1.1. Review by: H H Wu.
Mathematical Reviews MR0865650 (88e:53001).

The description of geometric phenomena is usually given by nonlinear partial differential equations. The monograph under review is the extended version of three lectures given at ETH-Zürich in 1981, giving a broad overview of the role of these equations in geometry. In addition, these lectures also emphasise the connection of geometry with physics, topology and algebraic geometry. The close relationship between physics and geometry is well known; almost all the commonly studied equations in geometry are related to problems in physics. Topology, on the other hand, provides the underpinning of any serious discussion in geometry, while algebraic geometry furnishes many natural examples in differential geometry.

The monograph is divided into six sections: \1. Eigenvalues and harmonic functions. \2. Yamabe's equation and conformally flat manifolds. \3. Harmonic maps. \4. Minimal submanifolds. \5. Kähler geometry. \6. Canonical metrics over complex manifolds. In the remainder of this review, an attempt will be made to give the barest of ideas about the many topics discussed in each of these sections.

2.1. From the Preface of the English translation.

In the Spring of 1984, The authors gave a series of lectures in the Institute for advanced study in Princeton . Professor K H Zhong of the Academic Sinica in Peking took the lecture notes. These lectures become the first four chapters of this book. Then in the academic year 1984-1985, we continued to give the lectures in San Diego where Professor Wei-Vue Ding of the Academic Sinica and Professor Kung-Ching Chang of Peking University took the notes. Parts of these lecture notes become the last two chapter of this book. Since all the notes were written in Chinese, they were initially published in Chinese. While they were widely circulated in China, it was clear that it will be useful for the book to appear in English. Professor Ding and Professor S Y Cheng were kind enough to translate them into English. We are very grateful to all of these mathematicians for their tremendous help, without which the book would never appear. This book has been widely circulated in Chinese since 1986. We hope the translation will be useful for the English speaking audience.

Since much of this material was presented in lectures given almost ten years ago, many of the discussions and conjectures are out of date. While we have done some updating of these discussions together with references, we have not done this systematically because we felt it better not to delay publication.

2.2. From a translation of the Chinese preface.

Among all branches of mathematics, mathematicians have had high respect to the study of geometry since ancient times. The reason is that in geometry one studies certain forms of natural phenomena, and the vivid perception provided by natural phenomena has always been an important source of inspiration for mathematicians. As a result, geometry has a very close kinship with other branches of mathematics. On the other hand, apparently it is advancing along with progress in natural sciences. Einstein's general relativity proposed in the early part of the century and Yang-Mills' gauge field theory studied in the past two decades are a perfect illustration of how geometry and physics meet.

A significant part of geometry is differential geometry. In modern differential geometry one studies the analytic structure of a manifold and various geometric properties derived from it. Its origin may be traced back to Gauss and Riemann. Soon after Riemann proposed the geometry which later bore his name, the study of local geometry prospered quickly. Tensor analysis was invented. At the same time, Klein made known to the public his celebrated Erlangen Programme where invariants of the transformation group of the space were studied from a group-theoretic point of view. Many different geometries were introduced. In addition, the study of Riemann surfaces was promoted by the uniformization theory in complex function theory. All these developments together with the classical theory of surfaces laid the foundation for differential geometry of the twentieth century.

Differential geometry progressed rapidly in this century. The progress may be described in the following four categories.

First, classification of Lie groups and Riemannian symmetric spaces was carried out by Cartan and Weyl. Cartan generalised the concept of a connection by combining Klein's theory and Riemannian geometry. Moreover I he further developed Cartan-Kahler theory by introducing exterior differentiation. All these efforts pushed forward the local theory of differential geometry in a big step.

Second, mathematicians including de Rham, Hodge, Kodaira, Hopf, Lefschetz, Whitney, Weil and Chern established a close relationship between differential geometry and topology and algebraic geometry, which were blossoming during this period. Global differential geometry started taking its form.

Third, along the vein of classical differential geometry, geometry of convex surfaces, synthetic and integral geometry advanced greatly under the leadership of Alexandroff, Cohn-Vossen, Pogorelov, Busemann, Rauch, and Santalo.

