<p>Well, of course, I've only been to one place, but I will offer my truthful opinion of things here:</p>
<p>The UofC is certainly probably one of the best places to study math. Even for the non-mathematically inclined, even the 130s math sequence gives students significant exposure to proofs and detailed analysis of E-d proofs (they claim that we're the only school to require all its calculus sequences to do it). But, since you are confident that you'll place into Honors Analysis, then we're talking about a whoooolllleee 'nother ball game.</p>
<p>Honors Analysis is taught by Paul Sally Jr., director of undergraduate studies in math at the university. Professor Sally is kind of a big deal in mathematics education, especially because of his effort in editing the Chicago Math Series, a set of math textbooks that are used in classes K-12. Sally has been teaching at the UofC for over 40 years, and in 2001 won the MAA award for excellence in teaching. Despite all of that, what makes Paul Sally Paul Sally is his demeanor in the classroom. I have never seen someone so excited about anything in my entire life. He absolutely loves teaching and loves his students. Now, hard to do that since he teaches Honors Analysis, the cream of the crop of Chicago mathematicians. However, this year he started to teach Intro to Analysis 199, a transition class for kids who took the lower two calculus sequence. I am in it, and his devotion to us is amazing. He'll have extra study sessions on Thursday nights, leave his office right open for us, really amazing person... and he's a really fun guy. So, my point is that if you are this certain of being placed into Honors Analysis, if you happen to visit before you decide, try to get in touch with Paul Sally. He loves prospective students and will answer any questions you have.</p>
<p>Generally speaking, I know of a few interesting programs that the school has for its math students - and not just concentrators. One such program is the DRP, or directed reading program. Open to students in all years, this program matches undergraduates with PhD candidate mentors. When you start the program, you decide on a given topic in math that you want to explore semi-independently, and then the department matches you with a graduate student who is either working in or close to the area of study you want to pursue. The program last one full quarter and at the end of it you are asked to give a presentation on what you've learned/done over the quarter. </p>
<p>There are also several summer opportunities if you want to do further research in math:</p>
<p><a href="http://www.math.uchicago.edu/%7Emay/VIGRE/%5B/url%5D">http://www.math.uchicago.edu/~may/VIGRE/</a></p>
<p>Also, if you have any questions, I'd suggest contacting Diane Herrmann. She's Sally's partner in crime, and would surely answer any questions you have a whole lot better than anyone around here.</p>
<p>P.S.: We did better than Columbia in Putnam this year LOL.</p>