Profile
Profile

Prof. Dr. Don Bernard Zagier

E-mail: don.zagier(at)mpim-bonn.mpg.de
Phone: +49 228 402 233
Fax: +49 228 402 277
Homepage: http://www.mpim-bonn.mpg.de/node/97
Location: Max Planck Institute for Mathematics
Institute: Max Planck Institute for Mathematics
Research Areas: Research Area C
Research Area DE
Former Research Area D
Former Research Area E
Date of birth: 29.Jun 1951
Mathscinet-Number: 186205

Academic Career

1969

Diploma of Advanced Mathematics, Oxford University, England, UK

1971

D.Phil., Oxford University, England, UK

1971 - 1984

Scientific Member, DFG Collaborative Research Center SFB 72 “Approximation”, University of Bonn

1975

Habilitation, University of Bonn

1979 - 1990

Chair Professor of Number Theory, University of Maryland, College Park, MD, USA

1990 - 2001

Professor, University of Utrecht, Netherlands

1990 - 1991

Professor, Kyushu University, Fukuoka, Japan

1992 - 1993

Professor, Kyushu University, Fukuoka, Japan

2000 - 2014

Professor, Collège de France, Paris, France

Since 1976

APL Professor, University of Bonn

Since 1984

Scientific Member, Max Planck Institute for Mathematics, Bonn

Since 1995

Director, Max Planck Institute for Mathematics, Bonn

Since 2014

Distinguished Staff Associate, International Centre for Theoretical Physics, Trieste, Italy

Research Profile

Modular forms, which are my main area of research, can be seen as part of both the theory of automorphic forms and of moduli spaces (Research Area DE), but are also of great importance in many parts of quantum field theory and string theory (Area C). My research in the last years has touched all these aspects, two examples being my work with Dabolkar and Murthy on applications of ''mock modular forms'' (as developed by my then student Zwegers, myself and others) to the string theory of black holes and my recent work with Möller on applications of the theory of modular and quasimodular forms to Teichmüller curves and to moduli spaces of flat surfaces. With Garoufalidis I have also been studying the arithmetic of quantum invariants of knots: we proved some cases of the conjectured ''quantum modularity'' properties of Kashaev invariants that I had discoved experimently ten years ago and, in joint work with Frank Calegari, found a construction of algebraic units from classes in algebraic K-theory having as an unexpected consequence a proof of Nahm's conjecture relating modularity to algebraic K-theory.

Recently I have become interested in the arithmetic and topology of differential equations (the subject of the ''Hirzebruch Lecture'' that I gave at the ECM 2016). Together with Vasily Golyshev, with whom I have already published one paper on the subject (proving the so-called ''Gamma Conjecture'' for all rank one Fano 3-folds) and others (in particular Masha Vlasenko and Spencer Bloch), I am studying the relation of ''motivic gamma functions'' (a kind of Mellin transform of solutions of Picard-Fuchs differential equations) and Hirzebruch-like characteristic classes of algebraic varieties. In other directions, I am studying together with Lin Weng the properties of the ''higher rank zeta functions'' of curves over finite fields that he defined some years ago (in particular, we proved the Riemann hypothesis for the genus one case and are working on the general case), and am also working with T. Ibukiyama to extend our theory of ''higher spherical polynomials'' to a theory of higher spherical functions. Finally, in collaboration with Martin Möller and others (recently Di Yang and Boris Dubrovin), we are extending our earlier work on combinatorial aspects of moduli spaces (Hurwitz numbers, graph counting, generalizations of the Bloch-Okounkov theorem, ...).