Finally, by the maturity of the theory of differential equations, geometers started using analytical methods to tackle problems in geometry. In the reverse direction, people discovered a large number of significant differential equations from differential geometry. New approaches or methods are very often required to solve these equations. Thus analysts also watched progress in geometry closely. Leaders in this aspect include Hadamard, Morse, Lewy, Morrey, Bochner, Nash, Moser, Nirenberg, and Efimov. Their works have formed the cornerstone for the application of nonlinear partial differential equations in geometry in the past twenty years.

We attempt to describe major achievements in all these aspects in this book. The reader will find that differential geometry is a whole subject where all these aspects interlace in a natural way. In this first volume we shall study differential equations on a manifold and derive results relating its curvature to its topology. Only a single equation will be treated. Systems of differential equations will be postponed to the forthcoming second and third volumes, where we shall discuss subjects including Hodge theory, minimal submanifolds , harmonic maps, gauge fields, Kahler manifolds, and Mong-Ampere equations. Relationship between geometry and topology, algebraic geometry, general relativity and high energy physics will also be discussed.

The present volume has six chapters. In the first chapter we treat the Laplace operator, the most important differential operator in differential geometry. Its importance lies in the fact that the linearisation of many important nonlinear operators gives rise to Laplace operators of some Riemannian metrics. In specific places we need to approximate nonlinear operators by linear ones. In this aspect estimations independent of the Riemannian metric are sought for. The function space we shall use either consists of bounded functions or square summable functions. As we know, the object of study in classical harmonic analysis is the class of harmonic functions in the Euclidean space or its bounded domains. The proper generalisation in geometry is a complete Riemannian manifold, where non-negative Ricci curvature tensor corresponds to the Euclidean space and negative sectional curvature corresponds to bounded Euclidean domains. In principle, all major results in classical harmonic analysis have their counterparts on manifolds. In the first two chapters of this volume we discuss some of these important generalisations. It is worthwhile to point out that very often new methods must be introduced to achieve this goal. We have also found that many geometric problems can be solved by these analytic methods. When the Laplace operator acts on the space of square summable functions, spectral analysis is the key issue. The spectrum is discrete on a compact manifold. We shall study eigenfunctions and eigenvalues in Chapter 3. In Chapter 4 we study the spectrum via the heat kernel. It is also desirable to look at the wave kernel for the obvious reason that it links the spectrum of a manifold to its geodesics. Very little is known about the spectrum of a noncompact manifold, especially when it is continuous. We hope to discuss it in a later occasion.

We study a nonlinear partial differential equation arising from conformal deformation in Chapter 5 and Chapter 6. Research in this direction has never ceased since Poincare. First we discuss the Yamebe problem, restricting to the compact case. The noncompact case has not yet been solved completely. We wish to ret urn to it in due course. Chapter 6 is devoted to conformally flat Riemannian manifolds. The reader will find from this chapter that a rather good understanding for these manifolds can be obtained when they have positive scalar curvature. However, the problem becomes formidable in the case of negative scalar curvature.

We have not been able to find a suit able text book on differential geometry in the large, especially a systematic one which is based on topology and algebraic geometry and uses analysis as its major tool. Our book may he regarded as an attempt in this direction.

This volume is based on a series of the author's lectures. The first four chapters came from lectures given at Princeton in 1983. The notes were taken and written up by Jiaqing Zhong. The last two chapter came from lectures given at San Diego in 1984 and 1985, respectively. Chapter 5 was written by Yichao Xu and Weiyue Ding. Chapter 6 contains results mainly obtained by the authors during this period; and it was writ ten by K C Chang. All these colleagues involved in writing up the manuscript are outstanding mathematicians in their own fields. Their valuable opinions incorporated in the text surely improve the book a lot. Additionally, the first author's students Gang Tian, Huaidong Cao, and Jun Li proofread the manuscript. We would like to take this opportunity to thank all of them for their assistance.

