Reacción a "The Abel Prize, the ‘Nobel Prize’ of mathematics, has been awarded to Gerd Faltings for his work on Diophantine equations"

The news that the Abel Prize has been awarded to Gerd Faltings is fantastic! Although he is being awarded the prize for his influence on arithmetic geometry, I must say that his work has also influenced algebraic geometry and mathematical physics.

Faltings, a Fields Medallist (1986) and Shaw Prize laureate (2015), is particularly well known for his proof of Mordell’s theorem, which states that for polynomial equations defining curves of genus greater than one (for example, in the case of smooth plane curves, those defined by equations of degree at least 4), the number of rational solutions is finite. The study of these rational solutions falls within the scope of the theory of Diophantine equations. These equations are particularly beautiful because they are formulated in an extremely simple manner, yet their behaviour is surprisingly complex and profound. Diophantine equations help us understand the arithmetic structure of numbers and also arise naturally in applications such as cryptography. The Mordell-Faltings theorem now shows that these solutions are strongly conditioned by the geometry of the associated curve.

But the influence of Faltings’ work extends far beyond this: not only did he prove conjectures that seemed utterly out of reach (such as those of Mordell and Shafarevich), but he also built part of the mathematical framework necessary for the progress of others. The ecosystem of arithmetic geometry, to which Faltings made a decisive contribution, forms part of the conceptual framework that made possible, amongst other advances, Andrew Wiles’ proof of Fermat’s Last Theorem.

Moreover, Faltings’ influence, through his work on abelian varieties (algebraic and complex generalisations analogous to the shape of a doughnut), extends to areas such as algebraic geometry in relation to mathematical physics. A concrete example is his proof of Verlinde’s formula, which allows the dimension of the space of conformal blocks to be calculated; in the language of physics, this dimension is interpreted as that of the Hilbert space (the space of all possible states) of certain quantum field theories, such as Chern–Simons theory following its quantisation.