Students returning to PROMYS Europe are able to revisit those parts of the number theory course that they didn't have time to explore fully as a first-year student, while also pursuing lectures and problems specifically aimed at returning students, and working on a research project.
In 2017, Vicky Neale will give a course on Graph Theory aimed at returning students.
The study of graphs, networks of vertices (or nodes) some of which are connected by edges, arguably began in the 18th century with Euler's celebrated solution of the Bridges of Königsberg problem, but much of the work in the area dates from the 20th and 21st centuries (so far). The theory of graphs has been developed significantly since Euler's early work, but the area is still characterised by problems that are easy to state and (in some cases) fiendishly difficult to solve, and very often the key to progress is hard work and ingenuity in problem solving rather than the application of sophisticated theory. The course will start with definitions and elementary results, and move on to sample some of the beautiful theory in different aspects of the study of graphs. Along the way, there will be lots of opportunity for working on problems and for playing around with the ideas.
We are also exploring the possibility of offering another course in algebraic number theory or some other topic of interest to returning students.
Returning students will 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 "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.