Weekend research talks on the work of Faltings

Gerd Faltings was recently awarded the Abel Prize, and there is now a number of nice video lectures explaining his work to more or less specialized audiences. I have chosen here three talks in increasing level of specialization, hoping that at least one of them will provide an interesting watch over the weekend, perhaps as a warmup to watching Sweden smash the Netherlands to pieces in the World Cup on Saturday.

Watching these talks, you will find many points of contact with the RH Saga. One such point is the idea that if we fix the degree $d$ and the conductor $N$, there should only be finitely many L-functions with this specific $d$ and $N$. In general, one might speculate that a good future theory of $\mathbb{F}_1$ might be a new tool for proving finiteness theorems in arithmetic geometry, and several major results by Faltings are representative examples of what a "finiteness theorem" is.

Kramer on the work of Faltings

The first talk is the most accessible one. Jürg Kramer will gently take you through the basic ideas leading up to the proof of the Mordell Conjecture that gave Faltings the Fields medal in 1986. Kramer covers not the original proof, but a proof found later by Bombieri (this is the same Enrico Bombieri who wrote the RH Millennium problem description and who also found a simplified version of the RH proof for curves over finite fields in the 70s).

The story told by Faltings himself

There is something unique about hearing the story of a major mathematical breakthrough from the person who did the original work. In his Laureate lecture from Oslo, Faltings tells the story of the Mordell conjecture, from its historical roots to his own work and beyond.

Zhang on Faltings heights and L-series

Finally, if you want to skip the Sweden-Netherlands game altogether and instead spend all of your Saturday evening on ideas emerging from the work of Faltings, you might enjoy this series of more technical talks on Carmin.tv, with more direct links to many RH Saga topics like L-functions and the abc conjecture. Unlike the previous talks, this is a specialized research talk for an expert audience.

Here is the first talk on YouTube; find the rest either on Carmin or by searching on YouTube.