Cognitive psychologists Dale Griffin and Amos Tversky (one of the discoverers of cognitive biases along with Daniel Kahneman) asked 24 of their colleagues choosing between jobs to estimate their probability of making each choice. Their average confidence in their choices was 66%, but 96% of them ended up choosing the job to which they had assigned a higher probability.

Isn't that terrifying? If you can guess where you're most likely going to college, you're almost certainly going there. We change our minds less often than we think.

Last semester I applied for an NSF Fellowship. Part of the application required that I give the name of my graduate institution. Of course, I didn't have a graduate institution yet; I wasn't even done applying! I knew that I was applying to MIT, Harvard, Columbia, UChicago, and UC Berkeley, and I'd looked up the professors at each school to find out which ones I might want to work with, but I hadn't visited them or anything (with the obvious exception of MIT). I did not feel prepared to predict which one I'd end up at. According to the application, I could change my mind later, but it sounded like an annoying process and I would save time by correctly predicting where I'd go right then and there.

Last semester, months before I formally made my decision to go there, I put "UC Berkeley" on the NSF Fellowship application.

Knowing that I don't change my mind as often as I think is pretty scary, but at least it explains some advice I'd received in high school about college decisions. I'll give it to you in the form of a short poem I found recently by Piet Hein:

Whenever you’re called on to make up your mind,

and you’re hampered by not having any,

the best way to solve the dilemma, you’ll find,

is simply by spinning a penny.

No—not so that chance shall decide the affair

while you’re passively standing there moping;

but the moment the penny is up in the air,

you suddenly know what you’re hoping.

Okay, so you've almost certainly made your decision. Possibly you made it months ago, before you knew all the things you know now. That could be bad, right? What can you do about that?

Well, to some extent, it doesn't matter. If you're good enough to get into MIT, you're probably good enough to get into other great schools too, and you'd probably be happy at any of them. Your experiences at each school will likely be very different, but in many of the most important ways you won't be able to predict those differences. Maybe if you go to one school you'll strike up a strong relationship with a professor who will encourage you to go into academia. Maybe if you go to another school you'll meet some interesting fellow students, drop out, and start a company together. And maybe if you go to a third school you'll meet the love of your life. Who knows?

If you're content with the decision you've already made, then congratulations! If you're worried that you should be attempting to change your mind, I don't have any easy answers. Reading some of the material on lesswrong.com might be a good idea, although I don't know how much good it'll do you in a few days. Start with the Sequences if you want to try; the relevant one is How To Actually Change Your Mind but I don't remember how much it draws on the others.

If that sounds like work, try reading *Harry Potter and the Methods of Rationality* instead. Actually, even if you don't listen to anything else I just said, you should try reading Methods anyway. It's good. It's really good.

- My documents (mostly math papers and math notes)
- My music (mostly old and kept for nostalgia value)
- My pictures (mostly nice desktop backgrounds)

Here's what I **didn't** have to back up before doing this:

- My bookmarks (synced by Chrome)
- My Chrome extensions (synced by Chrome as of 2010; I distinctly remember this not happening before)
- My contacts (stored by Facebook and/or Gmail)
- My calendar (stored by Google Calendar)
- My other documents (stored by Google Docs)
- My other music (stored by Spotify)
- My other other documents, including all of my homework and applications to various things (stored by Dropbox)
- My games (stored either by Steam or online at the Humble Bundle)

I love living in the future, you guys.

]]>What does that mean for you? Well, it means I not only have a lot of free time for writing blog posts, but I'm also feeling reflective. I have a lot of wisdom to share with you all, but I'm not really sure where to start, so I figure this isn't a bad way to find a starting point: ask me anything in the comments.

Go ahead. Don't be shy. This is a judgment-free zone.

