# Are there any high school level summer program that teaches Analysis?

All summer programs I know for highschool students focuses on number theory, combinatory, graph theory, logic, and all kinds of stuffs in discrete mathematics. (I am mainly interested in UK, US, Canada. Any summer program that uses English Language.)

Can anyone recommand some more continuous summer programs? That is to say, the summer schools that offers course in Analysis topic, Abstract algebra, Topology, and set theory. I am also interested in special functions, like $\Gamma$, $\Theta$ and $\zeta$ functions and elliptic integral. Just things more closely related to undergraduate level study.

Many thanks.

• Originally asked at Mathematics Stack Exchange, where someone suggested that the question be posted here. – Joel Reyes Noche Jan 22 '18 at 12:45
• "All summer programs I know" Programs for what country? – Joel Reyes Noche Jan 22 '18 at 12:47
• This is a very reasonable question. We should note that it is obvious that low-prerequisite topics will generally be more manageable, even for precocious kids. I do have to object to common presumptions that "number theory" is hardly more than elementary number theory, the latter being bereft of abstract algebra, complex analysis, and representation theory. Number theory in real life bears little relation to elementary number theory. At the same time Complex Analysis could be discussed much earlier... [cont'd] – paul garrett Jan 22 '18 at 23:50
• ... [cont'd] if it were not for inertial prejudices that declare it to be post-real-analysis. In fact, most of the worries that typical real analysis high-light do not arise in complex analysis. Everything works amazingly well. And complex analysis is a bare minimum to do anything not entirely trivial in number theory, for example. But the most-typical image people have of the curriculum doesn't allow us to get there even by the middle of a typical undergrad degree in math. This unfortunate inertia is part of the reason for lack of high school intros to such ideas. Sorry. – paul garrett Jan 22 '18 at 23:53
• I don't know of such a program, but, it would be quite interesting to attempt such a program. It would stand as an isolated point from the discrete programs (ha). Probably, if you feel strongly about this, if you have a problem. Find a good professor who you get along with and propose something. Forget about a program, think outside the box. There was a highschool student who worked with the theoretical physics professor I took GR from a few years back. Of course, forging such a relationship is easier said than done... – James S. Cook Jan 23 '18 at 2:51