Shing-Tung Yau
Richard Schoen

2.3. Review by: Man Chun Leung.
Mathematical Reviews MR1333601 (97d:53001).

As the authors note in their introduction, the book under review was written for the lecture series given at Princeton University in 1983 and at the University of California, San Diego, in 1984 and 1985. The book contains significant results in differential geometry and global analysis; many of them are the works of the authors. The main topics are differential equations on a manifold and the relation between curvature and topology of a Riemannian manifold. There are nine chapters in the book, with the last three chapters more like appendices, which focus on problems concerning different areas of differential geometry.
...
The book under review is very well written. Readers will find comprehensive and detailed discussions of many significant results in geometric analysis. The book is both useful as a reference book for researchers and as a course book for graduate students. With details of proofs and background materials presented in a concise and delightful way, the book provides access to some of the most exciting areas in differential geometry.

3.1. From the Publisher.

A presentation of research on harmonic maps, based on lectures given at the University of California, San Diego. Schoen has worked to use the Fells/Sampson theorem to reprove the rigidity theorem of Masfow and superrigidity theorem of Marqulis. Many of these developments are recorded here.

3.2. From the Preface.

There is a natural concept of energy associated to maps between manifolds. Critical points of such an energy functional are called harmonic maps. From the beginning harmonic maps were studied in connection with the theory of minimal surfaces. Bôchner was the first one to single out the theory of harmonic maps as generalised minimal surfaces. However, the major existence and regularity theory had to wait until C Morrey solved the famous Plateau problem in the late forties. Morrey's theory had deep influence on all of the later works on harmonic maps defined on two dimensional surfaces. This includes the fundamental work of Sacks-Uhlenbeck on minimal spheres and the works on incompressible minimal surfaces. In the mid-seventies the authors realised that much of the theory can be applied to the study of Teichmüller theory and Kähler geometry.

Part I of the book is devoted to harmonic maps defined on Riemann surfaces. While we include topics that we find interesting, we do omit a lot of important developments. Most notable is the subject of harmonic maps as exactly solvable model. The theory of harmonic maps defined on a higher dimensional manifold was not developed until the major breakthrough of Eells and Sampson in the mid-sixties. Instead of the variational approach, Eells-Sampson used a heat equation argument which has had deep influence on geometry. Regularity theory was developed much later when the target manifold is not necessary negatively curved.

In the first two chapters of Part II, we report on such regularity theory even when the target space need not be a nice manifold. In this setting, it was developed by the first author, and jointly with N Korevaar later. In the early seventies, the first author realised that the theorem of Eells and Sampson could be used to reprove the famous rigidity theorem of Mostow and superrigidity theory of Margulis. It was not until the late eighties that most of these goals were achieved. This is reported in a joint work by J Jost and the second author in the last chapter of Part II. In the mid-seventies the authors had already succeeded in applying the theory of harmonic maps to study topology of manifolds of negative curvature; these works are reported in Part II. We regret that many more applications are omitted because of time constraints.

In 1985, the authors presented a series of lectures at the University of California at San Diego on the subject of harmonic maps. Most of these lectures are collected here as Part I. Part II was added much more recently, and comprises part of the thesis of the first author, the work of N Korevaar and the second author, and applications to geometry by the authors. The final chapter is due to J Jost and the second author. The authors would like to extend their special thanks to Jiagyoung Choe for taking and preparing the copious lecture notes that became Part I of this book.

R. Shoen, Stanford University
S-T Yau, Harvard University

3.3. Review by: John C Wood.
Mathematical Reviews MR1474501 (98i:58072).

This is a very useful contribution to the literature on harmonic maps. It is not an elementary textbook on harmonic maps ... neither is it a comprehensive catalogue of known results in harmonic maps. It is rather a collection of some of the most important topics in, and applications of, harmonic maps, skewed towards the authors' interests. ... the book, especially Part I, is suitable for an advanced graduate course; some previous familiarity with harmonic maps would help but is not essential. It is mostly self-contained with important definitions given, though with reference to the literature for some harder proofs. There are some annoying instances of references which do not appear in the bibliographies, and results are not always credited, e.g. in Chapter 1. Most of the material has appeared in articles, but the authors have arranged the topics in a logical order with some chapters referring to previous ones for concepts. Each chapter has a very useful motivating introduction and the informal, yet precise, style of writing makes it a very nice book to read. In summary, this is a book that everyone interested in harmonic maps will benefit from reading.

