<p>Hello,</p>
<p>Could a Caltech student (math major) shed some light on the Mathematics option at Caltech? Questions like:</p>
<p>How good is Caltech in areas like Analysis and Algebra?
Any research opportunities?
How many students choose to major in math annually?</p>
<p>etcetc.</p>
<p>Thanks!</p>
<p>Hi. I'm a math major in the class of 2007.</p>
<p>How good is Caltech in areas like Analysis and Algebra?</p>
<p>Caltech has perhaps the top algebraist in the world, Professor Michael Aschbacher, who is widely credited with the key steps finishing the classification of the finite simple groups -- the biggest problem in algebra of the century. David Wales, who is also here, was another major player in this effort. Both these professors are marvelously accessible and often teach the first year algebra course. Those are the two individuals worth mentioning. Our number theorists also take a rather algebraic view, i.e. they really do algebraic geometry. So the bottom line is that Caltech is very very strong in algebra.</p>
<p>In Analysis, we have about 5 of the best professors and postdocs, largely from the Russian school (Moscow State and Leningrad) who are very very strong. Also Barry Simon, one of the world's top experts on analysis and the mathematical methods of physics, is here.</p>
<p>Our faculty in math is smaller than our peer institutions, but that means that the professors who are here are very carefully picked and top-notch (there's no ballast) and the ratio of math students to math professors is very low. Having been around the mathematics department at Princeton a lot, and knowing a lot of people at Harvard, I can honestly say that among the elite U.S. schools, research opportunities in mathematics at Caltech are by far the most easily available. Virtually every student who wants a math SURF project can get one: a summer research job with a $5000 stipend.</p>
<p>We have about 25 students in each class majoring in math, making it a fairly popular major.</p>
<p>Please ask if you have any more questions.</p>
<p>What is the "typical" first undergraduate course in mathematics at Caltech? Is there a Web link to the course description, syllabus, etc.? That's something I've looked up for some of the other superselective schools, and there seem to be some interesting differences in approach among schools that seek top math students to enroll as undergraduate math majors.</p>
<p>Ma 1a. You can get to the course page pretty easily through the math department's webpage</p>
<p>Math 1a is the required math course that all freshmen must take or place out of. Some math majors, especially the strong ones, take Math 5 their first year, which is a thorough year-long course in abstract algebra (groups, group classifications, rings, modules, representation theory, field theory, Galois theory), and the syllabus is essentially the entirety (minus four chapters) of the 19-chapter, 900-page textbook Abstract Algebra by Dummit and Foote.</p>
<p>Links to the websites for each of the three terms (trimesters) of the course are available at <a href="http://math.caltech.edu/courses.html%5B/url%5D">http://math.caltech.edu/courses.html</a>.</p>
<p>After having chosen Caltech over Princeton and Harvard to pursue a math major, I feel strongly that the math department's main feeder course here -- Math 5 -- is by far the strongest of the various courses at top universities which are taken by the strongest math students. It's main virtue is that it is long enough (a year) to do something serious, and that it does it in a thorough methodical way, building up steadily to huge, important theorems that you actually understand fully by the time you get to them.</p>
<p>I know that the "stronger than the others" claim is true for sure in comparison to Princeton, since I actually took their math major feeder courses when I was a high school senior. (Problems there: teaching quality haphazard, too-advanced material rushed through so that even the brightest students are lost, though Jordan Ellenberg's Math 214 was a well-known and beautiful exception -- but he's not there anymore.) And yes, I think Math 5 here is stronger even than Harvard's Math 55. While Harvard's famous course covers a lot of esoteric and advanced topics, it does so with very little unity and requires overwhelming amounts of outsdie reading so that even the best students miss 30% or so of the ideas. After a year and a half at Caltech, I knew everything that a Math 55 graduate knew, but various comments I've heard make it pretty clear that most of them come out with a "scattered" feeling -- they've been exposed to a lot but don't have a particularly unified picture. Math 5 keeps to a more manageable area and explores it more deeply, and so one comes away with some very tangible and coherent knowledge.</p>
<p>Those are my feelings on the subject.</p>
<p>Having taken math 5 this past year, I would say that while the material is very well, and methodically presented, the assigned problems could be more challenging (most depend only on the results just proved in class). The problems in Dummit and Foote tend to reiterate the results proved in the preceding section, rather than demonstrate their application to other algebraic problems (an approach taken, for example, by Artin's algebra textbook).</p>
<p>Ma 5 is a class required for ALL math majors, regardless of their background and so some, especially those who have had previous exposure to group/ring theory find it not challenging enough while others find it unduly difficult.
