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**Correspondence between Henri Cartan and André Weil (1928–1991).
(Correspondance entre Henri Cartan et André Weil (1928–1991).)**
*(French)*
Zbl 1225.01034

Documents Mathématiques 6. Paris: Société Mathématique de France (ISBN 978-2-85629-314-0/hbk). xiii, 720 p. (2011).

This volume presents the known correspondence between two eminent mathematicians of the 20th century: Henri Cartan and André Weil. The letters, written between 1928 and 1991, make up about 430 pages of this book. Then there are more than 220 pages of comments by the editor, a bibliography, and an extensive index.

Weil’s first letter is the reply to a proof by Cartan that \(\lim_{n \to \infty} \pi(n)/n = 0\), where \(\pi(n)\) is the prime number function. Two years later, Weil writes from India and comments on its system of higher education as well as on recent results by Siegel and Gelfond. There are no letters during the period from November 1933 and spring 1939, when Cartan and Weil both were colleagues in Strasbourg. In the subsequent years, Weil is busy with his proof of the Riemann conjecture for function fields of genus \(> 1\) as well as with the Bourbaki project. After several letters written during his imprisonment in Rouen, the correspondence resumes in November 1944, when Weil accepted a position in Brazil. In the years that follow, Weil and Cartan exchange their views on various topics related to the Bourbaki project, in particular distributions, topology, cohomology etc.

In 1956, Weil suggests that the founding members of Bourbaki should vanish gradually but in a finite amount of time (like the Cheshire cat in Alice in Wonderland), and the remaining letters up to the last one written by Cartan in 1991 contain almost no mathematics.

Michèle Audin has done historians of mathematics all over the world a huge favor by publishing this correspondence and by explaining most of the more cryptic remarks of Weil in her comments. The Société Mathématique de France also has to be thanked for publishing this volume in times when both society and politics have lost the power to direct the streams of money.

Weil’s first letter is the reply to a proof by Cartan that \(\lim_{n \to \infty} \pi(n)/n = 0\), where \(\pi(n)\) is the prime number function. Two years later, Weil writes from India and comments on its system of higher education as well as on recent results by Siegel and Gelfond. There are no letters during the period from November 1933 and spring 1939, when Cartan and Weil both were colleagues in Strasbourg. In the subsequent years, Weil is busy with his proof of the Riemann conjecture for function fields of genus \(> 1\) as well as with the Bourbaki project. After several letters written during his imprisonment in Rouen, the correspondence resumes in November 1944, when Weil accepted a position in Brazil. In the years that follow, Weil and Cartan exchange their views on various topics related to the Bourbaki project, in particular distributions, topology, cohomology etc.

In 1956, Weil suggests that the founding members of Bourbaki should vanish gradually but in a finite amount of time (like the Cheshire cat in Alice in Wonderland), and the remaining letters up to the last one written by Cartan in 1991 contain almost no mathematics.

Michèle Audin has done historians of mathematics all over the world a huge favor by publishing this correspondence and by explaining most of the more cryptic remarks of Weil in her comments. The Société Mathématique de France also has to be thanked for publishing this volume in times when both society and politics have lost the power to direct the streams of money.

Reviewer: Franz Lemmermeyer (Jagstzell)