<p>One certainly doesn’t need to do Olympiad as has been pointed out. Indeed, many excellent first tier mathematicians would find competitions to be against their grain. It is merely one path toward preparation, for those whose minds work in that style. And as my S and 4thFloor point out, Olympiad kids seem to like algebra (he would say because it is more creative but that is the flip side of not liking epsilon-delta reasoning). He found very few in 55 liked or were equally good at both (not that they didn’t do well in the course) --and as pointed out they sorted fairly neatly also based upon their pre-Harvard preparation.</p>
<p>^etondad,</p>
<p>Is Math 55 too difficult for freshman? When did your son take it? Freshman or Sophomore?</p>
<p>You can only do it as a freshman. And is too difficult for a freshman?-- some might think so !! This past year Siu used the Psets from his grad course on analysis for his 55 students–same number of questions per Pset too. It took my son 40-60/hrs a week to do them (and just about everyone works in groups to rough out proofs and then does the rigor and write-ups (on LaTex) on his or her own). I just asked him if 55 was worth it – he said that it certainly lets a student know his limits (or in those rare cases, not). He loved 55a and tolerated (barely) 55b. That seems to be the consensus of most of his friends from the class…however, as the course is very dependent upon which professor teaches it–that can change dramatically next year (Siu’s 55b is becoming the stuff of legends the same way Elkies’ 55 has become…)
If someone decides on doing 55, be prepared that you will be doing 55 and the rest of your courses will kinda be for show. As he said Siu=no summa Math 25=summa – for him he doesn’t care about grades per se (yet did quite well anyway) so it was an easy choice–but if you want a <em>great</em> transcript, 55 can be toxic…</p>
<p>So I discussed this olympiad vs. advanced coursework issue with my son, and after he looked at me strangely for a while, he said that last year there weren’t really any prominent olympians in 55. I don’t think he’d agree the rest of his coursework was for show, either. Several of them took quantum mechanics along with 55b.</p>
<p>^^^ Actually, 25 is no slouch either, unless you consider baby Rudin (which is often used as textbook in some second-tier math PhD programs) to be easy … but yes, 25 does not compare to 55, which is reputed to be THE hardest undergraduate math course in the country.</p>
<p>It’s one of those courses that if you can’t take it as a freshman, you won’t find it any easier as a senior (even if they opened it up to seniors).</p>
<p>^^^ Another typical companion course seems to be Mitzenmacher’s Algorithms course, which is also pretty meaty.</p>
<p>The issue isn’t that a 55’er needs to be a prominent Olympian, but rather the process of doing Olympiad style math training is one of the ways of preparing for it. And as I noted Olympiad training is just one way to prepare, and by no means the only way. But regardless of what way one prepares being very comfortable in proofs and abstract thinking is critical. </p>
<p>There is a reason that about 85 came to the first class of 55a–18 finished the semester and 15 stayed for 55b (actually 13 but 2 who took Physics 16 (which is one semester) joined 55b).</p>
<p>^^Re: toughest math course-- When my son told Paul Sally that he had decided on Harvard rather than Chicago – Sally said that Harvard was the only school he could say was as good as Chicago for undergraduate math. Then Sally asked him what he would be taking and he said 55-- Sally just smiled and replied that that proved that he was crazy. :-)</p>
<p>[Harvard</a> Mathematics Department : 21, 23, 25, or 55?](<a href=“http://www.math.harvard.edu/pamphlets/freshmenguide.html]Harvard”>http://www.math.harvard.edu/pamphlets/freshmenguide.html)</p>
<p>This is from the source, for 55:</p>
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</p>
<p>So it looks like Olympiad may be sufficient but not necessary. It looks as if advanced coursework that is proof based could also be sufficient. And as has been suggested, there is an overlap between the two groups. </p>
