Just as a random question, will you be at HMMT?

No, that's far too long a trip for us!

As a note before you read the rest of this message: ideally, I would like to do pure math research this summer, or at least something related to it. So you might say that I am looking at things from a slightly slanted point of view.

Do you know of any place that ranks some of these in terms of their level of challenge? I know most of them to be really great programs, but at the same time, having looked at some of their syllabuses (syllabi?), I keep getting the impression like "oh, we covered that at MOP already" and it would kind of not be cool to spend the summer learning something I've already learned. Also, I'm worried that some of them might concentrate more on contest-type math, which isn't really what I'm looking for.

Am I incorrect to think along these lines? Do programs like Ross and PROMYS go more in-depth into Number Theory than MOP does? Do they focus on research-type math or contest-type math? I am also currently taking introductory-level abstract algebra, and I'm worried that this might trivialize some of the Number Theory-type programs (which seems to be what a lot of these camps focus on; that makes a lot of sense, of course, as Number Theory seems to be the easiest topic to approach rigorously without completely confusing people with massive abstraction).

I considered, but did not apply to, RSI, on the basis that I didn't like the idea of being forced to do a project in a 6-week time frame, and also because it didn't seem to me that a theoretical project would benefit as much there. I'm told that Clay has some RSI-ish thing that is targeted specifically towards math. I looked on their site but only found stuff about Ross and PROMYS. Have you ever heard of this program, and if so, what is it called and what is your opinion of it?

I realize that I've just asked a ton of questions at once; I've sort of been fruitlessly searching for what to do with myself this summer, and your blog entry indicated that you probably had some good ideas, so I thought I'd ask. Thanks in advance.

EDIT: Could the Clay thing be this? (http://www.claymath.org/programs/outreach/academy/about.php) I certainly hope not, since it seems to have been cancelled for 2007.

Sadly, the Clay program is, indeed, canceled for this year. It also was quite short, and I'm not really sure it was what you're looking for, anyway. (It was mid-April, one week, and mainly a series of classes, as I understand it.)

As for programs that offer non-contest math, I would say that *most* high school summer programs are more research-oriented than contest oriented. Certainly there's math at MathCamp that is at essentially a graduate school level. There's not much structure at MathCamp, so if you need structure that might not be a great fit (there's more structure at Ross and PROMYS). But if you work well in an unstructured environment, you'll see a lot of fantastic deep non-contest math at MathCamp.

As for Ross and PROMYS, I believe (but am not 100% certain) that both camps have multi-year programs, and that you would likely be appropriate for later-year programs. You can certainly ask them directly what they will cover and judge for yourself if it will be a repeat for you. As I understand them, these are not contest-type camps. They may start in an area that is esentially overlap with math contests, if only to start students on familiar ground. But they definitely move on to problems that are a good deal deeper than you will see on most contests. (I'm assuming the programs haven't changed much in the past 5-10 years.)

Camps like AwesomeMath and MOP would not be what you're looking for. As I'm guessing you know, these are olympiad training grounds.

Unfortunately, if you're really looking for research, RSI is probably the best bet. However, a place like MathCamp will expose you to a lot of very high-level and abstract work. While they don't have a formal 'training for research' program there like they do at RSI, they do have a structure wherein you can work on a short research project with mentorship from very experienced and very able counselors.

Another vote in favor of MathCamp is that you'll find many kindred spirits - students who love pure mathematics, may be future research mathematicians, but are not so interested in contests.

Is this the program that you are referring to? http://www.mathcamp.org/index.php

It seems like a very good program. Thanks for the information!

Yes, that's it. I've attended it a couple times as a guest instructor, and found the quality of students and instructors very high.

Will you be there this year?

Probably not.

Will you be at AwesomeMath

Also, probably not. I'm trying to travel a little less. Someone from AoPS may be at each camp, but it won't be me this year.