The n-Category Café

Motivating Motives

Underlying the Riemann Hypothesis there is a question whose full answer still eludes us: what do the zeros of the Riemann zeta function really mean? As a step toward answering this, André Weil proposed a series of conjectures that include a simplified version of the Riemann Hypothesis in which the meaning of the zeros becomes somewhat easier to understand. Grothendieck and others worked for decades to prove Weil’s conjectures, inventing a large chunk of modern algebraic geometry in the process. This quest, still in part unfulfilled, led Grothendieck to dream of ‘motives’: mysterious building blocks that could explain the zeros (and poles) of Weil’s analogue of the Riemann zeta function. This talk by a complete amateur will try to sketch some of these ideas in ways that other amateurs can enjoy.

You can watch a video of this talk here. You can see the slides here — they’re a series of webpages, so click “Next” to move on to the next slide. You can also see the slides as a PDF, but without the animations.

I’ve also been writing a series of blog articles that goes into more detail, but these don’t reach all the topics in my talk:

• The Riemann Hypothesis (Part 1) - wave-like corrections to the prime counting function coming from nontrivial zeros of the Riemann zeta function.
• The Riemann Hypothesis (Part 2) - Hasse’s theorem on the number of points of elliptic curves over finite fields.
• The Riemann Hypothesis (Part 3) - Weil’s theorem on the number of points of curves over finite fields, and Weil’s conjectures generalizing this to smooth projective varieties.