29 June 1951 |
Heidelberg, West Germany
|Institutions||Max Planck Institute for Mathematics
Collège de France
|Alma mater||University of Bonn|
|Doctoral advisor||Friedrich Hirzebruch|
|Doctoral students||Svetlana Katok
|Known for||Gross–Zagier theorem
Don Bernard Zagier (born 29 June 1951) is an American mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany, and a professor at the Collège de France in Paris, France.
He was born in Heidelberg, West Germany. His mother was a psychiatrist, and his father was the dean of instruction at the American College of Switzerland. His father held five different citizenships, and he spent his youth living in many different countries. After finishing high school (at age 13) and attending Winchester College for a year, he studied for three years at M.I.T., completing his bachelor's and master's degrees and being named a Putnam Fellow in 1967 at the age of 16. He then wrote a doctoral dissertation on characteristic classes under Friedrich Hirzebruch at Bonn, receiving his PhD at 20. He received his Habilitation at the age of 23, and was named professor at age 24.
He collaborated with Hirzebruch in work on Hilbert modular surfaces. Hirzebruch and Zagier coauthored Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, where they proved that intersection numbers of algebraic cycles on a Hilbert modular surface occur as Fourier coefficients of a modular form. Stephen Kudla, John Millson and others generalized this result to intersection numbers of algebraic cycles on arithmetic quotients of symmetric spaces.
One of his most famous results is a joint work with Benedict Gross (the so-called Gross–Zagier formula). This formula relates the first derivative of the complex L-series of an elliptic curve evaluated at 1 to the height of a certain Heegner point. This theorem has many applications including implying cases of the Birch and Swinnerton-Dyer conjecture along with being a key ingredient to Dorian Goldfeld's solution of the class number problem. As a part of their work, Gross and Zagier found a formula for norms of differences of singular moduli. Zagier later found a formula for traces of singular moduli as Fourier coefficients of a weight 3/2 modular form.
Zagier found a formula for the value of the Dedekind zeta function of an arbitrary number field at s = 2 in terms of the dilogarithm function, by studying arithmetic hyperbolic 3-manifolds. He later formulated a general conjecture giving formulas for special values of Dedekind zeta functions in terms of polylogarithm functions.
- "Upon looking at these numbers, one has the feeling of being in the presence of one of the inexplicable secrets of creation." The First 50 Million Prime Numbers
- "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definitions and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision." The First 50 Million Prime Numbers
- Imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4, then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it's true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrödinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful." Mathematicians: An Outer View of the Inner World
- Mathematics is very creative, not just a mechanical procedure. It is very personal. Sometimes just from the statement of a result you can guess which mathematician did the work. In some sense, math is already there and is true whether we discover it or not - there is a real mathematical world and it is much wider than the physical world of ninety-two elements or sixteen elementary particles. When you find a result, it is not really yours, because it was already true, but you express your personality by the choices that you make in discovering and proving it. It's like chess, where the available moves are the same for everybody, but the novice and the expert make very different choices in how to proceed. Except that in chess there are only twenty moves at a given stage, but in mathematics there are infinitely many. The life of a mathematician is filled with a permanent sense of wonder, and one can never be bored. Mathematicians: An Outer View of the Inner World
- Zagier, D. (1990), "A One-Sentence Proof That Every Prime p ≡ 1 (mod 4) Is a Sum of Two Squares", The American Mathematical Monthly (Mathematical Association of America) 97 (2): 144, doi:10.2307/2323918, JSTOR 2323918. The First 50 Million Prime Numbers." Math. Intel. 0, 221–224, 1977.
- (with F. Hirzebruch) "Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus" Invent. Math. 36 (1976) 57-113
- Hyperbolic manifolds and special values of Dedekind zeta functions Invent. Math. 83 (1986) 285-302
- (with B. Gross) Singular moduli J. reine Angew. Math. 355 (1985) 191-220
- (with B. Gross) Heegner points and derivative of L-series Invent. Math. 85 (1986) 225-320
- (with J. Harer) The Euler characteristic of the moduli space of curves Invent. Math. 85 (1986) 457-485
- (with B. Gross and W. Kohnen) Heegner points and derivatives of L-series. II Math. Annalen 278 (1987) 497-562
- The Birch-Swinnerton-Dyer conjecture from a naive point of view in Arithmetic Algebraic Geometry (G. v.d. Geer, F. Oort, J. Steenbrink, eds.), Prog. in Math. 89, Birkhäuser, Boston (1990) 377-389
- Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields in Arithmetic Algebraic Geometry (G. v.d. Geer, F. Oort, J. Steenbrink, eds.), Prog. in Math. 89, Birkhäuser, Boston (1990) 391-430
- Frank Nelson Cole Prize in Number Theory, American Mathematical Society. Accessed March 17, 2010
- Zagier Receives Von Staudt Prize. Notices of the American Mathematical Society, vol. 48 (2001), no. 8, pp. 830–831
- Biography from the webpage of the Max Planck Society
- Don Zagier at the Mathematics Genealogy Project