Classifying varieties up to birational equivalence is one of the driving forces for modern research in algebraic geometry. Some of the greatest results in 20th and 21st Century algebraic geometry lie in birational geometry, with both Mori and Birkar awarded the Fields medal for their work in this area. Birational geometry has seen some very healthy and surprising interactions with number theory over the last decade.
Firstly, many of the existing techniques and results in birational geometry only work over the field of complex numbers, due to the use of transcendental methods. There has been a push to make these methods algebraic and make them work over algebraically closed fields of positive characteristic, or even non-algebraically closed fields. Number theory has also fed into birational geometry, with the realisation that rational curves should behave like rational points, i.e. solutions to Diophantine equations.
Recent developments suggest that there is much to be gained at the interface between number theory and birational geometry, and this workshop was one of the first attempts in the form of an international meeting to explore these interactions.
