Students returning to PROMYS Europe are encouraged to revisit those parts of the number theory course that they didn't have time to explore fully as a first-year student. There is always more to investigate, and returning students often welcome this opportunity to deepen their understanding. Students will be able to talk to their counsellor and to the faculty to design the most suitable individual programme for them.
In addition, returning students can pursue lectures and problems specifically aimed at returning students, and can work in a small group on a research project.
In recent years, courses for returning students have covered group theory, graph theory, abstract algebra. The courses fit well with the number theory course and are aimed at returning students but are also accessible to students participating in PROMYS Europe for the first time.
Returning students also have the option of working in a small group on a research project. Projects are designed by leading mathematicians, and give returning students an opportunity to explore an area of mathematics in great detail and potentially to work on open research questions. In recent years projects have included:
"Lower bounds on ɑ-Numbers of Artin-Schreier curves" (proposed by Jeremy Booher)
"Dickson's Theorem" (proposed by John Bergdall)
"Class Groups of Function Fields" (proposed by Erick Knight and Ananth Shankar)
"Permutation Weights" (proposed by Paul Gunnells)
"2-Torsion in class groups" (proposed by Erick Knight)
"Unimodular lattices and modular forms" (proposed by Victor Rotger)
"Extremal problems in ordered graphs" (proposed by David Conlon),
"Project 691" (proposed by Kevin Buzzard),
"Modular representations of GL2(Fp)" (proposed by Laurent Berger and Sandra Rozensztajn), and
"Graph colouring problems" (proposed by David Conlon).
Returning students working on research projects have been mentored by PhD students from the Oxford Mathematical Institute.