Want to study abroad and learn a lot? Does full immersion in mathematics sound fun to you?
Study abroad programs combine the instruction of college with the allure of foreign cultures. More and more students study abroad to fulfill their college's requirements, through first-hand experience, on history and language.
Unless your major is History or a foreign language, there are few chances for upper-level courses in your major whose credits will transfer effectively. That is, until Budapest Semesters in Mathematics (BSM) became popular in 1985.
Hungarians have a long-history of smarts and make no reservations in telling you about it – more Nobel Prize winners than any other country – many chess champions and the inventor of the Rubik's Cube – and the lingering legends of study-places of great mathematical problem solvers like Paul Erdos and G. Polya.
Budapest Semesters in Mathematics puts North American Mathematics majors into a great study environment, allowing them to take full semesters in only mathematics (as well as courses in Hungarian culture, history, and language). Students from research institutions like MIT come for courses with renowned professors, while students from liberal arts colleges come to supplement their education and test whether Graduate School is right for them with a semester of only math. Independent studies with other renowned local mathematicians were possible (I studied Quadratic Forms with Joseph Pelikan). Inexpensive weekend trips to the many nearby destinations (Prague, Krakow, Vienna, Brasov, Croatia, etc) were common.
More recently the Mathematics Advanced Studies Semester (MASS) program at Penn State University has opened up a similar opportunity for those who can't afford to go so far from home. MASS offers a seminar about the “must see” problems in math, a colloquium featuring such renowned speakers as John Conway (see my puzzles and game page), and three specialized graduate-style courses in the respective areas of Algebra, Topology/Geometry, and Analysis; these three courses vary widely: in 2001 Algebra concerned Combinatorics, Topology was Relativity Theory, and Analysis was Fluid Dynamics. In MASS one takes fewer courses than BSM (and has no choice of courses), but these well-taught courses give an excellent taste of graduate school. MASS adds to the anti with a well-payed summer Research Experience for Undergraduates (REU), and provides funding for those who cannot otherwise afford the semester program.
MASS was founded in part by a few Russian mathematicians who kept parts of the old system – the most daunting of which (though excellently done) are the oral exams. Their Russian colleagues in Moscow saw the success of BSM and MASS as a sign to start their own program: Math In Moscow (MIM). The Independent University of Moscow (IUM) was historically created to compete with bad political situations as the renowned Moscow State University (MSU). Excellent but persecuted professors and students chose the IUM for lack of other options; their friends and sympathizers joined them. Today many co-teach or co-study at MSU and IUM. Spring 2003 was one of the early years so my opinions may not be representative; with four students instead of today's 20, the courses were very small (1-3 students each) and student-teacher contact was extraordinary. The caliber of professors combined with the interactions was the best of all three programs. However, I would recommend against arriving in January as Russian Winters are renowned to be dreary (i.e. start your studies in the Fall).
To my knowledge, I am the first and only student to have attended all three programs. Would I recommend it? No. I recommend instead staying a year in one location and getting to know the professors well through research. In Mathematics, relationships with advisors are analogous to those with significant others: long-term relationships will get you further than short flings.
There was an article in the Notices of the American Math Society (AMS) comparing the programs, which can be found here.
To my knowledge, these are the only semester-long college math programs. There are, however, a number of advanced master's programs: Cambridge TRIPOS in England, Ecole Normale Superiur in France, and the Utrecht Master Class in the Netherlands.
Disclaimer: These are opinions and memories of the reader only. Facts may not be completely accurate. I appreciate any citations you may have.