Don Zagier

Don Zagier
Name	Don Zagier
Birth date	1951
Birth place	New York City
Nationality	American
Fields	Mathematics
Workplaces	Princeton University, Max Planck Institute for Mathematics
Alma mater	Harvard University, University of Cambridge
Doctoral advisor	John H. Coates

Don Zagier is an American mathematician noted for deep contributions to number theory, modular forms, and mathematical physics. His work spans connections between prime number theorem-related phenomena, elliptic curve arithmetic, and quantum field theoretic structures arising in string theory and conformal field theory. He has held positions at leading institutions and influenced generations of researchers through both foundational theorems and expository clarity.

Early life and education

Zagier was born in New York City and educated at Harvard University and the University of Cambridge, where he studied under John H. Coates. His early exposure to problems in analytic number theory, algebraic number theory, and modular forms guided graduate work that intersected with themes in Galois theory and the theory of L-functions. During this period he interacted with scholars from Princeton University, Massachusetts Institute of Technology, Institute for Advanced Study, and École Normale Supérieure.

Mathematical career

Zagier held appointments at Princeton University and the Max Planck Institute for Mathematics, collaborating with mathematicians at Harvard University, Cambridge University, University of Bonn, and ETH Zurich. He contributed to conferences such as the International Congress of Mathematicians and workshops at Mathematical Sciences Research Institute, linking work in representation theory with applications in statistical mechanics and knot theory. His collaborations included interactions with figures from Andrew Wiles-era research, researchers in Iwasawa theory, and contributors to the proof of the Modularity theorem.

Major contributions and research topics

Zagier's research spans multiple interconnected areas: he produced seminal results on the theory of modular forms, including explicit formulas for traces of singular moduli and relationships with Hecke operators and Maass forms. He developed important insights into L-functions and special values, connecting to conjectures of Birch and Swinnerton-Dyer and to regulators appearing in Beilinson conjectures. His work on Dedekind sums and Kronecker limit formula clarified classical analytic identities and their arithmetic implications. Zagier contributed to the theory of modular curves and Shimura varieties, providing computational techniques used in the study of elliptic curves and complex multiplication. He investigated quantum modular forms and their ties to quantum invariants of knots, bridging to researchers working on the Jones polynomial, Chern–Simons theory, and topological quantum field theory. His studies of heights, periods, and polylogarithm relations influenced developments in motivic cohomology and interactions with the Bloch–Kato conjecture.

Awards and honors

Zagier has received numerous recognitions including invitations to speak at the International Congress of Mathematicians and prizes associated with contributions to number theory and mathematical physics. He has been affiliated with the Max Planck Society and elected to academies alongside members from Royal Society, National Academy of Sciences, and other national scholarly bodies. His lectures and expository writings have been celebrated in venues such as the American Mathematical Society, London Mathematical Society, and Société Mathématique de France.

Selected publications

- Papers on traces of singular moduli and relations to class field theory and Hilbert class field computations, published in journals associated with Cambridge University Press and Elsevier-distributed series. - Expository articles linking modular forms to knot invariants and to applications in conformal field theory and string theory proceedings. - Works on polylogarithms, regulators, and special values of L-functions appearing in collections from Institute for Advanced Study and conference volumes of Springer-Verlag.

Influence and legacy

Zagier's blend of concrete computation and abstract theory shaped research agendas in number theory, algebraic geometry, and mathematical aspects of theoretical physics. His methods influenced the work of researchers studying the Langlands program, Iwasawa theory, arithmetic geometry, and computational aspects of modular forms in software projects guided by teams at SageMath, PARI/GP, and university research groups. Lectures and survey articles by Zagier continue to serve as standard references for graduate programs at institutions such as Princeton University, Harvard University, Cambridge University, and ETH Zurich.

Category:American mathematicians Category:Number theorists Category:Harvard University alumni