4.1. Note.

This was a text book on ordinary differential equations which was written when Yau was a second year college student in Hong Kong being taught by Stephen Salaff.

5.1. From the Preface.

Conformal geometry is in the intersection of many fields in pure mathematics, such as Riemann surface theory, differential geometry, algebraic curves, algebraic topology, partial differential geometry, complex analysis, and many other related fields. It has a long history in pure mathematics, and is an active field in both modern geometry and modern physics. For example, the conformal fields in super string theory and modular space in theoretic physics are research areas with very fast developments.

Recently, with the rapid development of three dimensional digital scanning technology, computer aided geometric design, bio-informatics, and medical imaging, more and more three dimensional digital models are available. The need for effective methods to represent, process, and utilise the huge amount 3D surfaces has become urgent. Digital geometric processing emerges as an inter-disciplinary field, combining computer graphics, computer vision, visualisation, and geometry.

Computational conformal geometry plays an important role in digital geometry processing. It has been applied in many practical applications already, such as surface repairing, smoothing, de-noising, segmentation, feature extraction, registration, re-meshing, mesh spline conversion, animation, and texture synthesis. Especially, conformal geometry lays down the theoretic foundation and offers rigorous algorithms for surface parameterisations. Computational conformal geometry is also applied in computer vision for human face tracking, recognition, expression transfer; in medical imaging, for brain mapping, virtual colonoscopy, data fusion; in geometric modelling for constructing splines on manifolds with general topologies.

The fundamental reason why conformal geometry is so useful lies in the following facts:

- Conformal geometry studies conformal structure. All surfaces in daily life have a natural conformal structure. Therefore, the conformal geometric algorithms are very general.

- Conformal structure of a general surface is more flexible than Riemannian metric structure and more rigid than topological structure. It can handle large deformations, which Riemannian geometry cannot efficiently deal with; it preserves a lot of geometric information during the deformation, whereas topological methods lose too much information.

- Conformal maps are easy to control. For example, the conformal maps between two simply connected closed surfaces form a 6-dimensional space, therefore by fixing three points, the mapping is uniquely determined. This fact makes conformal geometric methods very valuable for surface matching and comparison.

- Conformal maps preserve local shapes, therefore it is convenient for visualisation purposes.

- All surfaces can be classified according to their conformal structures, and all the conformal equivalent classes form a finite dimensional manifold. This manifold has rich geometric structures, and can be analysed and studied. In comparison, the isometric classes of surfaces form an infinite dimensional space, which is really difficult to deal with.

- Computational conformal geometric algorithms are based on elliptic partial differential equations, which are easy to solve and the process is stable. Therefore, computational conformal geometry methods are very practical for real engineering applications.

- In conformal geometry, all surfaces in daily life can be deformed to three canonical spaces: the sphere, the plane, or the disk (the hyperbolic space). In other words, any surface admits one of the three canonical geometries: spherical geometry, Euclidean geometry, or hyperbolic geometry. Most digital geometric processing tasks in three dimensional space can be converted to the task in these two dimensional canonical spaces.

The major goals for writing this book are twofold. First, we want to introduce the beautiful theories of conformal geometry to general audiences, and make the elegant conformal structures better appreciated. The major concepts in conformal geometry are profound and abstract, which mainly existed in the imaginations of professional mathematicians. Our conformal geometric methods can compute those concepts explicitly on all kinds of surfaces in daily life, and display them using modern computer graphics and visualisation technologies. Therefore, the students can see them, sense them, and accumulate intuition. Professional mathematicians can design experiments and use computers to help their exploration.

Furthermore, we would like to introduce the practical conformal geometric algorithms, and make them easily accessible for computer scientists and engineers. Therefore, the whole book is written to use less abstract mathematical reasoning, but more intuitive explanations and hands on experience. Major concepts and theorems are visualised by figures and computational algorithms are given. Students can implement the algorithms by themselves and see the abstract concepts represented as data structures on computers and create the images reflecting various geometric structures.

The book has two parts. The first part focuses on the theoretical foundations. It covers algebraic topology, differential exterior calculus, differential geometry, Riemann surface theory, surface Ricci flow, and general geometric structures. All of this knowledge is required for doing research in computational conformal geometry. Most of these topics are elementary, and some advanced topics are briefly touched with thorough references.