To give you some ideas, here are some things you could ask me about and expect a reasonably intelligent answer in return:

- Me (my 3.5 years at MIT, how I feel about them, what I loved, what I'd do differently)
- Math (my experiences with it, anyway; for actual math questions, you should probably go to math.stackexchange.com)

I don't feel terribly qualified to talk about anything else, but if you'd like to ask anyway, I'd be happy to attempt to answer questions about things like:

- Dealing with the college application process
- Relationships
- Feminism
- Disney movies
- Your fears about the future

Oh, and I suppose I should tell you about my semester before I go. Briefly: 18.821 ended up being more interesting than I expected, but I didn't get to devote quite as much time to the projects as I wanted because I was busy with other things. 18.03 was exactly what I expected. Concert Choir was fun but scheduled at an unfortunate time (7:00-9:30pm Mondays and Wednesdays); I never really got used to it. I learned a lot in 21M.302; I looked at some sheet music I'd tried to analyze a few years ago, and some of it makes a lot more sense now. 24.244 was an interesting look at how non-mathematicians study a mathematical subject.

And as you already know, I did this. It was fun! Highly recommended. Just wish I'd actually worn a bowtie.

**Edit:** okay, one exception: I am not going to answer the question "what are my chances?" Sorry, but there's definitely no way for me to know without seeing your entire application, and even if I did I don't know how admissions actually works.

Here's the deal: I'm having a pretty boring semester academically. After spending a year at Cambridge taking amazing classes that didn't fulfill any graduation requirements, every single class I'm taking right now is a class I absolutely need to graduate. (I won't bore you with the list.) Fortunately, that's it; after this semester, I'll be **done**. I am graduating in **February**. I will be spending the Spring semester at MIT but **not taking any classes**. How awesome is that? I know someone else who did exactly this, and he had the time of his life.

I fully intend to have the time of my life too. But I need your help! What are some awesome things I could be doing?

We spend so much time on these blogs telling you about all the awesome things we're doing, but if you're prospective MIT students, shouldn't you have something to say about all the awesome things **you're** doing (and that I could be doing too)? Help me out here! Here's some basic stuff I've thought of so far:

- Math blogging. (I've been neglecting this for grad school applications.)
- Learning me a Haskell for great good.
- Learning to cook.
- Learning to swim, for real. (I passed the swim class but I can't really do anything except a terrible backstroke.)
- Learning to play the harmonica.

Not bad, but it could use a little more awesome. What do you have that's better?

The only mild constraint I'll add is that I don't currently want to do anything that requires that I be any particular place, including MIT, on a regular basis (so I'm leaning away from a UROP, but if someone can convince me that that's the best use of my time I'll think about it), so I can travel for long periods of time if I want. I mean, when am I going to get this kind of freedom again?

]]>Fortunately, you live in more civilized times. Nowadays, when you want to know something and have access to the Internet, you can Google it, which will probably lead you to a Wikipedia article. Depending on the subject, you might subsequently find an online article, blog, or forum dedicated to the subject that will suggest further directions for investigation or provide an opportunity for you to shoot off questions to helpful strangers. And you can do all of this without ever leaving your room.

What are you going to do with all that power?

I can't answer that question for you. What I can do is give you an idea of what's possible using the resources that you, as a person reading this on the Internet, have access to. I can tell you what I did: I learned a lot of math! And I didn't even have to be at MIT to do it.

**Art of Problem Solving**

My first encounter with Internet mathematics occurred when my high school calculus teacher directed me to ArtOfProblemSolving.com (hereafter abbreviated AoPS). AoPS has some great resources, including collections of problems from a lot of competitions, but by far the most valuable resource for me was the AoPS forum. Some of the best and brightest students around the world gathered here to post and discuss problems from competitions of varying difficulty. It was very easy for me to find problems just a bit harder than what I was comfortable with, and I got a lot of practice in problem-solving and proof-writing.

