<p>As a prospective Math and Physics applicant to UChicago, I have researched UChicago's Math curriculum, and the general consensus is that Honors Analysis is considered the hardest Math (Or even overall) course in the United States. I know that I am being extremely naive here as I haven't done anything remotely close to it in Mathematics before, but at the core of it, isn't Math technically following some rules while using creativity? I know that the course is proof based, so can anyone who has taken this course tell me what makes it extremely difficult? I am not doubting its abstractness, I'm just curious as to what makes the problem sets so tough. Do professors give seemingly out of the blue questions with no guidance at first? My naivety tells me that anything that is based on logic and rules can be done given enough time.</p>
<p>This is probably a more general question about proof based classes in total then it is about Honors Analysis in Rn I-III, but it's interesting to me because if I do end up here (low chance), I might take it as a sophomore (or maybe regular analysis) if I place into 160s Math as a freshman.</p>
<p>Our son completed that sequence not too long ago and he claims that it was the most rewarding experience in college. If you are good at managing time then they shouldn’t be too hard. I think what make them hard are the degree of difficulty of problem sets given and the higher level of time pressure one feels in a quarter system. </p>
<p>In the past, there were a couple of Honors Analysis students who were regular posters on CC, and there were several threads in which people who had taken that course compared member-measurements with people who were familiar with Harvard’s Math 55 to see which was the harder course. Lots of specifics and links to problem sets. Incomprehensible to me and, I suspect, almost everyone else. You can probably find some of the threads by searching; I’m not going to do it for you.</p>
<p>It’s worth considering that, at least as far as I know, the math faculties at MIT and Princeton, peers of those at Chicago and Harvard, choose not to offer their own version of the Crazy Elite Freshman Math Course. In other words, it’s maybe a little pedagogically questionable. In fact not everyone who is offered the opportunity to take Math 207-9 or Math 55 accepts it, and it’s not because they are lazy or wussies or don’t like math.</p>
<p>Honors Analysis is what most other universities would call graduate analysis with a spin. For comparison, the first quarter is roughly equivalent to the entirety of MIT’s graduate measure theory course, but the problem sets are more involved and in general more difficult. (To respond to JHS’s comment above, this is why Princeton/MIT choose not to offer crazy difficult UG freshman classes - their introductory graduate classes aren’t terribly difficult, so advanced freshmen hop right into them. This is basically unheard of at Chicago, where we cover the first 10 chapters of Big Rudin in the first quarter of Grad Analysis.)</p>
<p>The class structure of Honors Analysis varies from year to year, but when I took the course, we started with lebesgue measure theory and Lp functions and quickly moved into general measure theory (and everything it involves, e.g. Radon-Nikodym derivatives, Fubini’s theorem). The course would then “backtrack” and cover topology, Hilbert spaces, and functional analysis, in addition to covering some more advanced topics in metric spaces, compactness, and completeness. In the first two quarters, we covered basically all of Kolmogorov/Fomin’s Introductory Analysis and Royden’s Real Analysis. Sally took over 3rd quarter and started teaching us all kinds of stuff, starting with group theory, topological groups, and manifold theory, and ending with some introductory representation theory and p-adics.</p>
<p>Although I ended up with good grades in the sequence and found it very rewarding, it was definitely a struggle. Dedicating 25 hours a week to the course wasn’t an unusual occurrence, and I definitely spent more hours in the library than I should have that year. The reason it has a reputation for being so difficult is because it’s horribly abstract, and if you don’t already have a very solid background in analysis (via Baby Rudin or at the very least Spivak’s Calculus), it’s basically impossible to keep up. That’s a pretty demanding requirement of first and second years.</p>
<p>@Phuriku Does 16100-16300 use Spivak’s or would one have to self-study it? Though I don’t know whether you took 20700-20900 as a freshman or sophomore and if you took 160s Calculus. I might see if I can get my hands on one senior year just to become stronger analytically.</p>
<p>EDIT: Wait never mind I Googled and it does use it, or at least they used it one year.</p>
<p>MATH 161-163 usually uses Spivak. I took 207-209 as a first year, but I used Spivak (and to a lesser extent Rudin) to prepare for the test. Ultimately, what’s important is that you have a good understanding of proofs, limits (and the useful sup/inf), the construction of the real numbers (via Cauchy sequences), and the abstract construction and fundamental meaning of differentiation and integration.</p>
<p>For an OP who is interested in both math and physics, honors analysis isn’t the way to go. Spivak’s Calculus book is highly acclaimed and regarded. However, Rudin’s Real and Complex is overkill for a physics undergrad. There are better math books for people interested in physics.</p>