The second part focuses on computational algorithms, and is completely written in computer science language. It covers the computational algorithms for surfaces, which can be easily generalised to 3-manifolds. Then the algorithms on computing conformal structures for surfaces using various methods are explained in detail. Finally, algorithms for computing hyperbolic structure, and projective structure using Ricci flow method are examined. All algorithms are accompanied by pseudo-code, which is extremely easy to convert to programming language. We hope students can build the software system from scratch, and follow the book to implement various algorithms. The algorithms described in the book have already been applied in industrial applications.

The major content of the book is summarised from our research projects during the last several years. This textbook has been taught in graduate level courses in the Mathematics Department at Harvard University and the Computer Science Department at the State University of New York at Stony Brook. The theory part takes one semester, the computer science part takes one semester. The problem sets and programming exercises are valuable for students to improve their understanding and build their practical skill for developing geometric processing software. More teaching materials, coding samples and geometric surface data sets are available from the authors by requests.

5.2. Review by: Lyuba S Alboul.
Mathematical Reviews MR2439718 (2010m:68165).

This textbook provides a broad introduction to the emerging field of computational conformal geometry. The book consists of two parts: the first is dedicated to theoretical foundations of conformal geometry and related concepts and the second focuses on algorithms and adaptation of conformal geometry methods to computer-aided applications.

Conformal geometry is a well-established field within pure mathematics, and is closely related to physics. The authors therefore provide an elementary introduction to various special areas related to conformal geometry, such as differential geometry of surfaces, combinatorial and algebraic topology, theory of Riemannian surfaces and geometric structures, and harmonic maps, with emphasis on the interplay among them. As the applications covered mostly concern digital processing of three-dimensional data, many theoretical results are given in three-dimensional settings. Whenever appropriate, references to further reading are provided.

The second part of the book is dedicated to computational conformal algorithms. The main idea is based on the Riemann uniformization theorem, namely its special case that states that every simply connected Riemann surface is conformally equivalent either to the Riemann sphere, the complex plane or the open unit disk. This allows most digital geometric processing tasks in R3 to be converted into two-dimensional problems on the plane.

At the end of the book the authors present discrete Gaussian curvature and discrete surface Ricci flow. However, they do not mention the so-called total absolute discrete curvature, which has also been used for triangular mesh processing and optimisation

6.1. From the Preface by Yau.

Mathematics is often called the language of science, or at least the language of the physical sciences, and that is certainly true: Our physical laws can only be stated precisely in terms of mathematical equations rather than through the written or spoken word. Yet regarding mathematics as merely a language doesn't do justice to the subject at all, as the word leaves the erroneous impression that, save for some minor tweaks here and there, the whole business has been pretty well sorted out.

In fact, nothing could be further from the truth. Although scholars have built a strong foundation over the course of hundreds - and indeed thousands - of years, mathematics is still very much a thriving and dynamic enterprise. Rather than being a static body of knowledge (not to suggest that languages themselves are set in stone), mathematics is actually a dynamic, evolving science, with new insights and discoveries made every day rivalling those made in other branches of science, though mathematical discoveries don't capture the headlines in the same way that the discovery of a new elementary particle, a new planet, or a new cure for cancer does. In fact, save for the proof of a centuries-old problem from time to time, they rarely capture headlines at all.

Yet for those who appreciate the sheer force of mathematics, it can be viewed as not just a language but as the surest path to the truth - the bedrock upon which the whole edifice of physical science rests. The strength of this discipline, again, lies not simply in its ability to explain physical reality or to reveal it, because to a mathematician, mathematics is reality. The geometric figures and spaces, whose existence we prove, are just as real to us as are the elementary particles of physics that make up all matter. But we consider mathematical structures even more fundamental than the particles of nature because mathematical structures can be used not only to understand such particles but also to understand the phenomena of everyday life, such as the contours of a human face or the symmetry of flowers. What excites geometers perhaps most of all is the power and beauty of the abstract principles that underlie the familiar forms and shapes of our contemporary world.