<p>My kid may be able to take 230 at Yale, a proof based course on multivariate calculus, vector analysis, and vector analysis–which I’ve heard requires circa 20 hours per week on the problem sets. That and a couple of electives, and if he does well, he’d be good to go as an advanced course type–whether or not he puts a big effort into competition. </p>
<p>(Coincidentally, the day after I posted on this thread, my son’s calculus teacher emailed me that “he wanted to talk about Olympiad, etc.” How will the poor kid find the time to do that and play the trombone, cello, and lacrosse?)</p>
<p>Question: Any thoughts about how Honors Analysis at Chicago compares to 55?</p>
<p>[AoPS</a> Forum - Is anyone taking math 55 at harvard this semester? • Art of Problem Solving](<a href=“http://www.artofproblemsolving.com/Forum/viewtopic.php?f=143&t=120138&hilit=math+55]AoPS”>http://www.artofproblemsolving.com/Forum/viewtopic.php?f=143&t=120138&hilit=math+55), which does not really answer your question of Honors Analysis at Chicago. I have a student going to Chicago next year and I hope to be able to provide more information on this then.</p>
<p>CC poster phuriku took Math 207-09, Honors Analysis, at Chicago in 2007-08, and engaged in a lot of discussion with Harvard people and others about the Chicago-Harvard comparison. The two courses are clearly very similar in approach and design. Phuriku claimed that the Chicago course was more difficult, but I don’t know how true that is. Like Math 55, Honors Analysis dominates the life of those who are taking it, but not it seems to the 50-60 hours/week extent Math 55 does.</p>
<p>Also, sometime poster CountingDown (I think) had a son who was invited to take 207, but opted instead to take more standard advanced math courses (starting with an inquiry-based section of Analysis), and was very happy with that choice. She wrote about it a fair amount in the Chicago forum.</p>
<p>There are some differences in structure: Only first-year students can take Math 55, and the department lets tons of students try it out then drop back to Math 25 or 23. So 55 can start with 30+ students, and usually gets down to 10-15 by the end of the semester. The Chicago course is strictly by invitation, so starts out small, and includes some second-year students who performed well in the last quarter of Honors Calculus or Introduction to Analysis and Linear Algebra. (It’s not that hard to negotiate an invitation for students who are in the ballpark as candidates.)</p>
<p>That link had this noteworthy comment:</p>
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</p>
<p>That you could be an IMO medalist without knowing integration. Guess the overlap between Olympiad types and advanced course types is far from perfect. But this person is obviously very talented to be able to learn it while getting through 55.</p>
<p>Must have had one devil of a year!</p>
<p>Well, there is integration like x^(n+1)/(n+1), and there is integration like Lebesgue integration …</p>
<p>I think the claim was Chicago’s Honors Analysis covered more advanced material than Math 55 but it only covers analysis instead of both algebra and analysis like 55 so it’s still less work.</p>
<p>^^^ Can anyone post the syllabus and textbook for Chicago’s Honors Analysis, so we can take a look ourselves?</p>
<p>According my son, Chicago is a bit slower and has less breath even within analysis (now part of this may be that this past year 55 was taught by Siu who is an analyst and essentially taught his graduate seminar again in 55) but as there is more time there is very good depth.</p>
<p>As pointed out as well, Sally’s course has sophomores as well as freshmen-- which changes the classroom dynamics significantly.</p>
<p>My s corrected me that Sally didn’t call him “insane” for choosing 55-- he called him “psychotic” a subtle distinction, to be sure.</p>
<p>^^^ that agrees with I’ve heard too (the first paragraph).</p>
<p>Two of my friends made USAMO and got into Harvard, and they took Math 25. This was primarily because of the difficulty level. Math 55 covers topology, real and complex analysis, abstract algebra, and differential geometry - most of which is done at the graduate level elsewhere.</p>
<p>Well, here’s what I could glean from the UChicago website:
This highly theoretical sequence in analysis is intended for the most able students. Topics include the real number system, metric spaces, basic functional analysis, and the Lebesgue integral.</p>
<p>They probably use Rudin.</p>