But I got more than practice: I got feedback! If I did something wrong, someone would quickly point out my mistake. If I wrote something unclear, someone would ask me to clarify it. If there was a nicer way to get a certain conclusion, someone would post the appropriate proof. Often a problem would be too hard for me, but someone else would post a short solution using a clever method that I'd never seen before, and I picked up some very clever methods this way.

In total, I wrote about 12,000 posts on the AoPS forum. When people ask me how I learned mathematics, I want to say I used some clever trick, but the truth is that I just invested a lot of time into it. It certainly helped that posting on AoPS was a fun way to learn math. I never felt at any point that I was doing work; in fact, I used to log onto AoPS to procrastinate on my actual work.

About a year after I joined, I started the first incarnation of my blog, Annoying Precision, on AoPS. My initial goal was to share some interesting ideas I'd learned about at PROMYS, and more generally to share interesting techniques for solving problems that I often found I needed to refer to on the AoPS forum. Annoying Precision gradually grew into another valuable tool for my mathematical education: I kept writing posts about a subject and learning new things about it while writing, and I also found the process of writing valuable as a way to sharpen my thoughts. Annoying Precision was also a good way to let other people know what I was up to: Richard Rusczyk, the founder of AoPS, was impressed enough with what I'd been writing to offer me a job writing handouts for one of AoPS's online classes.

**Math Blogs**

During my freshman year at MIT, I discovered a second amazing online math resource: other math blogs! I no longer remember how this happened, but at some point I heard about Fields Medalist Terence Tao's blog, What's new. Reading through his archives was a revelation. A blog post isn't anything like a paper or a textbook; due to the more informal setting, Tao could explain his intuitions and big-picture ideas instead of just writing down proofs. There's plenty of interesting stuff going on in a mathematician's head between the time that he starts thinking about a problem and the time that he writes a paper explaining his solution, but a lot of it never finds its way into print. Tao's blog posts contain all sorts of insight into the mathematical thought process that it would be difficult to find in any other medium, and all of it was freely available online.

I was entranced, so I clicked through Tao's blogroll looking for more blogs like his. I found plenty of fascinating stuff, but let me single out Tim Gowers' (also a Fields Medalist) and John Baez's blogs in particular. Like Tao, Gowers is also a great writer, and I also gained a great deal of insight into the mathematical thought process from reading his posts. John Baez is ~~actually a physicist, but his~~ a mathematical physicist whose blog contains all sorts of interesting mathematics explained in a big-picture way, without a lot of details, and it greatly expanded my mental conception of what mathematics could be about.

Reading all of these amazing blogs inspired me to take my own blog more seriously, so I moved Annoying Precision to WordPress, where it's been ever since.

**MathOverflow and math.StackExchange**

During my sophomore year, something very exciting happened in the world of mathematics. A group of UC Berkeley graduate students and postdocs started a website called MathOverflow (MO) as part of StackExchange 1.0 (SE 1.0). A basic problem in mathematical research is that a question will come up which is probably not difficult, but which lies outside of your area of expertise. If you knew someone in the appropriate area in your department, you could ask them, but if you didn't, or if that didn't work, it might take months to figure out the answer on your own. MO is at least in part for asking these kinds of questions: post your question on MO and an expert will probably find it and post an answer that makes everything clear. The general idea is to harness cognitive surplus to accelerate the rate of mathematical research.

MathOverflow was an incredible boon for me. I had a lot of questions, and until the advent of MO I didn't have a good place to ask them. With MO around, I could get answers to my questions from world-class experts. Just to give one example that's particularly stuck with me, I was struggling to understand something I read in a textbook, so I posted about it on MO, and the author of the textbook answered me with an explanation!

math.StackExchange (math.SE), on the other hand, began as part of StackExchange 2.0 as a general-purpose Q&A site for mathematical questions at all levels. Around this time, I had stopped posting on AoPS, and math.SE became my new place for answering rather than asking questions. At least for my purposes, math.SE is a major improvement over AoPS for several reasons. First, it draws from a much wider audience (such as people from other SE sites) of people curious about math but lacking a good resource for addressing their questions, and these people sometimes ask much more interesting questions than anyone would ever ask on AoPS. Second, for reasons that I don't completely understand, SE sites are very visible on Google, so answering a question well on math.SE makes the answer available to a potentially large audience of future questioners curious about the same thing on Google. Finally, math.SE was boosted from the beginning by an influx of users from MO, mostly professional mathematicians, and it is very interesting to see their answers to even relatively elementary questions.

