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

To be honest, when I heard the news, I was really disappointed not to see the name of a female mathematician. I am convinced that many would be worthy of the award and, moreover, it would help raise the profile of women’s role in mathematics in the media. At least for a few days. And such visibility is increasingly necessary.

But there is no doubt that Gerd Faltings is one of the most important mathematicians working today. Proof of this is that he already won the Fields Medal forty years ago. The field in which he has made his greatest contributions is arithmetic geometry, a discipline that applies techniques from algebraic geometry to problems in number theory. Above all, arithmetic geometry focuses on Diophantine problems, the study of rational points on algebraic varieties.

One of his most notable (and, in my opinion, most beautiful) results is the proof of Mordell’s conjecture. This conjecture establishes a relationship between the rational solutions of a curve and its form, more specifically with the number of holes in its representation. Specifically, Mordell conjectured in 1922 and Faltings proved in 1983 that a curve of genus (number of holes) 2 or more cannot pass through infinitely many rational points.

This relationship between the topology of a curve and its rational solutions represents a major breakthrough in one of the fields that the great Gauss described as ‘the queen of mathematics’.

His contributions to Hodge theory are also fundamental; once again, this involves the use of tools from branches of mathematics that are, in principle, very distant from one another, to obtain results in another branch. Specifically, he obtains results in algebraic topology (cohomology of geometric varieties) using partial differential equations.

In summary, Faltings has made highly significant advances in areas of mathematics where results can be remarkably difficult to achieve, devising highly innovative approaches, using tools from other branches of mathematics, demonstrating the interconnection between disciplines that may seem very distant, and providing tools that have proven to be of great utility. A well-deserved prize.