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

Can anyone recommend some more continuous summer programs? That is to say, are there summer schools that offers course in Analysis, Abstract algebra, Topology, and/or set theory? I am also interested in special functions, like $\Gamma$, $\Theta$ and $\zeta$ functions and elliptic integrals: just things more closely related to undergraduate level study.

Many thanks.

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    $\begingroup$ Originally asked at Mathematics Stack Exchange, where someone suggested that the question be posted here. $\endgroup$ – Joel Reyes Noche Jan 22 '18 at 12:45
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    $\begingroup$ 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] $\endgroup$ – paul garrett Jan 22 '18 at 23:50
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    $\begingroup$ ... [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. $\endgroup$ – paul garrett Jan 22 '18 at 23:53
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    $\begingroup$ 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... $\endgroup$ – James S. Cook Jan 23 '18 at 2:51
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    $\begingroup$ Since when has abstract algebra been described as a "continuous" math topic?? $\endgroup$ – amWhy Dec 2 '19 at 22:07
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The answers in the comments are probably the best (in particular @James S. Cook's advice to talk to a professor at a local university if you have such a connection) but here's at least a list to get you started: Given that you've got quite a range of interested there I would suggest going through the MAA's list of Math programs:

https://www.maa.org/programs-and-communities/member-communities/students/student-resources/high-school

In particular HCSSIM offers some sessions on continuous topics, but the focus isn't "Learn Analysis." Johns Hopkins Center For Talented Youth has a topology course but that might be a little lower level then you're interested in:

https://cty.jhu.edu/summer/grades7-12/intensive/catalog/math.html#topo

Otherwise, if you're already thinking about these topics, you might want to look at a local university and see if they're offering any summer courses or math camps. Finally, if you are on your way to college at the end of the summer you should check if your school has an early entry program for declared majors.

This definitely does sound like something that should exist.

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