The Wall Street Journal on the Tate Conjecture

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.

Milne has recently posted an article on the arXiv (also available on his web-site here) that corrects this, noting

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:

  • The space of algebraic cycles in the variety, modulo some equivalence relation
  • Certain cohomology groups associated to the variety
  • The order of a pole in the zeta-function of the 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.

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    8 Responses to The Wall Street Journal on the Tate Conjecture

    1. Clark says:

      Thank you for your reference to the talk of Prof. Milne, to whom it is always worth paying attention .

      I note from this source that he is not convinced of the validity of the Tate Conjecture, at least in full generality:

      “ASIDE 6.4. The Hodge conjecture is known for divisors, and the Tate conjecture is generally expected to be true for divisors. However, there is little evidence for either conjecture in higher codimensions, and hence little reason to expect them to be true. On the other hand, Deligne expects his conjecture to be true. “

    2. JP says:

      Tate conjecture?

      Seems like esoteric mathematics … probably doesn’t have much to do with theoretical physics. But thanks anyhow for the post.


    3. Matt Daws says:

      “they rarely get hired by hedge funds”. Yep, as you say, the writer clearly has never heard of Jim Simons for example.

    4. Thomas Love says:

      Since I had already read this entry, I decide to check out the links you provide in the right hand column. I went to each of the sites under String Theory Weblogs and found discusisions of The Bionic Woman, How to do slide shows, The Beijing Jazz festival, A proposal for a new theme new website, The Cosmic Climate link, a site last updated on Dec 5, 2006, The fun of cleaning up, another
      site last updated on Dec 5, 2006, and a site in Spanish, which I don’t know. I have to conclude that there is not much progress being made in String Theory. Surprise, Surprise, Surprise.

    5. Jim Clarage says:

      The video link to Dan Freed’s talk didn’t work for me. The link to it from the main Clay video page:

      does appear to work.

      ps,. thanks for the respite from the landscape. These popular/survey links are always appreciated.

    6. woit says:

      Thanks Jim,

      Link fixed. Now if only my postings on things like the Tate conjecture would draw 100 comments…

    7. SnarkFest says:

      Jim Simons wasn’t hired by a hedge fund. He started one. Right?

    8. Tom Whicker says:

      Instead of String Theory, it should be “The Susskind Conjecture”.

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