There are many summer programs in the US, aimed at motivated undergraduate students of mathematics, that offer lectures and mini-courses on a variety of topics. However, as far as I know, most programs require that the applicant is American or a permenant resident in the US.

My question:

Are there any such summer programs in the US, Europe, Asia, South America, or Africa to which a European student can apply?

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    Just to note: similar questions regarding the REUs (rather than lecture courses) were discussed at math.SX and MO math.stackexchange.com/questions/16254/… mathoverflow.net/questions/13847/… Also while it is not exactly summer exchange, in Europe you may be eligible for Erasmus student exchanges (see e.g. esn.org/content/erasmus-programme ) – just-learning Dec 25 '14 at 16:40
  • @just-learning Thank you for pointing out those threads. – user23758 Dec 25 '14 at 16:42
  • Are you sure this is required in the US? Of course, there are visa issues, but a polite email to check takes two minutes to write and can be sent out a hundred times ;) – Calchas Mar 10 '15 at 4:10
  • Also, unless you are a first year, do not confine your interest to "undergraduate" programmes. There are a lot of introductory graduate programmes that actually would be fine for a bright undergraduate. I went on such a summer school before I applied for my PhD position and at a minimum it gave me something to talk about in the interview. – Calchas Mar 10 '15 at 4:13
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    @Calchas A lot of things are funded by US government agencies, which often require you be a US citizen or permanent resident. So if you aren't, funding would have to come from elsewhere. – cpast Apr 13 '15 at 4:52

Park City summer program organized by the IAS (Institute for Advanced Study, Princeton) is worth trying. In particular, they have Undergraduate Summer School. Here is their Application procedure.

However, non-US permenant resident/citizens may have to pay more as in their web page

Undergraduates funded by PCMI may receive lodging, travel, meals and a stipend (stipends are awarded to U.S. Citizens or U.S. permanent residents only).

Please note that the application deadline for 2015 is already passed (January 31, 2015).

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