USAMO and College Admissions

JHS
Mon, 28 May 2012 20:03:45 +0000

Here’s a decent thread discussing Honors Analysis and 55, with links to other threads, and a link to a Math 207 mid-term:
http://talk.collegeconfidential.com/university-chicago/551878-math-uchicago.html?highlight=55

Here’s a link to some first quarter Honors Analysis problem sets: [Index of /~ryzhik/MATH207-07](http://www.math.uchicago.edu/~ryzhik/MATH207-07/)

And here’s a couple of fairly concise summaries phuriku wrote a few years ago:



In Honors Analysis, you start with Lebesgue measure theory and work your way up through a rigorous development of integration and differentiation (trust me, this is stuff that you haven’t seen before), L^p spaces, and then you get into abstract measure theory, signed measures, Radon-Nikodym derivatives, topology, manifolds, differential forms, Hilbert spaces, self-adjoint/compact/closed operators, etc. I don’t think the methods of MVC/ODEs/LA are needed anywhere here, except for a familiarity with vector spaces. Most of the material taught in MVC/ODEs/LA classes are calculation-based anyway and aren’t particularly helpful in lending the student familiarity with abstract principles of thought. Most important would be linear algebra, but getting acquainted with linear algebra isn’t terribly difficult. Learn the axioms of a vector space, get familiar with linear operators (i.e., matrices), etc.





Chicago’s honors undergraduate courses seem to be about equivalent to the graduate courses at peer institutions.

For instance, many (by which I mean ~5-10 or so) MIT first-year math majors opt to take 18.125: Measure Theory and Integration (MIT OpenCourseWare | Mathematics | 18.125 Measure and Integration, Fall 2003 | Home), which is a graduate course. Although it is technically a graduate course, the course content resembles very closely that of Honors Analysis at Chicago, which is an undergraduate course taken by about 5-7 first-years and 10-12 second years each year. 18.125 covers the first 3 chapters of Rudin in the first semester, and in Honors Analysis, the first quarter covers Chapters 1-6 of Royden. The former is slightly more abstract (using abstract measure theory instead of just Lebesgue measure theory, although the presented proofs don’t differ by much), but the latter covers more material (for instance, a more rigorous theory of differentiation and the (abstract) fundamental theorem of calculus). I have compared problem sets from both classes, and I see no tangible distinction.

Chicago’s Graduate Analysis class, on the other hand, is significantly more advanced and covers about 5x more material than 18.125. It covers the first 9 chapters of Rudin, along with the professor’s (rather long, I might add) notes on rigorous probability theory. So although yes, Chicago’s undergrads take fewer graduate courses, it seems that the undergraduate honors courses are just as rigorous, and that Chicago’s ‘beginning’ graduate courses are about equivalent to other schools’ more advanced graduate courses (which most incoming graduate students probably start out from anyway).