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 started with definitions and elementary results, and moved on to sample some of the beautiful theory in different aspects of the study of graphs. Along the way, there was lots of opportunity for working on problems and for playing around with the ideas.
One of the key aspects of advanced mathematics is the process of abstraction. We find that seemingly unrelated situations share certain properties, so we unpick exactly what it is that those situations have in common that lead to those properties. If we can then prove results assuming only those properties, then they will apply to all the situations with those properties and so we get many results for the price of one. One absolutely key example of this process is the group, examples of which can be found in many, many mathematical settings. In this course, there was discussion of what a group is, presentation of a number of important examples, and exploration of some properties of general groups and of particular types of group. It followed on very nicely from the Number Theory course that returning students had taken in their first year, but it was also accessible to students participating in PROMYS Europe for the first time
This course drew on the book Visual Group Theory, by Nathan Carter.
David Conlon (Oxford, Wadham College)
A five week reading course on the probabilistic method, based on two sources, the book The Probabilistic Method, by Noga Alon and Joel Spencer, and the notes of an undergraduate lecture course at Oxford.