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Talk - The Weil Conjectures and Topos Theory

07 Dec 2021 - Tags: my-talks

This quarter I took a class on analytic number theory with Dr Lapidus, and as a final project I wrote up a survey about the Weil Conjectures, which were the impotus for Topos Theory, an extremely deep subject that I’m always trying to learn more about. For bonus points, we could give a talk on our project, and since I’m an attention whore always happy to give a talk, I went for this option.

This was definitely one of the most difficult topics I’ve ever had to research for a talk, and I spent a lot of time on it. I’m glad I did, because my knowledge of toposes, cohomology, and their interaction has gotten much more solid. Of course, now I’m more intimately aware of just how much more about this I have to learn. It’s daunting and exciting in equal parts!

As for the talk itself, I wanted to give some very concrete examples showcasing what the Weil Conjectures say, as well as how we can use them to solve concrete problems. I read a lot of papers and watched a lot of videos about them, and I didn’t see anybody giving these kinds of concrete examples. I’ll almost certainly write up a blog post soon giving some1, because the computations aren’t super difficult and I think they’re worth showing.

In the associated paper, I spend some more time talking about toposes and how they relate to the Weil Conjectures through their cohomology theory. The subject is so deep that it’s hard to know exactly what to include, but I gave what I hope is a good introduction for beginners.

Both the paper and the talk are based heavily on this excellent talk by Sophie Morel (for the information about the Weil Conjectures) and on this excellent talk by Colin McLarty2 (for the historical development of toposes). There’s also Milne’s excellent paper here, and you can find a more complete bibliography in the paper3. At one point, I cite some geometric facts about the Grassmannians $\mathsf{Gr}(2,4)$, and while these are well known4, you can find some references for the computation here, here, and here.

Of course, I wasn’t the only speaker today! There were other presentations as well. Notably Will Hoffer gave a nice talk on explicit formulas, which you can find here. Then Adam Richardson gave a talk generalizing these “explicit formulas” to the setting of fractal geometry. This is one of the major subects that Dr Lapidus studies, and I think both of them are working with him, so it makes sense. You can tell that they collaborated, and Will’s talk was a great setup for Adam’s. As far as I know, Adam’s slides are not available online, but if he posts them somewhere I’ll be sure to link them! Next was Khoi Vo, who proved the prime number theorem. He’s planning to upload his project here sometime soon. Lastly was little old me. It was kind of a shame, because the talks were running almost 45 minutes behind schedule by the time it was my turn, which means I would have needed to fit my talk into a 15 minute slot if we wanted to end on time. Of course, that’s not possible, so almost everyone had to leave either before or during the talk. Thankfully, we recorded it so that people who are interested will still be able to watch ^_^.

The Weil Conjectures and Topos Theory

The Weil Conjectures are a set of conjectures governing the number of solutions to diophantine equations mod $p^n$. Surprisingly, a certain generating function (due to Hasse and Weil) for the number of solutions is intimately related to the geometry of the complex solutions to the same equations! Grothendieck’s famed “toposes” were instrumental in the formalization and solution of these conjectures, and remain an independently interesting topic of study to this day. In this talk we explain concretely what the Weil Conjectures are, then survey how the theory of toposes was used to solve them.

You can find the slides here, the paper here, and a recording below:

  1. Though saying I’ll write up a post on something seems to be a quick way to guarantee I never actually do it, haha. 

  2. The Weil Conjecturs, from Abel to Deligne and Nevertheless One Should Learn the Language of Topos respectively 

  3. It’s not directly related to the Weil Conjectures, but one great paper I found while looking into this stuff is Connes’s An Essay on the Riemann Hypothesis, available here

  4. On this note, I really need to spend some time playing around with Schubert Calculus… It seems like it has the nice blend of algebra and combinatorics that I’ve always found appealing.