<p>what makes princeton math special? (I have been told its analysis series is amazing)</p>
<p>How nurturing is the department?
I am a "very" accomplished kid with many impressive awards who has already exhausted his offerings at HS, but I did not learn calc when I was 12 years old and just cannot compete with someone who already knows manifolds. I need to learn more, but I am worried that if I go to a top department, I would be passed over in favor of other students who aren't necessarily smarter, but have just learned more.</p>
<p>what grad schools do the "2nd tier" math students at princeton go? Is it a linear slope down (from best to worst) or is it more like a extreme distribution curve, where students go either very high or 4th tier grad schools.</p>
<p>Does anyone here want to comment on the Williams math dept? no one over there can really say anything.</p>
<p>Basically, at the end of the day, you have to teach a lot of this stuff to yourself. If you look at the huge gulf between what is covered in undergrad classes at Princeton and the kind of background you need to write a JP or senior thesis in math, you’ll see why maybe 10-20 students out of 1300 major in math. Basically, to do research in math past the expository stage you really have to have a mastery of the area you’re discussing… and a good place to develop and apply that expertise is in an REU, which hopefully you can start to participate in from freshman to sophomore summer. </p>
<p>The Princeton analysis sequence was designed by Prof. Stein who is absolutely amazing obviously, both as a researcher and as a teacher. His concern for his students and genuine enthusiasm for teaching are just incredible. The rigor is about the same as you would find at Chicago, Columbia, Stanford … i.e. does not have the manic quality of Math 55. Yet hardly any Princeton students take the analysis sequence to completion because there are definitely paths of lesser resistance. What you’re left with is very, very strong students who mess up the curve for the very strong students haha…</p>
<p>One thing about upper level undergrad math at Princeton is that most finals are take homes and the course grades depend heavily on homework. So while the problems are really tough at times, you have the option of researching questions online and at the library and learning more as you go along. I’d like to contrast that with undergrad math at UCB, UCLA, Columbia SEAS and most engineering schools- that mandate closed book midterms and finals that count substantially for all of your grade (emphasizing the need for an excellent memory over reasoning skills).</p>
<p>That said, the case against Williams is if you exhaust the school’s offerings and want access to graduate-level math in a structured, classroom setting. Though if Williams was good enough for Curt McMullen I guess it’s good enough for just about anyone.</p>
<p>is there a lot of collaboration among the students in the math department? i have heard from friends that since the introduction of grade deflation, students have become much more reluctant to work with another. Does this hold true in the math (and physics) depts?</p>
<p>Looking at the math department’s courses at Williams, I would assume that you would exhaust the courses pretty quickly if your goal is to do a Ph.D. in math. The fact is that most people at top Ph.D. programs have taken at least one graduate sequence in math while still at their undergraduate institution (i.e. the analysis, algebra or topology sequence). At the very minimum they have gone to REUs and studied advanced stuff on their own.</p>
<p>I also wouldn’t really go with the McMullen argument. I think McMullen would have done well and got into a top grad school no matter which school he chose. One of the Fields Medalists (Edward Witten) even majored in history and got a minor in linguistics, so I guess at a certain level it really doesn’t matter what you do, because your abilities will still get noticed. However, I wouldn’t recommend that path even if you have what it takes. :)</p>
<p>eof: really? Williams has more math classes than princeton. Maybe I am surprised because I do not know what a sequence is and whether Williams has any. After looking at Williams math dept course offerings, what do you see their weaknesses as?</p>
<p>It seems to me that no matter where I went, I could do an REU.</p>
<p>Nice trivia about Witten.</p>
<p>Also, if you look at it, both princeton and williams have exactly 1 fields medalist from their undergrad programs. So, I still have no idea about Williams’ math</p>
<p>OK, so I might add that I did not do my undergrad in the US and will be starting Ph.D. studies in math next fall, so you might take what I say with a grain of salt (meaning I am not an expert on US undergraduate degrees). What I meant is that the number and names of courses aren’t everything.</p>
<p>If you’re a gifted undergrad, you basically have two choices if you’re sure you want to do a Ph.D. in math:</p>
<ol>
<li><p>Take a bunch of different undergraduate math courses in e.g. complex analysis, analytic number theory, algebraic number theory, algebraic geometry, differential geometry, algebraic topology… Williams seems to offer pretty much all of these even as a LAC.</p></li>
<li><p>Place out of all the trivial stuff like calculus and dive straight into real analysis, point-set topology, abstract algebra as a freshman. After this you are in principle able to handle graduate courses by your 2nd or 3rd year.</p></li>
</ol>
<p>To understand what I mean by graduate course sequences, you could see e.g. the following page:</p>
<p>So these courses have similar names as corresponding undergraduate courses, but they cover a lot more material. The homework is also a lot more demanding. The second option is something you only have at a research university. However, it’s hard to say which approach is better. Many people can for example take a graduate course in algebraic geometry and plow through something like Hartshorne’s Algebraic Geometry and still have no idea how to apply the theory into practical problems relating to curves etc. Thus, for many it’s a better approach to first study the material at a more down-to-earth level and then go for the modern approach in grad school, because it helps them see the motivation behind all the abstract concepts.</p>
<p>Thanks for your post. Where are you coming from and where are you headed for grad school (I can’t tell if it is pton or uchicago or neither)? Which of your two tracks did you take?</p>
<p>Do you know anything about Williams (math dept or otherwise)? I am always interested in hearing international points if view. </p>
<p>Something really cool about Williams is that the math dept has won 5 teaching awards! The next most awards in a math dept is 2.</p>
<p>I am from Europe and I am heading for Penn. Regarding Williams, I had never even heard the name of the school before I started to look at US schools last year. I knew which schools were among the Ivies + MIT, Stanford, Caltech, Chicago, Berkeley, UCLA and a few others, but LACs don’t really have an international reputation.</p>
<p>The strong point with Williams is obviously the teaching. I would assume that a professor would get the boot if he/she can’t teach as teaching is their main priority. I followed option 2, but it also had to do with the fact that our undergraduate courses weren’t as demanding as I wanted. If the courses would have been more demanding I might have opted with option 1. However, I did the mistake of taking a course in measure theory as a freshman, before knowing epsilon-delta proofs well enough. I spent probably 75% of my time working on this single course… so too hard isn’t good either.</p>
<p>You said that you already know calculus i.e. have worked through something like Stewart’s Calculus (so you also know the multivariate stuff)? I would suggest that you pick up the following two books and try to read them (in parallel, Axler is probably easier):</p>
<ol>
<li>Rudin, Principles of Mathematical Analysis</li>
<li>Axler, Linear Algebra Done Right</li>
</ol>
<p>If you can actually learn the material (meaning that you can prove the results in the text even a few days later, without looking at the text), then you should have no problem following option 2.</p>