Each summer, a group of very enthusiastic teenage mathematicians come to spend six weeks in Oxford, working intensively on mathematics. They are participants in the PROMYS Europe programme, now in its fourth year and modelled on PROMYS in Boston, which was founded in 1989. One of the distinctive features of the PROMYS philosophy is that the students spend most of the programme discovering mathematical ideas and making connections for themselves, thereby getting a taste for life as a practising mathematician.

Mornings start with a number theory lecture followed by a problems sheet, which sounds very traditional. But at PROMYS Europe, the lectures are always at least three days later than the material comes up on the problems sheets! This allows the students to have their own mathematical adventures, exploring numerical data and seeking patterns, then proving their own conjectures before the ideas are discussed in a lecture. Another crucial part of PROMYS Europe is the community feel. This year there are 21 students participating for the first time, and six who have returned for a second experience. In addition, there are eight undergraduate counsellors, who mentor the students. Each counsellor gives daily individual feedback to their three or four students, allowing each student to progress at their own rate and to focus on their own particular interests. The counsellors are also working on their own mathematics - this year they are teaching themselves about p-adic analysis. The returning students are working in small groups on research projects, and this year are also exploring group theory. The PROMYS Europe faculty are also available to the students for much of the time, reinforcing the supportive and collaborative nature of the programme.

The occasional guest lectures give the participants glimpses of current research mathematics and of topics beyond the programme. So far, in the first two weeks of the 2018 programme students have learned about Catalan numbers and quivers from Konstanze Rietsch (King's College London), and Andrew Wiles (University of Oxford) spoke about using analysis to solve equations.

As Andrew said: "PROMYS has done very impressive work over many years in creating an environment in Boston in which young mathematicians from all over the United States can immerse themselves in serious mathematical problems over several weeks, without distraction. It is an exciting development that PROMYS and the Clay Institute have now opened up the same opportunity in Europe."

The programme is very intensive, and students spend a great deal of time grappling with challenging mathematical ideas through the daily problem sets. At the weekends, students have extra-long weekend problem sets, but also have time to explore Oxford and the surrounding area. So far this has included a tour of Oxford colleges, the chance to go punting, and a visit to Bletchley Park and the National Museum of Computing.

As in previous years, this year's group is very international, coming from 15 countries across Europe. Students have to demonstrate a sufficient command of English when they are applying, and the international language of mathematics soon transcends linguistic and cultural differences once participants arrive!

Students apply to attend PROMYS Europe, and are selected based on their mathematical potential, as displayed in their work on a number of very challenging problems. This year there were more than 200 applications for around 21 places: the students who are invited to participate have produced exceptional work on the application problems, and displayed significant commitment and mathematical maturity. The programme is dedicated to the principle that no student should be unable to attend PROMYS Europe due to financial need, and is able to provide partial and full financial aid to students who would otherwise be unable to participate.

Alumni of PROMYS in Boston have gone on to achieve at high levels in mathematics. More than 50% of PROMYS alumni go on to earn a doctorate, and 150 are currently professors, many at top universities in the US. PROMYS Europe alumni are also proving to be dedicated to pursuing mathematical studies, with several now studying at the University of Oxford.  Of this year's eight counsellors, seven previously participated in PROMYS or PROMYS Europe as students, and four are Oxford undergraduates.

PROMYS Europe is a partnership of PROMYS, Wadham College and the Mathematical Institute at the University of Oxford, and the Clay Mathematics Institute.  The programme is generously supported by its partners and by further financial support from alumni of the University of Oxford and Wadham College, as well as the Heilbronn Institute for Mathematical Research.

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