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The shape of a life. One mathematician’s search for the universe’s hidden geometry. (English) Zbl 1435.32001

New Haven, CT: Yale University Press (ISBN 978-0-300-23590-6/hbk). xvi, 293 p. (2019).
S.-T. Yau received, in 1983, probably the most important prize in mathematics, the Fields Medal, for his contribution to global geometry and elliptic partial differential equations, with applications in three-dimensional topology and in general relativity, besides other recognitions: Veblen Prize in 1981, Carfoord prize in 1994, Wolf Prize in 2010.
We mention few of the major contributions of Yau: solution of the Calabi conjecture on Kähler-Einstein metrics, work on metrics on the moduli space of curves, rigidity of manifolds and harmonic maps, solution to the positive mass conjecture in general relativity, Schwarz lemma for complex manifolds, etc. Also, he initiated important methods: the use of minimal surfaces for the topology of manifolds, the positive mass in relativity theory, uniformization of complex manifolds, methods of differential and algebraic geometry to applied mathematics such as computer imaging. Among the open problems proposed by Yau we mention the study of relations between Kähler-Einstein metrics on complex manifolds with positive first Chern class and the stability of manifolds in algebraic geometry.
The book surprises the reader first by its nature, a fascinating story of S.-T. Yau, and, second, by the very natural style of the exposition.
The well-known mathematician provides great talent to write his biography and make it so captivating not only for mathematics lovers, but for everyone.
The title itself is remarkable, in both versions: “The shape of a life” and “One mathematician’s search for the universe’s hidden geometry”.
After the preface, the book has 12 chapters, followed by an epilogue (containing two poems written by Yau, “Poincaré’s dream” and “Ode to space-time”) and an index. Photographs are included in the middle of the book.
Chapter 1, entitled “Itinerant youth” starts with a short historical presentation of Yau’s birth origin, of his family, strongly influenced by the Hakka culture, an ethnic group thought to have originated in the Yellow-River Valley of northern China. His father graduated at an university in Japan, his mother did not continue her studies beyond high school, where she worked as a librarian.
The others chapters are entitled: “Life goes on”, “Coming to America”, “In the foothills of Mount Calabi”, “The march to the summit”, “The road to Jiaoling”, “A special year”, “Strings and waves in sunny San Diego”, “Harvard bound”, “Getting centered”, “Beyond Poincaré”, “Between two cultures”.
Each chapter contains, generally following a chronological order, the steps of Yau’s life, between China, Hong Kong and the United States.
As Yau wrote in the Preface, “Conjectures have been made during these excursions, open problems raised and various theorems proved”.
S.-T. Yau had the chance to meet great mathematicians and physicists, which influenced his way in mathematics and life: S. S. Chern, his former advisor, A. Borel, E. Calabi, H. Hironaka, L. Nirenberg, J. Milnor, D. Mumford, R. Osserman, I. Singer, S. Hawking and many others.
From mathematical point of view, below are written few of the problems which the author mentioned in this book (in the appearance order): the Dedekind cut, the strong relation between geometry and topology, the curvature role in physics, the theorem of Preissman and its extension, the Calabi conjecture, the Monge-Ampère equation, the Dirichlet problem, the Poincaré conjecture, Dehn’s lemma, the Euler conjecture, Calabi-Yau manifolds, string theory, the characteristic (Euler) number, mirror manifolds, mirror symmetry, the SYZ conjecture, etc.
Indeed, S.-T. Yau was able to present his exciting journey and many pleasant surprises. The details of the previous list give in fact the charm of the volume and must be discovered by the readers having the chance to keep this wonderful book in their hands.

MSC:

32-03 History of several complex variables and analytic spaces
32Q25 Calabi-Yau theory (complex-analytic aspects)
01A70 Biographies, obituaries, personalia, bibliographies
01A60 History of mathematics in the 20th century
00A09 Popularization of mathematics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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