That's probably worth rephrasing: at math.SE, professional mathematicians might answer your (interesting) mathematical questions regardless of level. How amazing is that?

My activities on math.SE eventually got me noticed by StackExchange, and they offered me a summer internship which it would be going too far off-topic to describe now.

**Concluding Remarks**

It's hard for me to overstate the impact that having access to such amazing mathematical resources has had on my life, and it's also hard for me to resist pointing out that **MIT had nothing to do with it**. It's great that MIT has a lot of amazing educational resources, but most of those resources (with the notable exception of OCW) are only available to people in the MIT community, and at the end of the day that's not a very large community. But even people who don't have access to elite institutions and only have access to the Internet still have access to amazing educational resources, even if they don't always know it.

One more time: what are you going to do with all that power?

]]>A lot of people, even at MIT, don't really like math. The story I hear too often is that they loved math up to a certain point, then got a terrible math teacher, then it stopped making sense to them and they hated it after that. It's a sad story. Math very much builds on itself, and if you miss a vital piece of foundation, then your math is going to be fragile and prone to collapse. You can probably take literature classes in college without taking literature classes in high school, but good luck trying to take math classes in college without taking math classes in high school.

The saddest part, though, is that most people never get to the *good stuff*! Most of what gets taught in grade school doesn't really deserve to be called "math." It's really closer to what mathematician John Allen Paulos calls numeracy. It's important to distinguish mathematics from numeracy in the same way that it's important to distinguish literature from literacy. Literature is art; literacy is a basic skill. And even the stuff that isn't numeracy – trigonometry, for example – is absurdly old. Haven't you ever wondered what mathematicians have been up to since then?

I think the good stuff is beautiful – some of the most beautiful stuff in human history – and I want more people to at least know what it looks like. So I'd like to give some non-technical descriptions of the courses I took last fall at the University of Cambridge through CME. (I'd describe the courses I'm taking now, but most of them are graduation requirements.) It's a little harder to do this for math classes, especially purer math classes, than other classes because I can't just give short, easily-understandable descriptions like

**2.665**: Build robots.

**2.666**: Build robots that shoot lasers.

I have to explain at least roughly what some abstract concept is, and also why studying it is interesting. I haven't really tried to do this before, but I might as well start now, right? So let's see how I do. Feel free to ask in the comments for clarification!

**Galois Theory**

The quadratic formula tells you how to find the roots of a quadratic polynomial in terms of square roots. There's also a cubic formula for finding the roots of a cubic polynomial in terms of square roots and cube roots, although it's huge and impractical to use. There's even a quartic formula, which is even huger and more impractical. You might expect, based on this pattern, that there's a quintic formula that takes pages and pages to write down.

But something much more interesting is true: there is no quintic formula! I bet you're wondering why. The modern explanation, in terms of Galois theory, goes something like this: the roots of a polynomial are not as different from each other as they seem. In fact, in certain situations you can swap around some of them, and it doesn't really matter. In other words, the roots have certain symmetries, which are mathematically described using the notion of a group. Rather than try to explain what this means, I'll give some examples: think of the rotational and reflectional symmetries of a regular polygon, or of a Platonic solid.