This is probably an unavoidable shortcoming of having a small school, with relatively few math majors and a small (in number, certainly not in quality) mathematics faculty.</p>
<p>I really appreciate the comments about Caltech math courses. Now I'm off to read the information posted on the Caltech Web site.</p>
<p>your paragraph about specific Math courses at Caltech (Math 5), Harvard (Math 55), and Princeton (Math 214) is very informative. They seem to cover different material, however. In particular, Harvard's Math 55, which is an Analysis/Linear Algebra course, would not correspond to Princeton's Math 214, which (in the current catalog) is titled "Numbers, Equations, and Proofs". Would you say that the comparable Princeton courses would actually be Math 215/217/218, or perhaps Math 315/217?</p>
<p>4thfloor -- I apologize for the confusion. I should have made it clear that I was comparing courses in the category of </p>
<p>"main feeder course(s) taught in the math department to the strongest potential math majors."</p>
<p>At Caltech, that's Math 5 (abstract algebra), at Harvard 25/55 (linear algebra and analysis), at Princeton Math 214/215/217 (number theory/analysis/linear algebra). Many of these courses cover completely different fields, so the only thing they all have in common is their function as "feeders."</p>
<p>The reason I mentioned 214 specifically was that it was an exception to the complaints about Princeton's feeders being irregular/haphazard/rushed.</p>
<p>If we were comparing head-to-head for material, of course we would choose all the courses that teach the same thing and then compare those (so we'd be comparing Math 5 to a 300 level course at Princeton which wasn't open to freshmen when I was there). But it seems like many math students want to know what their first year in math will be like, since that determines your experience to a great extent, so that was the purpose of my comparison.</p>
<p>Thanks, Ben, for the clarification. Would you say, then, that for Real Analysis, the following courses should be comparable: Math 108a at Caltech, Math 25a/55a at Harvard, and Math 315 at Princeton? (I think they all use Rudin.) If so, how would you compare them?</p>
<p>Yes, I'd say that's correct. I didn't take 315 (freshmen & high school students can't take 300+ courses at Princeton), but my feeling from the 200-level courses was that Princeton was strong in analysis.</p>
<p>25/55, (especially the latter) from a lot of opinions I've heard over the years, take an immense amount of work just to keep in touch with some broad picture of what's going on; a lot of the background is relegated to appendices/notes/etc. to read on your own time, which friends have told me becomes infeasible. </p>
<p>108 was quite nice because it split analysis into three pieces over the course of a year -- we read Rudin, Carothers (both real analysis) and Elias Stein's complex analysis book. This allows the theorems to be developed in a great deal of depth (without rushing) and for important results to come up several times from new perspectives.</p>
<p>I think in analysis Princeton and Caltech are quite close, with Harvard in my view not giving enough time to really absorb things. But in other areas the reverse may be true.</p>
<p>Thank, Ben, much appreciated.</p>
<p>Ben Golub, you rock! Caltech is pretty much my top choice now.</p>
<p>I am aware of Math 55, a.k.a the so-called hardest undergraduate math class in the country, at Harvard, and yes, I can attest to what you say above, from what I hear. </p>
<p>Do you have any say on MIT's theoretical math program?</p>
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Do you have any say on MIT's theoretical math program?
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<p>Here is where I demonstrate my honesty -- whereas I have enough experience/connections at Princeton and Harvard to have a pretty extensive idea about those programs, I really don't have enough firsthand accounts to say what MIT math is "really" like. I'll give you some quite haphazard observations instead.</p>
<p>One thing I notice is that the way to serious algebra is somewhat cluttered -- the "real math" sequence is 18.701 then 18.702, which has either 18.700 or 18.100b as a prerequisite. In practice, that means it's quite hard to actually complete a full course in serious algebra before the end of your sophomore year, even if you try hard, and a friend of mine (the one good MIT source I do have) won't be able to get to it until junior year, despite having a strong math background out of high school. </p>
<p>Since algebra has been so key in my mathematical development, this seems rather inconvenient... but perhaps other aspects of the MIT experience, like early emphasis on analysis, compensate for this.</p>