<p>
</p>
<p>Getting into HA as a first-year is difficult, but feasible. You must have a very strong background in pure mathematics. One of the first things you’ll learn when/if you come here is that math contests are, well, completely useless if you actually want to become a successful mathematician. Mathematical ability itself means very little, and very few have the mental capacity to deal with the abstract nature of the topics of Honors Analysis.</p>
<p>I’ve found that the most valuable thing to success at math at the University of Chicago is being able to learn mathematics independently. I had only taken MVC, LA, Diff Eq before coming to Chicago, but I taught myself Analysis over the summer, which meant more to my ability than any course I had taken previously. Subsequently, I was able to test into HA as a first year, primarily by showing the test graders that I was familiar with concepts in metric spaces, topology, and other areas of analysis.</p>
<p>What kind of mathematical ability should you have for HA? The ability (not necessarily knowledge) of a first-year graduate student at a top-50 university in mathematics. Because that will be the level of content.</p>
<p>As to the OP: The distinctions between Chicago and other top universities in mathematics are very clear. Firstly, it seems that Chicago is rather disconnected from “mainstream mathematics” in the sense that we are rather apathetic toward math competitions such as the Putnam. This gives us a much more serious tint - if you want to be a serious researcher in mathematics, Chicago is a good place to come. Second, we are a very test-focused school. If you go to MIT, for instance, most of the classes will be graded 100% according to how well you do on the psets. At Chicago, 70-80% of your grade will be decided on test results. Third, Chicago probably has harder classes than other top universities. HA is the obvious example, but some of our teachers are really rather insane. For instance, in a class I’m taking this quarter called Differentiable Manifolds, the professor is approaching the subject matter via sheaf theory, a rather advanced theory usually not even taught in graduate school. In addition, as far as I’ve seen, our graduate classes are easily the hardest in the nation. I’ve seen graduate course material at Harvard, Stanford, Berkeley, and MIT, and I can say without hesitation that the course content at Chicago is significantly more difficult than at any of those schools. This will be very important to the more serious researchers who will get into graduate classes by their 3rd year or so.</p>