These are my notes for a talk at the The Tate Conjecture workshop at the American Institute of Mathematics in Palo Alto, CA, July 23--July 27, 2007, somewhat revised, expanded, and updated. The intent of the talk was to review what is known and to suggest directions for research. They are also available as arXiv:0709.3040.
The following was originally published in The Wall Street Journal, August 1, 2007; Page B4.
One is tempted to feel sorry for mathematicians. In contrast to, say, physicists, mathematicians don't have their own Nobel Prize; [0] they rarely get hired by hedge funds; [1] 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.
But save your pity, as this crowd has a blast doing what it does. How else can you explain 30 or so renowned mathematicians spending all of last week sitting happily in barely comfortable chairs inside a joyless conference room working 9-to-5 on a dense math problem, without stopping so much as to check their BlackBerrys?
The scene was in Palo Alto, Calif., at a workshop sponsored by the American Institute of Mathematics. The institute is funded by in large part by John Fry, the owner of a chain of computer and electronics stores. Mr. Fry is a math buff, but also something of a Thomas Pynchon figure, in that he declines all requests for interviews. Among its activities, the institute sponsors weeklong workshops on important math problems that are designed to allow for more give-and-take than occurs at traditional academic conferences.
The topic of last week's workshop was the Tate Conjecture put forward in the 1960s by John Tate, a famed American mathematician who, now 82 years old, was in attendance front and center. His conjecture is closely connected with the Hodge Conjecture, one of the seven well-publicized math problems for which the Clay Mathematics Institute, founded by another rich American businessman, is offering a $1 million prize apiece to solve.
There is enough of a link between Tate and Hodge that Dinakar Ramakrishnan of CalTech, who along with USC's Wayne Raskind planned the conference, joked that the person solving Tate should get $500,000. [2]
Mathematicians are slightly sensitive about their reputation as loners. But not only were the ones in attendance last week well-socialized, they also told some good jokes at their own expense. For example, how can you spot the extroverted mathematician? He's the one staring at the other person's shoes. [3] And while their shop talk had a tendency to begin with phrases like, "Fix a set, S, such that ... " they also worried as much as anyone else about prosaic matters, like what gifts to take home afterward for the kids.
So, what exactly is the Tate Conjecture? Unfortunately, even for a layperson who keeps up with math -- who recognizes names like Fermat, Riemann and Poincar\'e -- math at this level is all but impossible to grasp. The author of a book on the Clay prizes suggested that confused lay readers just skip the chapter on Hodge.
One explanation of it is to say that mathematicians often find it useful to study the solution to a complicated equation by transforming it into a shape. The Tate Conjecture provides guidance on how closely that shape corresponds to the numbers in the original solution. To someone who knows the field, though, that's about as useful as explaining baseball by saying it involves a bat and a round object.
In addition to "What is it?" the other question for which mathematicians are braced, but which they also usually don't enjoy answering, is "What's the practical application?"
In truth, many mathematicians like to work on things simply because they find them interesting. Having said that, the field has a pretty good track record over the centuries for being useful.
In this vein, it is fitting that the institute's current headquarters is in a building that also houses one of Mr. Fry's computer stores. Take a tour of the store with a mathematician like Steven G. Krantz, the institute's deputy director, and you embark on a history of math through the prism of Silicon Valley technologies. The aisles of CDs and DVDs, for instance, lead to the tale of Joseph Fourier, the 19th-century French mathematician whose ideas about modeling the spread of heat would one day lead to digital music and movies.
Over the five days of the conference last week, the mathematicians did what people do at most any workshop. They listened to presentations, broke up into small groups, gossiped over drinks before dinner, took a group picture and celebrated with a banquet at a local Chinese restaurant.
In the end, they didn't prove or disprove the Tate Conjecture. But no one was expecting them to, because a solution might not arise for many years. Despite nearly 24 centuries of trying, mathematicians still don't know the truth or falsehood of the Twin Prime Conjecture, which holds that there are infinitely many prime numbers that, such as 5 and 7 or 41 and 43, are two apart.
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 the 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.
Alternately, you might simply want to mull another mathematician joke, this one courtesy of some number theorists. What should you do if you meet a tiger in the forest? Nothing; the tiger will do everything himself.
Email me at Lee.Gomes@wsj.com.
------------------[0] We now have the Abel Prize instead. Unlike the cynical economists, who named their prize after Nobel, "who became rich by finding ways to kill more people faster than ever before", we named ours after the great Norwegian mathematician Henrik Abel (1802-1829).
[1] Actually, they do. Check out Renaissance Technologies.
[2] He botched this. The joke was that, since solving the Hodge conjecture is worth one million dollars and the Tate conjecture is harder than the Hodge conjecture, it should be worth two million dollars.
[3] Actually, mathematicians hate this hackneyed "joke". Why does the WSJ think it is OK to insult and stereotype mathematicians in this way?
Despite the article's flaws, most mathematicians I spoke to were grateful for any mention in the mainstream press.
------------------Kahn writes (page 7. of The full faithfulness conjectures in characteristic $p$, to appear in the actes de l'école d'été franco-asiatique de géométrie algébrique et de théorie des nombres (IHÉS, juillet 2006), vol. 2):
Milne writes [this article, Section 1] that Tate already formulated conjecture 3.1.2 [Beilinson's conjecture] orally at the Woods Hole seminar 1964.I don't and Tate didn't.