Selected Publications

[1] V. V. Golyshev, D. Zagir
Proof of the gamma conjecture for Fano 3-folds with a Picard lattice of rank one
Izv. Ross. Akad. Nauk Ser. Mat. , 80: (1): 27--54
2016
DOI: 10.4213/im8343
[2] Martin Möller, Don Zagier
Modular embeddings of Teichmüller curves
Compos. Math. , 152: (11): 2269--2349
2016
DOI: 10.1112/S0010437X16007636
[3] D. Zagier
Evaluation of the multiple zeta values ζ(2,ldots,2,3,2,ldots,2)
Ann. of Math. (2) , 175: (2): 977--1000
2012
DOI: 10.4007/annals.2012.175.2.11
[4] Tudor Dimofte, Sergei Gukov, Jonatan Lenells, Don Zagier
Exact results for perturbative Chern-Simons theory with complex gauge group
Commun. Number Theory Phys. , 3: (2): 363--443
2009
DOI: 10.4310/CNTP.2009.v3.n2.a4
[5] Don Zagier
Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann)
Séminaire Bourbaki. Vol. 2007/2008
Astérisque (326): Exp. No. 986, vii--viii, 143--164 (2010)
2009
ISBN: 978-285629-269-3
[6] J. Lewis, D. Zagier
Period functions for Maass wave forms. I
Ann. of Math. (2)
, 153: (1): 191--258
2001
DOI: 10.2307/2661374
[7] Benedict H. Gross, Don B. Zagier
Heegner points and derivatives of L-series
Invent. Math. , 84: (2): 225--320
1986
DOI: 10.1007/BF01388809
[8] J. Harer, D. Zagier
The Euler characteristic of the moduli space of curves
Invent. Math. , 85: (3): 457--485
1986
DOI: 10.1007/BF01390325
[9] Martin Eichler, Don Zagier
The theory of Jacobi forms
of Progress in Mathematics : v+148
Publisher: Birkhäuser Boston, Inc., Boston, MA
1985
ISBN: 0-8176-3180-1
DOI: 10.1007/978-1-4684-9162-3
[10] F. Hirzebruch, D. Zagier
Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus
Invent. Math. , 36: : 57--113
1976
DOI: 10.1007/BF01390005

Publication List

Editorships

• Journal of Number Theory (since 1981)
• Selecta Mathematica (since 1994)
• The Ramanujan Journal (since 1994)
• Kyushu Journal (since 1998)
• Ergebnisse der Mathematik und ihrer Grenzgebiete (since 1998)

Awards

1984

Carus Prize, Schweinfurt

1987

Frank Nelson Cole Prize in Number Theory

1996

Prix Elie Cartan, Académie des Sciences

2000

Chauvenet Prize of the Mathematical Association of America

2001

Karl Georg Christian von Staudt Prize

Selected PhD students

Georgia Triantafillou (1977): “Equivariant rational homotopy theory”,
now Professor, Temple University, PA, USA

Winfried Kohnen (1980): “Modular forms of alf-integral weight”,
now Professor, University of Heidelberg

Robert Sczech (1982): “Summation of L-series”,
now Associate Professor, Rutgers University, NJ, USA

Svetlana Katok (1983): “Modular forms and closed geodesics”,
now Professor, Pennsylvania State University, PA, USA

Nils-Peter Skoruppa (1984): “Jacobi and modular forms of half-integral weight”,
now Professor, University of Siegen

Maxim Kontsevich (1991): “KdV hierarchy and moduli spaces”,
now Permanent Professor, Institut des Hautes Études Scientifiques (IHÉS), France

Herbert Gangl (1994): “Functional equations of polylogarithms”,
now 1H Course Director, Reader, Durham University, England, UK

Sander Zwegers (2002): “Mock theta functions”,
now Professor (tenure), University of Cologne

Anton Mellit (2008): “Higher Green's functions for modular forms”,
now Postdoc, International School for Advanced Studies (SISSA), and Consultant, International Centre for Theoretical Physics (ICTP), Italy

Maryna Viazovska (2013): “Modular Functions and Special Cycles”,
now Postdoc, Berlin Mathematical School, HU Berlin

Danylo Radchenko (2016): “Higher cross-ratios and geometric functional equations for polylogarithm”,
now Postdoctoral Fellow, International Centre for Theoretical Physics (ICTP), Trieste, Italy

Supervised Theses

  • PhD theses: 25, currently 3
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