Some Information on Honors Analysis
Honors Analysis is a very tough, invitation-only math course at the University of Chicago. Invitation is extended annually to about 10 incoming first-years and about 15-20 rising second-years.
What's so special about it? Mainly that if you test into it, you'll have the opportunity to study with the best minds not only in the University, but in the entire world of mathematics. Most of your classmates will go directly into top 5 PhD departments after attaining their undergraduate degrees. Additionally, most first-years testing into Honors Analysis turned down the other big math schools (Harvard, MIT, Princeton, et al.) to come to the University, and a big motivation for students to turn down such schools was having the opportunity to take Honors Analysis.
There are rumors circulating that Honors Analysis takes anywhere from 40 to 60 hours a week of work to perform decently. This is an exaggeration. The average amount of work per week is probably 25-30 hours, but it could be greater or lesser depending on individual ability. Of course, this is still an enormous amount, especially when there are 2-3 other courses in which you must be registered to be enrolled as a full-time student.
To get invited to Honors Analysis as a first-year, you must take and pass the mathematics entrance examination with flying colors. In addition to a multiple-choice portion of the test that measures knowledge of the AP Calculus BC curriculum, there is a second half of the test for those with more advanced ability. This part of the test will primarily involve definitions and proofs. Do you know the specific definitions of the key concepts of calculus and analysis? Can you prove basic theorems of analysis? This is what the second half of the entrance examination intends to measure. If you want to get into Honors Analysis, it's best to go over the elementary notions of real analysis and get a very strong grasp of them. Mathematical ability, not knowledge of advanced topics (e.g., topology, functional analysis, etc.) will be the determining factor in whether or not you test into Honors Analysis.
I sometimes get requests on how a student should best prepare oneself to test into Honors Analysis or for the class itself. To test into Honors Analysis, best get Rudin's Principles of Mathematical Analysis and go over the key concepts, do all of the proofs, etc. Spivak's Calculus is also good for this. Boosting your mathematical ability should be the first thing on your mind. Now, I know that a lot of people will tell you not to study for the placement exam, but I disagree with this sentiment. Basically, the purpose of the test is to measure mathematical ability. If you are serious about mathematics and want to boost your mathematical ability over the summer, why not? Once again, the placement exam is about ability, not knowledge. And ability does not oscillate like knowledge does.
My senior year of high school, I took a bunch of classes like Multivariable Calculus, Differential Equations, and Linear Algebra. With this background, I knew I wouldn't get into Honors Analysis since I hadn't been introduced to rigorous mathematics that's usually introduced in a Real Analysis course. The summer before I entered the University, I decided to study Rudin's Principles of Mathematical Analysis, and boosted my mathematical ability significantly because of it. Because of that, I was able to test into Honors Analysis as a first-year, and I attained an A under Paul Sally. Clearly, studying for the test did me no harm. It shouldn't with other people either, as long as it is understood that ability is what should be emphasized.
The content of Honors Analysis varies per year, so it's harder to give an assessment as to how to prepare for the course once you test into it. Personally, I think that if you test into it, you're probably well-prepared for the class already. However, if one desires to prepare himself for the ideas that will be presented in the course, it's probably best to begin with Royden's Real Analysis and look over all of the material up through L^p spaces under Lebesgue measure. This is the content which is usually focused on during the first quarter. Roughly, in the year I took it, the second quarter explores functional analysis, metric spaces, and topology. The third quarter focused on p-adics and algebra. Like I said, though, it varies from year to year, so the curriculum will largely depend on who the instructor is for that quarter.
The difficulty also varies per instructor. However, I think that usually, the course's difficulty is approximately the same difficulty as Harvard 55, on which course there is a lot of information online. (I actually got this comparison confirmed by a teacher who has taught both H55 and HA.) The course content is different though, and in my opinion, Honors Analysis covers slightly more advanced (read: prototypically graduate-level) topics while Harvard 55 seems to teach the basic concepts of analysis with tangent projects that are fun and advanced but sometimes a bit obscure. Of course, p-adics is a very obscure topic in its own right. From my experience, it seems that Honors Analysis has more structure than Harvard 55; of course, whether or not this is a positive or negative thing is up to the individual.
In any case, I hope this helps anyone who is searching for info on the course. In my opinion, the lack of information surrounding HA is a bit worrisome, especially to an incoming first-year. If you have any addition questions, you can e-mail me at the address on the homepage.
