John Tate 1925-2019

John Tate, who was responsible for some of the most important developments in number theory and arithmetic geometry during the second half of the twentieth century, has passed away at the age of 94. Tate was a faculty member in the Harvard math department when I was an undergraduate there, moving on to UT Austin in 1990, then retiring from there in 2009.

The work that Tate is famous for includes “Tate’s thesis”, his 1950 doctoral thesis, which may be the most influential doctoral thesis of modern mathematics. For a book-length explanation of Tate’s thesis, see Ramakrishnan and Valenza’s Fourier Analysis on Number Fields. The later generalization of the GL(1) case of Tate’s thesis to the non-abelian GL(n) case is one of the founding pillars of the Langlands program.

Tate was the Abel Prize laureate in 2009, and one can learn a lot more about him from an interview conducted around the time of the award. For an extensive discussion of Tate’s mathematical work, see this article from James Milne, or this review by Milne of Tate’s Collected Works.

From Milne’s web-site, some stories about Tate:

A mathematician was explaining his work to Tate, who looked bored. Eventually the mathematician asked “You don’t find this interesting?” “No, no” said Tate, “I think it is very interesting, but I don’t have time to be interested in everything that’s interesting”.

As a thesis topic, Tate gave me the problem of proving a formula that he and Mike Artin had conjectured concerning algebraic surfaces over finite fields. One day he ran into me in the corridors of 2 Divinity Avenue and asked how it was going. “Not well” I said, “In one example, I computed the left hand side and got p13; for the other side, I got p17; 13 is not equal to 17, and so the conjecture is false.” For a moment, Tate was taken aback, but then he broke into a grin and said “That’s great! That’s really great! Mike and I must have overlooked some small factor which you have discovered.” He took me off to his office to show him. In writing it out in front of him, I discovered a mistake in my work, which in fact proved that the conjecture was correct in the example I considered. So I apologized to Tate for my carelessness. But Tate responded: “Your error was not that you made a mistake — we all make mistakes. Your error was not realizing that you must have made a mistake. This stuff is too beautiful not to be true.”

During a seminar at Harvard, a conjecture of Lichtenbaum’s was mentioned. Someone scornfully said that for the only case that anyone had been able to test it, the powers of 2 occurring in the conjectured formula had been computed and they turned out to be wrong; thus the conjecture is false. “Only for 2” responded Tate from the audience. [And, in fact, I think the conjecture turned out to be correct except for the power of 2.]

Tate’s father, John Torrence Tate Sr., was a physicist, editor of the Physical Review between 1926 and 1950. In one famous story, Tate Sr. stood up to Einstein by insisting that one of his papers be refereed in the usual way. Einstein was outraged (but it turned out the paper was incorrect). A few years ago I was at a talk here in New York at the Simons Foundation, during which the speaker put up a slide referring to Tate (Jr.)’s work, with a picture of Tate. After a moment, from the back of the room we heard “that’s not me, that’s my father!”.

Update: Kenneth Chang has an obituary of Tate at the New York Times.

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One Response to John Tate 1925-2019

  1. Michael Weiss says:

    My favorite memory of John comes from the fall of 1977. Prof. Tate was frustrated by our undergraduate class’ not understanding the special properties of normal subgroups, and so he had the entire class recite — aloud and in unison — the central theorem. Prof. Raoul Bott’s office was right across the hall. Hearing the chant-like recitation, Bott knocked on the door and asked in a stage whisper, “John, where is the revival meeting?” John relished the interruption. It was clear to us that these two legendary senior figures had a special bond based on a shared love for the beauty and depth of mathematics.

    Based on comments from Barry Mazur and Dick Gross in recent years, I do not think that John ever repeated this experiment in undergraduate education. Yet whether the recitation worked in any didactic sense, we loved John for his passion, clarity and kindness. (John years later filled in for Bott as interim Master of Dunster House, depriving Adams House students of his generous advice at meals.)

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