### Motivating Motives

#### Posted by John Baez

I gave an introductory talk on Grothendieck’s ‘motives’ at the conference Grothendieck’s Approach to Mathematics at Chapman University in late May.

Now the videos of all talks at this conference are on YouTube — including talks by Kevin Buzzard, Colin McLarty, Elaine Landry, Jean-Pierre Marquis, Mike Shulman and other people you’ve heard about on this blog.

Motivating MotivesUnderlying 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.

## Schubert varieties

I gotta object to a terminological issue. What you’re calling Schubert varieties are just the Schubert cells. The Schubert varieties are the closures thereof. For example, lots of people are interested in the singularities of Schubert varieties, and they wouldn’t be if those were vector spaces.

You may want to take the Next off of the (final) p27 of the HTML.