Colloquium, Mathematical Sciences

Abstract: In 1983, Gerd Faltings proved the celebrated Mordell conjecture. In simplified form, it says that a (smooth) curve defined by a polynomial f(x,y)=0 with rational coefficients of degree at least 4 can only have finitely many solutions where both x and y are rational numbers . This spectacular breakthrough, for which Faltings was awarded the Fields medal in 1986 and most recently the 2026 Abel Prize, transformed arithmetic geometry and opened up entirely new approaches to some of the oldest questions in mathematics.

In this talk, I will explain the history of the problem, some of the ideas behind Faltings' theorem, and describe how the search for quantitative versions continues to shape modern arithmetic geometry. No prior knowledge of arithmetic geometry will be assumed.