For me, the study of mathematics and my specialty, geometry, has truly been an adventure. I still recall the thrill I felt during my first year of graduate school, when - as a twenty-year-old fresh off the boat, so to speak - I first learned about Einstein's theory of gravity. I was struck by the notion that gravity and curvature could be regarded as one and the same, as I'd already become fascinated with curved surfaces during my undergraduate years in Hong Kong. Something about these shapes appealed to me on a visceral level. I don't know why, but I couldn't stop thinking about them. Hearing that curvature lay at the heart of Einstein's theory of general relativity gave me hope that someday, and in some way, I might be able to contribute to our understanding of the universe.

The Shape of Inner Space describes my explorations in the field of mathematics, focusing on one discovery in particular that has helped some scientists build models of the universe. No one can say for sure whether these models will ultimately prove correct. But the theory underlying these models, nevertheless, possesses a beauty that I find undeniable.

Taking on a book of this nature has been challenging, to say the least, for someone like me who's more comfortable with geometry and nonlinear differential equations than writing in the English language, which is not my native tongue. I find it frustrating because there's a great clarity, as well as a kind of elegance, in mathematical equations that is difficult, if not impossible, to express in words. It's a bit like trying to convey the majesty of Mount Everest or Niagara Falls without any pictures.

6.2. Review by: Nigel Hitchin.
Notices of the American Mathematical Society 58 (2) (2011), 311-312.

"The End of Geometry": a curious title for the final chapter of a book whose first author is perhaps the best-known differential geometer in the world. And this from someone who confesses that geometry is "the field closest to nature and therefore closest to answering the kinds of questions I care most about." Yau's concluding theme concerns the challenge to find the mathematical language to describe both general relativity (which differential geometry has done very successfully) and quantum physics (whose language in recent years has penetrated algebraic geometry). It is string theory and its practitioners that have brought these new ideas into geometry, it is they who use and christened Calabi-Yau manifolds, and it is they who are demanding from us a new geometry to suit their needs. The book aims to tell us how this came about via two themes: one is a personal story about Yau himself and his Fields Medal-winning proof of the Calabi conjecture, the other a description for the lay reader of string theory and the role of Calabi-Yau manifolds in its attempt to describe the universe.
...
In his earlier years Einstein commented that "since the mathematicians have invaded the theory of relativity, I do not understand it myself." It may be that some of us geometers have the same feeling about the incursion of ideas from physics today, but it is a fact that is not going to change. So how do we now take that final chapter - is the death of geometry real or simply greatly exaggerated? "Geometry as we know it will undoubtedly come to an end," say the authors, but it seems more likely a statement about string theory. For, despite Yau's energetic pursuit of string-related mathematical problems, despite his important facilitation of interactions between physicists and mathematicians, and despite his encouragement of students and postdocs on these problems, his oeuvre contains highly influential results in many other branches of geometry. In the book he steps aside ("one of the luxuries of being a mathematician") from the controversial discussions of the landscape or of multi-universes. Another such luxury is to accept the mathematical challenges of problems from whatever source, so long as they intrigue us, and I suspect the first author will continue to do this.

6.3. Review by: Michele Rossi.
Mathematical Reviews MR2722198 (2011m:00001).

The shape of inner space is a book meant to give a popularisation of the deep ties relating theoretical physics (and in particular superstring theory) and mathematics (and in particular complex geometry). From Nadis's preface: "Broadly speaking, this book is about understanding the universe through geometry. … The book tries to present some of the ideas from geometry and physics needed to understand where Calabi-Yau manifolds came from and why some physicists and mathematicians consider them important." From Yau's preface: "The shape of inner space describes my explorations in the field of mathematics, focusing on one discovery in particular that has helped some scientists build models of the universe." Such a particular discovery is the proof, given by Yau in 1978, of the Calabi conjecture stating that a compact Kähler manifold has a unique Kähler metric in the same class whose Ricci form is any given 2-form representing the first Chern class. In particular if the first Chern class vanishes there is a unique Kähler metric in the same class with vanishing Ricci curvature. These are called Calabi-Yau manifolds. Their importance in superstring theory was discovered by P. Candelas et al. (1985), creating an entirely new research field. In fact they showed how Calabi-Yau threefolds naturally arise as internal spaces of superstring vacuum configurations. Hence determining the shape of inner space is precisely studying the geometry of Calabi-Yau manifolds and understanding which Calabi-Yau threefold models the geometry of the universe's hidden dimensions.

