<p>What are the usual courses a math major takes at Caltech each year? Is there a rigorous analysis course all freshmen take?</p>
<p>Math 1a is "baby analysis", required of everyone, and more rigorous than any other required intro math course anywhere. Math 108, taken by sophomores and advanced freshman is the real deal, with classical topology of metric spaces, Weierstrass, Heine-Borel Theorems, Hilbert spaces, Fourier theory (in the general setting), etc. Also multivariate analysis and complex analysis (second two terms). See <a href="http://math.caltech.edu/general/bsmath.html%5B/url%5D">http://math.caltech.edu/general/bsmath.html</a> for an "easy" version of the schedule. Most math majors take many more courses than this, and somewhat earlier: see <a href="http://pr.caltech.edu/catalog/05_06/courses/listing/ma.html%5B/url%5D">http://pr.caltech.edu/catalog/05_06/courses/listing/ma.html</a> for a detailed listing of courses.</p>
<p>Question, Ben:</p>
<p>How would you compare the undergradute Mathematics department of Caltech with that of Harvard and Princeton? By this, I mean, is there a difference in underlying philosophy? Curricular approach/sequence? Quality of students?</p>
<p>Would the department at MIT be closer?</p>
<p>The AOPS forum might also offer answers, but there aren't too many Caltech people there.</p>
<p>
[quote]
Math 1a is "baby analysis", required of everyone, and more rigorous than any other required intro math course anywhere. Math 108, taken by sophomores and advanced freshman is the real deal, with classical topology of metric spaces, Weierstrass, Heine-Borel Theorems, Hilbert spaces, Fourier theory (in the general setting), etc. Also multivariate analysis and complex analysis (second two terms). See <a href="http://math.caltech.edu/general/bsmath.html%5B/url%5D">http://math.caltech.edu/general/bsmath.html</a> for an "easy" version of the schedule. Most math majors take many more courses than this, and somewhat earlier: see <a href="http://pr.caltech.edu/catalog/05_06/...isting/ma.html%5B/url%5D">http://pr.caltech.edu/catalog/05_06/...isting/ma.html</a> for a detailed listing of courses.
[/quote]
The hilarity of this is awesome. The bare minimum Cal Tech graduate has more math training and proficiency than probably 99%+ of undergraduate engineers, physics majors, chemistry majors, etc. It's just so awesome.</p>
<p>Gallo_Pallatino -- good question. I actually have a slightly unusual perspective on this question, since I had the good fortune of taking math courses at Princeton while I was a high school senior through Princeton's generosity before becoming a math major at Caltech.</p>
<p>In any case, I've found the Caltech curriculum to be somewhat more organized and complete. There is a definite triad of three courses, each a year long --- in algebra, analysis, and topometry (topology and geometry) that everyone takes. Since they're year-long courses, there is a certain thematic integrity to the material. On the plus side, it's easier to mix and match at Princeton and Harvard (with semesters), but it's also easier to leave gaps in your knowledge.</p>
<p>Math 55 at Harvard is certainly the hardest and most breakneck-paced course of the introductions offered anywhere. I haven't taken the course, but I have read portions of the notes. I have mixed feelings about this. On the one hand, being exposed to truly advanced concepts right away is a lot of fun and a good challenge, but can also be confusing and demoralizing. Personally, I prefer the extreme thoroughness of Math 5 and Math 108 to the beautiful quilt of everything that is Math 55. After two years of algebra and two years of analysis, any talented math student will know enough to teach Math 55, so the real question is what pace do you prefer -- do you like a whirlwind tour followed by more depth (55 approach) or a more meticulous step by step approach? This is a matter of taste.</p>
<p>Needless to say, the math majors at Harvard, Caltech, MIT, and Princeton are of very high quality (excluding yours truly). Harvard tends to attract a disproportionate number of Olympiad winners (though Caltech also has plenty of IMO gold medalists). I haven't found the level of students at Caltech to be lower than that of the students at Princeton -- in some places the Techers seem to work a little harder, and the Princetonites were a little more fun to talk to :-)</p>
<p>That's my brief stab at a comparison. Let me know if you have more questions.</p>
<p>Following up on the comparisons, is the size of the department limiting?
Silly example: Say you are working on a certain algebraic entity. If you go next door will Joe geometrist be like "Oh yeah, that's a heart shaped bundt pan, we can apply this theorem and this theorem to it."</p>
<p>So I guess I'm asking about the collaboration factor. Is the collective knowledge of the department diverse and intense enough so that good ideas can spark chain reactions? I know you have really smart people there, I'm just curious if a critical mass of top profs who attract top grad students has been achieved(for the math department specifically).</p>
<p>It depends on the field. In analysis and logic, there are very active seminars and an enthusiastic group of people who collaborate a lot (also joint seminars with UCLA). The geometers also seem to do a lot of collaborative work. Algebraists and algebraic geometers are more solitary. There are excellent graduate students, but the graduate program at Harvard and Princeton is stronger in math than at Caltech. </p>
<p>In any case, I don't think the level of collaboration and seminar attendance by the faculty should influence your decision of where to go as an undrgraduate... realistically, even though Princeton has a bigger faculty and more different kinds of math going on, the probability of being able to do some math (through an undergraduate research project) is much higher at Caltech... on an odd and slightly related note, Harvard's permanent faculty in math is only 3 or so professors biggger than Caltech's.</p>
<p>
[quote]
Needless to say, the math majors at Harvard, Caltech, MIT, and Princeton are of very high quality (excluding yours truly).
[/quote]
</p>
<p>If you only got a 7 on the AIME, would you be smart enough to successfully major in math at Caltech?</p>
<p>I never got more than a 3 and I'm doing fine.</p>
<p>He certainly is :-)</p>
<p>happyentropy = math</p>
<p>It makes me sad when nobody asks about UChicago. <em>cries</em></p>
<p>It makes me sad when you post! :-P</p>
<p>This coming spring i have the choice to take Diff equation and application of linear algebra or Diff equation and Number theory. Which combination will help me more if i want to major in biochem/chme e?</p>
<p>Number theory is, as far as is known, useless for biochem/chem e. Linear algebra is very, very useful. No contest, take the diffie q and linear algebra course</p>
<p>thx ben. I would like to take more math classes, but the local college doesn't offer anymore than the above 3 and a seminar in spring. Should i even bother to take the seminar if it's only 1 credit(meet once a week)</p>
<p>Sure, take it! What's the harm?</p>
<p>The thing is this college only allow high school students take up to 12 credits/semester. But since my mom works here, they said i can take 1 more credit if i got an approval from my current teacher.
After choosing all the classes i wanted to take, i have 3 credits left. I can either take a math seminar(1 credit) or a geology class (3 credits). I don't know if it is worth the time to take the seminar.</p>
<p>my dad used to teach math here</p>