Galois discovered an amazing relationship between these symmetries and writing down generalizations of the quadratic formula. It turns out that our ability to write down generalizations of the quadratic formula for a given polynomial depends on how complicated the symmetries of its roots are. For quadratic polynomials, the only interesting case is where you can swap the two roots, which is a very simple symmetry. For cubic polynomials, you can either cyclically permute the three roots, or you can in addition swap two of them: think of the rotational (then reflectional) symmetries of a triangle. For quartic polynomials, there are a few more possibilities: think of the rotational (then reflectional) symmetries of a square, then of a rectangle, then of a tetrahedron. It turns out that none of these are particularly complicated in the sense above, which is why we can write down quadratic, cubic, and quartic formulas.

For quintic polynomials, it can happen that the symmetries are too complicated: they can look like the rotational and reflectional symmetries of an icosahedron! And this turns out to be too complicated to allow for a quintic formula to exist.

Galois theory is related at least by analogy to a wide swath of modern mathematics, and in particular complicated descendants of Galois theory were fundamental to Wiles' proof of Fermat's Last Theorem.

*Rough MIT equivalent*: Studied in 18.702.

**Graph Theory**

Put six people into a room. Then either three of them will all be friends with each other or three of them will all be strangers. A sociologist once observed this and thought he might have made some deep sociological discovery, but he consulted some mathematicians first and learned that what he had observed instead was pure mathematical fact: what I just said is true regardless of which people are friends with which other people!

The relevant structure here is that of a graph, a collection of nodes connected by edges. Above, the nodes are the six people and the edges indicate who is friends with who. Another example of significant practical importance is the graph whose nodes are all websites on the internet and where an edge between two nodes means one links to the other. A surprising number of questions in mathematics can be phrased as questions about graphs, and there are all sorts of interesting questions you can ask about them that turn out to have interesting answers. Algorithms that deal with graphs are also extremely important in computer science and have many applications, both practical and theoretical.

*Rough MIT equivalent*: Studied in 18.304 and 6.042.

**Linear Analysis**

More commonly known as functional analysis, linear analysis is roughly speaking the study of infinite-dimensional vectors and matrices. The study of many interesting differential equations can be phrased as the study of properties of certain infinite-dimensional matrices, and differential equations are a powerful tool in both pure and applied mathematics, so functional analysis finds applications everywhere.

*Rough MIT equivalent*: Somewhere between 18.100B and 18.102.

**Logic and Set Theory**

It's difficult to explain what the point of this class is without explaining something called the foundational crisis in mathematics. Here is a very rough summary of what happened: mathematicians discovered that certain naive ways of constructing mathematical objects led to logical contradictions. To explain the kind of problem that mathematicians ran into, let me use the Grelling-Nelson paradox, which goes like this: some adjectives have the funny property that they don't describe themselves. For example, "monosyllabic" doesn't describe itself because it is polysyllabic. Let's say that such words are heterological.

Is "heterological" a heterological word?

If it is, then it doesn't describe itself, so it isn't. But if it isn't, then it describes itself, so it is! The mathematical version of this, called Russell's paradox, allows you to write down a mathematical object which both does and doesn't have a certain property. This is a contradiction, which is bad news; if you allow yourself a contradiction, you can prove anything. I had fun doing this in middle school by writing down proofs of statements like "Mr. Black [my teacher] is a carrot" starting from a "proof" that 0 = 1.

Anyway, this and other developments convinced mathematicians that they were being too naive about how they constructed mathematical objects, so they tried to write down rules that would allow them to construct all the objects they wanted without leading to contradictions. These rules are, roughly speaking, the subject of set theory. The study of how rules like the rules of set theory work is the subject of logic.

*Rough MIT equivalent*: 18.510.

**Probability and Measure**

Flip a coin a bunch of times. Approximately what proportion of them will be heads? About half, okay, but how does the deviation from exactly half behave? As it turns out, the deviation from exactly half looks like a Bell curve: mathematicians say it is (approximately) normally distributed, and gets more so the more coins you flip.