As far as I know it seems that among the popular books treating post-Einstein universe space-time and string theory the present book is probably the most mathematically oriented. It aims to describe the nature and evolution of the ideas and the research in geometry which drove Yau to prove the Calabi conjecture and to produce many other ideas which considerably contributed to developing knowledge of the universe. And probably Yau and Nadis are the most appropriate authors for this task. In fact on the one hand Yau is a mathematician who was awarded the Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampère equations. On the other hand, Nadis is a contributing editor to Astronomy magazine and an experienced science writer who has contributed to several popular scientific books.

The first part of the book (more or less Chapters 1 to 3) is in some sense historical, giving an introduction to the main geometric ideas underlying the current geometric model of a 10- or 11-dimensional space-time. A second part (Chapters 4 and 5) is biographical, giving an interesting account of the development of Yau's proof of the Calabi conjecture. The remaining chapters (6 to 14) are devoted to understanding the central role played by Calabi-Yau spaces in superstring theory. The book is strewn with valuable interviews with many scientists, both mathematicians and physicists, who have contributed to the development of string theory, Calabi-Yau's geometry and mirror symmetry. In addition, it is also endowed with a useful Glossary in which many of the unavoidable technical terms littering the text are explained.

6.4. Review by: Peter MacGregor
The Mathematical Gazette 98 (542) (2014), 370.

Shing-Tung Yau was awarded the Fields medal in 1982 for his work on partial differential equations and differential geometry. In this book he teams up with a science writer, Steve Nadis to tell the story of 'string theory and the geometry of the universe's hidden dimensions'. Passages from Yau's perspective are mixed with passages from Nadis based on his interviews with other researchers. This joint authorship works well, particularly in allowing Yau to explain about the Calabi- Yau manifolds with neither false modesty nor self-aggrandisement.

The book is intended for non-specialist readers interested in the borderland between mathematics and modern speculative physics. The organisation is along historical rather than logical lines, bringing together human interest stories, quotes and photographs of the main players and informal explanations of the mathematics. Yau's early years makes remarkable reading - from running with a ragamuffin street gang in Hong Kong to winning a place at Berkley and devouring the maths library shelf by shelf. Describing this stage of his life he quotes Confucius with approval -

I have spent a whole day without eating
and a whole night without sleeping
in order to think,
but I got nothing out of it.
Thinking cannot compare with studying.

In 1973 Yau's imagination was captured by an unproved conjecture - 'the total mass of the universe must be positive' - and the book shows how his exploration of this problem, via complex manifolds, Ricci curvature, Chern classes and so on led inexorably to the birth of string theory. In many passages Yau's single-minded enthusiasm shines through allowing us to imagine what it is like to work at the leading edge of mathematical research.

You should not expect to learn mathematics from this book, (I don't think it contains a single differential equation), but as geeky holiday book or bedside reading it will certainly win your admiration for Shing-Tung Yau and might well inspire you to greater efforts in your own field.

7.1. From the Publisher.

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard's mathematics department was at the centre of these developments. A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics - in algebraic geometry and topology, complex analysis, number theory, and a host of esoteric subdisciplines that have rarely been written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose.

The story begins in 1825, when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics - an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce's successors--William Fogg Osgood and Maxime Bôcher - undertook the task of transforming the math department into a world-class research centre, attracting to the faculty such luminaries as George David Birkhoff. Birkhoff produced a dazzling body of work, while training a generation of innovators - students like Marston Morse and Hassler Whitney, who forged novel pathways in topology and other areas. Influential figures from around the world soon flocked to Harvard, some overcoming great challenges to pursue their elected calling.

A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond.

7.2. Review by: Gerald B Folland.
The American Mathematical Monthly 122 (5) (2015), 508-510.

The story of the Harvard mathematics department's growth into one of the world's premier centres of mathematical research is of interest not only in its own right but as a microcosm of the development of the larger American mathematical community. A collaboration between a scientific journalist (Nadis) and an eminent Harvard mathematician (Yau) has now brought us a very readable account of this history.

