Weekend research talk: Sarnak on different approaches to GRH

In June 2018 a conference was held in Bristol called Perspectives on the Riemann Hypothesis. The talks from this conference are not on YouTube, but can be found on the conference webpage.

Many of these talks are excellent, but today I specifically want to recommend the talk by Sarnak, called Commentary and comparisons of some approaches to GRH (slides here, video here), This talk complements very well his more recent talk posted last week. Among other things it highlights the central role of positivity which plays a key role in all serious approaches to GRH.

In the analogy between the number field world and the function field world, perhaps the most important difference of all is that in the latter, there are geometric tools for proving positivity theorems, which then lead to proofs of the Riemann Hypothesis in that context. Grothendieck originally envisioned that positivity should come from his famous standard conjectures, but this was never fully realized, and in the end other routes to positivity were found. We'll come back to all of this. But check out Sarnak's talk and perhaps also some of the other talks from the conference! Among many other speakers you will find Booker and Rodriguez Villegas, two of my all-time favourite mathematical authors.