# 18.S190 by Paige B. '24

that's .0093 more than 18.S097!

For the second year in a row, I am teaching a class at MIT. A real, for-credit, in-the-catalog-of-classes class, at MIT. That I teach. On math.

Let’s back up a moment.

The (Loose) History/Background

In the math department, there are four versions of one class known as real analysis: 18.100A, 18.100B, 18.100P, and 18.100Q. A few things of note: i) real analysis seeks to, in addition to other pedagogical goals, teach how one proves that calculus works, and ii) P and Q are meant to just be communication intensive versions of A and B but this is fairly besides the point. One might ask, why are there so many? To which the indirect answer would be to say that there didn’t use to be.

There used to be (from my understanding of the history of the course) just one version of the class that all pure mathematics majors are supposed to take. This course was deep, and rigorous, and really hard. Like, it’s meant to be a class you can in theory take after 18.02,01 Multivariable Calculus  but it’s a huge leap. In fact, it is one of the largest leaps in math conceptually.

This may be for a few reasons. One may be that there didn’t exist a class on proof writing.02 at least until last year, when a class was created motivated by one class run by the UMA and various people around the mathematics department So students were expected to 1) learn hard material, 2) learn how to prove hard material, and 3) try to maintain their sanity, all at the same time.

Another reason, simply, might be that the class is just so abstract. The classic “we aren’t in Kansas” level abstract. In 18.02, mathematics is done over Euclidean space.03 for our purposes, you can just picture this as a flat plane which is two-dimensional Euclidean space  But the techniques and topics used to prove facts about calculus that students are vaguely expected to understand by this point can be used in much, much, higher generality.

So only having one class meant, almost consistently, under serving some subset of the class, whether that be students who want to learn everything at the difficult abstract level, or those who may want to learn those topics later, but who needed/wanted more time with the introductory simple examples.

Thus, over time, the class was split into two classes: real analysis over Euclidean space, and real analysis over metric spaces.

This division, however, can occasionally create its own problems for students.

What’s a Metric?

A metric is, at its core, an understanding of distance. We measure distance everyday, but in weirder ways than one might first imagine. Take for instance, Euclidean distance. This notion of a distance, is essentially the Pythagorean theorem: it gives the shortest path from point a to point b. However, you can also measure distance slightly differently. This tool is deeply important in multivariable calculus as it gives a notion of  magnitude for vectors.04 in particular, this is given by the Euclidean distance from a point <em>a</em> to the origin

Take for example, the taxicab metric, named so after grid-like cities such as New York City. Suppose I want to go from the corner on 1st and 42nd to the corner on 2nd and 43rd. Reasonably, I cannot walk through walls to get to my destination quickly and effectively; rather, I need to follow the grid. Like I’m in a taxicab that can only drive on the road– you add the number of blocks (in this case, two) as opposed to the absolute shortest distance.

So again, a metric is conceptually just a distance, a function, d( , ), which we want to satisfy certain properties:

1. Positive-definiteness: For all x,y in your space, we want d(x,y) to be bigger than or equal to 0. Furthermore, if d(x,y)= 0, then x=y.
In layman’s terms: We want our distance function to spit out non-negative values (naturally), and if the distance between two points is zero, we want those two points to be the same.
• Reflexive: For all x,y, d(x,y)=d(y,x).
This is saying that the distance from me to you, should be the same as the distance from you to me.
• Finallly, the Triangle Inequality: For all x,y,z, we have d(x,z) is less than or equal to d(x,y)+d(y,z).
This is simply saying, for example, that the distance from me to you should be less than the distance from me to Wide Tim plus the distance from Wide Tim to you.

These three properties define a metric, and vaguely make sense by how distance works in our real world. It isn’t the most insane definition, but it’s powerful. And the implications of these simple assumptions are used everywhere in future analysis courses.

So some students would get to the next class and feel behind immediately. Like they were missing some crucial piece of information that everyone is treating as obvious when it isn’t. And this, is a problem. One my class tries to remedy partially.

Enter My Class, Circa IAP 2022

Fall of my sophomore year, I started cold-emailing people in the math department to see if a class like this could be offered. Early on in the process, I was told that it might be a good idea for me to teach it, and as someone who loves education, I jumped at the chance. So first, I had to draft a syllabus, including a list of pre-requisites, a lecture schedule, and textbooks used. This part of the class took some time, sure, but really it was just a matter of sitting down and breaking up a chapter’s worth of material into four lectures plus two introductory ones. In this process, I also had to find a professor to be an advisor for this class (as of course, I am not a professor, yet), and my UROP advisor Larry Guth graciously agreed to be that person!

Then, I had to wait. Specifically, I had to wait for an education committee to sign off on the class being offered that IAP. This part took a few nerve-wracking months. Then, late in November, I got an email from Larry saying “P.S. Paige, your class is officially approved!” The class was numbered 18.S097: Intro to Metric Spaces. Then, I had to prepare.

Over the course of the two weeks between winter break and the beginning of IAP, I typed up nearly 40 pages of lecture notes, created 4 problemsets, and learned basic HTML to create this class website. Okay to be perfectly honest, I took the basic outline for the website from Professor Dyatlov’s site I had used that fall. But still! It was made. And I taught the class. For free, might I add, but there isn’t a world in which I would give up the opportunity to teach this class. It was an amazing experience.

And what came afterwards, was comparably amazing.

Post-18.S097

First, I advocated for the class to be put on OpenCourseWare (OCW). This was a little bit irregular, in that the class had only been taught once, but I thought it was really important to get this material out there. It can give students the opportunity to learn analysis at their speed and have all of the resources at their disposal to succeed.

A few months later, and the course was made available. One of my lifelong goals was to teach a math class and have it on OCW. I just didn’t think it would happen my sophomore year.

Also, besides this, I also won an award for teaching the class.

So, What Now?

Now, I’m gearing up to teach the class again! The number has changed (as the topics ones are prone to do) to 18.S190, but the basic material has stayed the same. There is a new module on what is known as a completion of a metric space, but this is neither here nor there for this blogpost. And this year, I’m being paid! All in all, really cool things at MIT.

I also want to use this post (as well as one coming soon) to emphasize how many opportunities can come out of MIT from your own imagination. You can work with OCW, you can teach a class, you can make your ideas a reality. The largest and most difficult step in my opinion of this process is simply emailing people. You need to know the right people, and if you don’t know the right people, you need to know who might know the right people and need to reach out to them.

The result can be something you’re really proud of.

1. Multivariable Calculus
2. at least until last year, when a class was created motivated by one class run by the UMA and various people around the mathematics department
3. for our purposes, you can just picture this as a flat plane which is two-dimensional Euclidean space
4. in particular, this is given by the Euclidean distance from a point a to the origin