But this isn't just a fact about coins; analogous statements are true if you replace coins by dice or by even more complicated random phenomena. This fundamental result, known as the central limit theorem, explains at least heuristically why Bell curves appear in nature: many (but by no means all!) natural phenomena occur due to the accumulation of a large number of independent, essentially random phenomena, such as properties of organisms controlled by a large number of genes.

The central limit theorem, as a mathematical fact, also has applications in mathematics; however, in mathematics, the random phenomena that need to be considered are very general. To handle them, mathematicians invented measure theory, the general study of "measures" (such as volumes, but also such as probabilities). This is a somewhat technical subject, but a very valuable tool: it gives us, among other things, a very flexible notion of integration.

*Rough MIT equivalent*: 18.125, with some material from 18.440/18.175.

Rushing isn't an easy job, and popular culture doesn't make it any easier. Many MIT freshmen, myself included, come to MIT with all sorts of misconceptions about fraternities. We're all boorish jocks or insufferable rich snobs, or maybe both. We're actively antagonistic towards women. We wear pink polos with popped collars. These kinds of misconceptions probably prevented some freshmen from going through Rush at all, at least for more than a few flashy events. Those who did give Rush a fair try may have been surprised to learn that MIT fraternity men are both diverse and intellectual, respect women (at least in my experience), and don't wear pink polos with popped collars any more than non-fraternity men. There are legitimate reasons not to pledge a fraternity (for example, because you have too many extracurriculars on your plate), but tired stereotypes shouldn't be one of them.

Now, if you've gone through Rush enough to get a bid, you probably already know this. Even so, you might have a parent or girlfriend (or boyfriend!) who doesn't, who is skeptical about fraternities, and whose opinion matters to you. To them, I say: please don't let the availability heuristic get to you. MIT fraternities are very different from fraternities in general, and are also very different from what you've heard about fraternities. I would never have joined one if it had been any other way.

If you or someone close to you have any questions about pledging, ask someone about them. The brothers of any fraternity who gave you a bid want you to make a well-informed decision as much as you do. They're not trying to trick you: it doesn't help anyone if you pledge a fraternity that's not right for you. You can trust them. If you want another perspective, though, you can also ask rush girls for advice. Parents of brothers are also sometimes available to talk to.

The worst thing you can do is not ask at all and just have a vague worry hanging over your head that could be dispelled by a short conversation. ("Oh, so you *don't* actually make us walk over Doritos?") This is MIT, after all. Get curious. Collect information. Then make your decision. Good luck!

Anyway, I guess I'm supposed to tell you some stuff about myself. I'm a Course 18 (math) major, and I write a math blog, Annoying Precision. I'm interested in a lot of things besides math, but I feel weird saying that. When most people say they're interested in something, it seems to imply that they know a lot about it, too. I'm interested in physics, cognitive science and musical theater, for example, but I know very little about all three. Mostly what I know a lot about is math.

I almost wrote one of my MIT essays about throwing myself into a waterfall, but decided MIT didn't need to know I could make decisions that dumb. (I couldn't swim at the time, you see.)

I get the feeling that it isn't completely typical for people to apply to be bloggers as seniors, so I should explain. The short story here is that Rachel F. '12 (as you know her) convinced me that it would be a good idea. The slightly longer story is that, after an amazing first two years at MIT, I decided to spend my junior year abroad at the University of Cambridge for a change of pace. Now that I'm back, I want to rekindle the flames of my romance with MIT, and blogging for MIT seems like a pretty good way to do it.

I may not blog a lot about MIT as an explicit topic. I'd like to be more precise about what I will blog about, but I learned from my math blog that it's hard to keep blogging promises. I blog about what's on my mind, and I can't predict what will be on my mind in the future. In any case, I hope my posts will indirectly give you insight into MIT, like a planet wobbling from the gravitational effects of the other planet that is MIT, or perhaps like a crime scene betraying the psychology of the murderer that is MIT, or… well, you get the idea.

]]>