Diophantine Approximation and Algebraic Curves

The first topic of the workshop, Diophantine approximation, has at its core the study of rational numbers which closely approximate a given real number. This topic has an ancient history, going back at least to the first rational approximations for π. The adjective Diophantine comes from the third century Hellenistic mathematician Diophantus, who wrote an influential text solving various equations in integers and rational numbers (and whose name is now also attached to such “Diophantine equations”). The subject rose to prominence in the last century beginning with the seminal work of Thue, and it has been the source of a large number of deep and influential results, including far-reaching work of Siegel, Roth, Baker, Schmidt, and Faltings, among others. It remains a highly active area at the forefront of number theory and mathematics. The second topic of the workshop, algebraic curves, has long been implicit and in the background of Diophantine problems. In this direction, the fundamental role of algebraic curves was cemented in the influential 1922 conjecture of Mordell that an algebraic curve of genus at least two possesses only finitely many rational points. The interplay between geometry and arithmetic has increased rapidly since that time, and the use of increasingly advanced tools from algebraic and arithmetic geometry has led to the solution of many outstanding and previously inaccessible problems, including the resolution of Mordell’s conjecture (by Faltings). The workshop centered on the interplay between Diophantine approximation and algebraic curves, with interconnections to a diverse array of topics in algebra, geometry, analysis, and logic, among others.