##
**The Abel Prize 2013–2017.**
*(English)*
Zbl 1419.01001

The Abel Prize. Cham: Springer (ISBN 978-3-319-99027-9/hbk; 978-3-319-99028-6/ebook). xi, 774 p. (2019).

This is the third volume dedicated to the work of Abel prize winners; the first volume [Zbl 1247.01025] covered the years 2003–2007, the second [Zbl 1282.01002] the period 2008–2012, and the present volume the years 2013–2017. The recipients of the prize are Pierre Deligne (2013), Yakov G. Sinai (2014), John Nash and Louis Nirenberg (2015), Andrew Wiles (2016) and Yves Meyer (2017). Each mathematician has contributed a mathematical autobiography, and there is an additional curriculum vitae and a list of publications for each.

Luc Illusie describes Deligne’s work in “Pierre Deligne: a poet of arithmetic geometry”. His early work deals with spectral sequences, duality theorems, and Hodge theory. Deligne’s best-known contribution probably is his work on Weil conjectures. Among other topics he has worked on, the Weil-Deligne group, local constants of \(L\)-functions, and motives and periods are mentioned.

Carlo Boldrighini and Dong Li describe Sinai’s work on dynamical systems in fluid dynamics; Leonid Bunimovich explains Sinai’s ideas on mathematical billiards, and F. Cellarosi gives a survey on Sinai’s contributions to number theory. The next articles are by E. Gurovich (“Entropy theory of dynamical systems”), K. Khanin (“Mathematical physics”), Y. Pesin (“Sinai’s work on Markov partitions and SRB measures”), N. Simányi (“Further developments of Sinai’s ideas: the Boltzmann-Sinai hypothesis”), and D. Szász (“Markov approximations and statistical properties of billiards”).

Sylvia Nasir describes John Nash and his life, and C. De Lellis explains “The masterpieces of John Forbes Nash Jr.”, which seem to have been eclipsed by his work in game theory. R. V. Kohn discusses “A few of Louis Nirenberg’s many contributions to the theory of partial differential equations”.

Christopher Skinner takes the readers through “The mathematical works of Andrew Wiles”: explicit reciprocity laws, elliptic units, the Coates-Wiles theorem, Iwasawa’s main conjecture, Galois representations, modular elliptic curves and Fermat’s last theorem.

Finally, A. Cohen takes the readers on “A journey through the mathematics of Yves Meyes”.

Luc Illusie describes Deligne’s work in “Pierre Deligne: a poet of arithmetic geometry”. His early work deals with spectral sequences, duality theorems, and Hodge theory. Deligne’s best-known contribution probably is his work on Weil conjectures. Among other topics he has worked on, the Weil-Deligne group, local constants of \(L\)-functions, and motives and periods are mentioned.

Carlo Boldrighini and Dong Li describe Sinai’s work on dynamical systems in fluid dynamics; Leonid Bunimovich explains Sinai’s ideas on mathematical billiards, and F. Cellarosi gives a survey on Sinai’s contributions to number theory. The next articles are by E. Gurovich (“Entropy theory of dynamical systems”), K. Khanin (“Mathematical physics”), Y. Pesin (“Sinai’s work on Markov partitions and SRB measures”), N. Simányi (“Further developments of Sinai’s ideas: the Boltzmann-Sinai hypothesis”), and D. Szász (“Markov approximations and statistical properties of billiards”).

Sylvia Nasir describes John Nash and his life, and C. De Lellis explains “The masterpieces of John Forbes Nash Jr.”, which seem to have been eclipsed by his work in game theory. R. V. Kohn discusses “A few of Louis Nirenberg’s many contributions to the theory of partial differential equations”.

Christopher Skinner takes the readers through “The mathematical works of Andrew Wiles”: explicit reciprocity laws, elliptic units, the Coates-Wiles theorem, Iwasawa’s main conjecture, Galois representations, modular elliptic curves and Fermat’s last theorem.

Finally, A. Cohen takes the readers on “A journey through the mathematics of Yves Meyes”.

Reviewer: Franz Lemmermeyer (Jagstzell)

### MSC:

01-06 | Proceedings, conferences, collections, etc. pertaining to history and biography |

01A65 | Development of contemporary mathematics |

01A61 | History of mathematics in the 21st century |

01A70 | Biographies, obituaries, personalia, bibliographies |

01A60 | History of mathematics in the 20th century |