Fields medalist Gerd Faltings is here in Heidelberg for the HLF, and while he’s not giving a talk he will be participating in other activities such as visits by the laureates to local schools. He’s a legendary figure in the math community about whom I had heard a lot over the years, and I was glad to finally get a chance to meet him and talk for a while over lunch.
Faltings was awarded the Fields Medal in 1986, primarily for his proof of the Mordell conjecture. This was announced in a ceremony at the Berkeley ICM, which I snuck into with help of some of my mathematician friends (I was a physics postdoc at the time, not registered for the conference). Since then Faltings has had a distinguished career, first in Princeton, then at the Max Planck Institute for Mathematics in Bonn, where he is now co-director.
His subfield of mathematics is arithmetic algebraic geometry, which studies the solutions of polynomial equations by, for instance, integers or rational numbers, using geometrical and topological methods. This is a field with a long history that includes as its targets some of the most famous problems in mathematics. It has had a remarkable amount of success in recent decades, using methods often of a high degree of sophistication and abstraction.
Two of the latest dramatic developments in the field are ones he has some significant connection to. His ex-student Shinichi Mochizuki has since 2012 claimed to have a proof of the abc conjecture, a proof involving a very complex set of new concepts that he has developed. Faltings and other experts have been unable to follow this proof, leading to a very unusual situation. No one seems to be able to identify the new ideas of the proof that could succeed where previous attempts couldn’t. At the same time the written version of the proof spans many hundreds of hard to follow pages, resisting the efforts of experts to understand and check what is going on. Faltings attended last year’s workshop at Oxford on this, but came away discouraged that more progress was not made in understanding and evaluating Mochizuki’s work.
A second new development that Faltings has some connection to is the ground-breaking work over the past few years of Peter Scholze. Scholze is also in Bonn, where he was first a student, and is now a professor at the university. His work has generated a lot of excitement, and one of its aspects is that it pushes much further lines of research developed earlier by Faltings, which he finds gratifying.
Among topics we discussed at lunch were differences between the way people worked in the US and in Europe. For instance, his experience at Princeton was that people were often making a big point of how hard they were working, something that those in Bonn would be much less likely to admit to.
Over the years I had heard various stories about how demanding Faltings was as a teacher, with one story that he had started out an undergraduate course at Princeton in such a way that the students needed to know (or learn very quickly) homological algebra. This always seemed to me hard to believe, likely a tall tale, and he assured me that it was not true.