There are several summer math camps that would have fit in the theoretical math post, but they focus only on number theory, so I decided to separate them out. These are all programs for high school students. Details on each below the fold!
Ross
Ross Mathematics Program is a six week program for 14-18 year olds. There's a focus on proof writing and spending a lot of time thinking about problems. Only eight hours a week are spent in class, but most time out of class is spent on math, as well. There are also some counselor-organized recreational activities and field trips. All the first-year students study the same topics, and returning students have advanced seminars.
Ross is always held at Ohio State. There are about forty first-year students each year. In 2013, the program is from June 17 to July 26. The application involves a recommendation, some essay questions, and solutions to a set of math problems. Admissions is rolling, with a May 1 deadline. The website is http://www.math.osu.edu/ross/index.html
PROMYS
PROMYS is a six week program for pre-college students who are at least 14 years old. PROMYS is very similar to Ross in setup -- not many hours per week are spent in class, but students are expected to put a lot of time into problem sets. There are guest lectures and counselor-taught mini-courses. Outside of math, like at Ross, there are counselor organized activities and field trips, but again, PROMYS really stresses putting a lot of focus on math. There are advanced seminars for returning students.
PROMYS is always held at Boston University. There are about sixty first-year students each year with twenty returning students. In 2013, the program is from June 30 to August 10. Admissions is rolling, with a May 1 deadline, and the application involves short essays, solutions to math problems, and a recommendation. The website is http://www.promys.org/.
Mathworks
Honors Summer Math Camp is a six week program for high school students. Most first-year students are 9th or 10th graders. Each day, students have a class on number theory, an honors seminar, a Mathematica class, and a problem session. They work on problem sets in small groups in the evenings. There's a focus on proof-writing and teamwork. There are activities on weekends and guest lectures followed by picnics on Fridays.
For returning students, Mathworks isn't as number theory focused. Students work on research projects and take classes in combinatorics, analysis, abstract algebra, or topology.
Mathworks is always held at Texas State University - San Marcos. There are sixty total students, with thirty-two first-year spots. In 2013, the program is from June 23 to August 3. The application involves an essay and a teacher recommendation. The website is http://www.txstate.edu/mathworks/camps/hsmc.html.
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