This is not a very timely posting, since my readers let me down by not telling me about this when it came out. Last month the Wall Street Journal ran a piece by Lee Gomes about a workshop on the Tate conjecture held recently at AIM, the institute now housed in Palo Alto behind Fry’s Electronics, at some point to move to its own castle. The piece was entitled Math Whizzes at Conference Prove Just How Exciting The Tate Conjecture Can Be, and it gave a good feel for what a math workshop looks like to an outsider. The full piece is not available on-line, but the MAA Math News has an article that quotes much of it.
I noticed two inaccuracies in the piece. It begins with:
One is tempted to feel sorry for mathematicians. In contrast to, say, physicists, mathematicians don’t have their own Nobel Prize; they rarely get hired by hedge funds; they don’t have grand toys like particle accelerators to play with; and their work is usually so recondite that not even their families understand it.
This is pretty accurate except for the part about hedge funds. I know quite a few mathematicians who have gone to work for them, and at some of them mathematicians form a sizable fraction of the people holding so-called “quant” jobs.
At the end of the piece there’s the news:
Progress, though, was made. V. Kumar Murty, of the University of Toronto, said that as a result of the sessions, he’d be pursuing a new line of attack on Tate. It makes use of ideas of J.S. Milne of Michigan, who was also in attendance, and involves Abelian varieties over finite fields, in case you want to get started yourself.
This becomes more-or-less correct when you replace “Tate” with the “weak rationality conjecture”.
Milne’s article is actually a write-up of his talk at the AIM workshop, and it does an excellent job of surveying the state of what is known about questions related to the conjecture.
I was going to try and put together some explanation of what the Tate conjecture says and how it relates to other parts of mathematics, but since this is a tricky business, and since experts who really understand this have already done a better job elsewhere than I could ever do here, I’ll mostly just provide links.
The Tate conjecture is an analog for varieties over finite fields of one of the Clay Millennium problems, the Hodge conjecture, which deals with the case of varieties over the complex numbers. For a popular discussion of this, there’s a nice talk by Dan Freed on the subject (slides here, video here). In the number field case there’s another Millennium problem analog, the Birch and Swinnerton-Dyer conjecture. For a popular discussion of this, there’s a video of a talk by Fernando Rodriguez-Villegas (who has a blog here).
These conjectures all revolve around the idea that it should be possible to relate three apparently different mathematical objects associated with an algebraic variety:
There’s no evidence we’re close to a proof of these conjectures, but there are many partial results and the conjectures can be proved in certain special cases. Experts seem convinced of the truth of these conjectures despite the lack of proof, one reason being that they fit nicely into the general philosophy of “motives” first promulgated by Grothendieck. One expert on the Tate conjecture, when asked about the probability of it not being true, responded something like: “Don’t be silly. It’s true.”
For more about the Tate conjecture, there are two documents put together for the AIM workshop that may be helpful: an expository piece for a wide audience here, and a technical summary of the workshop here.