs!

One of my students recently said to me, "I'm not good at math because I'm really slow." Right then and there, she had voiced what is one of many misconceptions that folks have about math.

But friends, speed has nothing to do with one's ability to do mathematics. In particular, being "slow" does not mean you do not have the ability to think about, understand, or enjoy the ideas of math.

Let me tell you....

Last weekend, I spent seven (SEVEN!) hours trying to understand two-and-a-half (TWO-AND-A-HALF!) sentences of a page-long proof. (My brain just couldn't make it to the end of the third sentence.) And this past weekend, it took me two (TWO!) full days (DAYS!) to understand three (THRE--okay, I'll stop shouting) little equations.

But now I finally understand the proof, and now I know what those little equations really mean. Even better, I understand them both thoroughly because of (not in spite of) all those hours that I spent with them.

So, as I wanted to tell my student:

Who cares how long it takes you?

We can never focus on and enjoy the mathematical scenery around us
if we're too busy concerned with racing our neighbors.

Don't forget, dear friends:

Speed is not equivalent to mathematical ability.

On a very related note, I am currently preparing for my oral exam*, so free time is a rarity these days. For that reason, I'm going to take a little break from blogging over the next few months. But I do hope to have lots of cool things to write about when my exam is over! In the mean time, I plan to continue sharing math-y things on Facebook, Twitter, and Instagram. See you in a few months!

*This means I'm learning about a specialized topic now, and will soon give an oral presentation (a seminar talk) in front of some faculty members. Afterwards, I'll be asked a bunch of questions about the topic to ensure I'm ready to move on to the dissertation stage of grad school. No pressure, really ;)

Comment

# crumbs!

Physicist Freeman Dyson once observed that there are two types of mathematicians: birds -- those who fly high, enjoy the big picture, and look for unifying concepts -- and frogs -- those who dwell on the ground, find beauty in the scenery close by, and enjoy the details.

Of course, both vantage points are essential to mathematical progress, and
I often tend to think of myself as more of a bird.
(I'm, uh, bird-brained?)

Well one day I complained to my advisor, John Terilla -- you can see him doing cool math in that video above! -- about one of my classes which at the time felt a bit froggy:

Me: I'm having a hard time enjoying this class. I don't want to do a bunch of detailed computations on these objects. I'd rather spend my time learning category theory. I want to soar high and study the full landscape!

John: Well, yes, the sky is nice. But the ground -- and the ocean -- is nice, too. In fact, the best place to be is where the sky and the ocean meet, where the tips of the waves turn white.

That's the only place where you can go surfing.

That's where you want to be.

Snap, y'all.

I think I've a new respect for the little details.

Let's go surfing!

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# Group Elements, Categorically

On Monday we concluded our mini-series on basic category theory with a discussion on natural transformations and functors. This led us to make the simple observation that the elements of any set are really just functions from the single-point set {✳︎} to that set. But what if we replace "set" by "group"? Can we view group elements categorically as well?

The answer to that question is the topic for today's post, written by guest-author Arthur Parzygnat.

# What is a Natural Transformation? Definition and Examples, Part 2

Continuing our list of examples of natural transformations, here is: Example #2 (double dual space of a vector space) and Example #3 (representability and Yoneda's lemma).

# What is a Natural Transformation? Definition and Examples

I hope you have enjoyed our little series on basic category theory. (I know I have!) This week we'll close out by chatting about natural transformations which are, in short, a nice way of moving from one functor to another. If you're new to this mini-series, be sure to check out the very first post, What is Category Theory Anyway? as well as What is a Category? and last week's What is a Functor?

# crumbs!

I was at the grocery store earlier today, minding my own business, and while I was intently studying the lentil beans (Why are there so many options?) a man came down the aisle, pushing a cart with him. He then stopped in front of me, turned, looked me directly in the eyes and said...

# What is a Functor? Definitions and Examples, Part 2

Continuing yesterday's list of examples of functors, here is Example #3 (the chain rule from multivariable calculus), Example #4 (contravariant functors), and Example #5 (representable functors).

# What is a Functor? Definition and Examples, Part 1

Next up in our mini series on basic category theory: functors! We began this series by asking What is category theory, anyway? and last week walked through the precise definition of a category along with some examples. As we saw in example #3 in that post, a functor can be viewed an arrow/morphism between two categories.

# Introducing... crumbs!

Hello friends! I've decided to launch a new series on the blog called crumbs! Every now and then, I'd like to share little stories -- crumbs, if you will -- from behind the scenes of Math3ma. To start us off, I posted (a slightly modified version of) the story below on January 23 on Facebook/Twitter/Instagram, so you may have seen this one already. Even so, I thought it'd be a good fit for the blog as well. I have a few more of these quick, soft-topic blurbs that I plan to share throughout the year. So stay tuned! I do hope you'll enjoy this newest addition to Math3ma.

# What is a Category? Definition and Examples

As promised, here is the first in our triad of posts on basic category theory definitions: categories, functors, and natural transformations. If you're just now tuning in and are wondering what is category theory, anyway? be sure to follow the link to find out!

A category 𝖢 consists of some data that satisfy certain properties...