America had no real scientific community until the late 1800s. Before that time, its few contributions to science were made by a handful of talented self-taught amateurs who pursued scientific inquiry as an avocation while making their way in the world by other means. Benjamin Franklin is the best-known example. Another one is Nathaniel Bowditch (1773-1838), who went to sea in his twenties, wrote a definitive text on techniques of navigation, then produced some significant contributions to astronomy and mathematics - including a copiously annotated translation of Laplace's Mécanique Céleste - while building a successful career in the insurance business.

Benjamin Peirce, Harvard's first professor of mathematics to achieve an international reputation, falls into this category in spite of his academic position because he had no formal training beyond the bachelor's degree (he also benefitted from some mentoring by Bowditch) and because the university did not consider his research activities to be part of his job. After graduating from Harvard in 1829, he taught at a prep school for two years then returned to Harvard as a mathematics tutor. He was immediately made the acting head of the department since the senior professor was on leave, a job that was made permanent when he was given a professorship two years later. The university administration evidently recognised that they had a talented man here, but they paid him to be a teacher and provided little support or encouragement for his original scientific work. The latter included contributions to number theory, astronomy, statistics, and geodesy, a text on analytical mechanics that won favourable notices in Europe, and - at the end of his life - a major paper on linear associative algebra. (His other notable gift to the world was his son, the philosopher and logician Charles Sanders Peirce.)

With Peirce's death in 1880, the Harvard mathematics department sank back into intellectual dormancy. Meanwhile, however, the academic environment was beginning to change. Under the presidency of Charles W Eliot, Harvard had begun to expand its educational mission beyond the baccalaureate level - its first Ph.D. was granted in 1873 to William E Byerly, a student of Benjamin Peirce - and the nation got its first research-oriented university with the founding of Johns Hopkins in 1876. The Harvard math department got back into the game with the appointment of William Fogg Osgood in 1890 and Maxime Bôcher a year later; both were Harvard graduates who had gone to Germany to get their doctorates, and both were respected researchers in mathematical analysis.

The ball really got rolling, though, with the appointment of George David Birkhoff in 1912. Celebrated for his work in dynamical systems and differential equations, Birkhoff was one of the first illustrious American mathematicians who was "home- grown," that is, who was educated entirely in the United States. (He got his Ph.D. at Chicago under the direction of E H Moore, having been influenced by Bôcher as an undergraduate at Harvard.) Birkhoff's influence on mathematics at Harvard came not only through his own research but also through his Ph.D. students, no fewer than five of whom became distinguished members of the next generation of Harvard faculty. In chronological order, they are Marston Morse, Joseph Walsh, David Widder, Marshall Stone, and Hassler Whitney, a group that represents a breathtaking diversity of research accomplishments in classical and abstract analysis, geometry, and topology. The department was further strengthened by the appointments of Birkhoff's son Garrett and Stone's student George Mackey.

At the end of World War II, the Harvard math faculty was almost entirely American born and American trained. Over the next 15 years, it became truly cosmopolitan with the addition of four Europeans: Lars Ahlfors, Oscar Zariski, Richard Brauer, and Raoul Bott. The journeys that brought these men to Harvard were all quite different, but they all had to do in one way or another with the cataclysm in Europe. It was not only at Harvard or in mathematics, of course, that Europe's loss was America's gain; here, too, the story reflects the larger history of science in America.

Nadis and Yau's book fleshes out the history outlined in the preceding paragraphs, largely by means of accounts of the lives and work of the principal personalities involved. Most of the people mentioned above get several pages of material, as do Saunders Mac Lane, Andrew Gleason, and John Tate; the rest are accorded at least a paragraph or two. The selection of featured persons and mathematical developments is not comprehensive, and one might argue with the authors' choices at some points. I, for one, think that Marshall Stone got short-changed; in spite of his several claims to mathematical immortality, he is given only one paragraph plus a couple of other brief mentions.

There is also a nice collection of photographic portraits. But I don't know why photos of two of Zariski's students who were never Harvard professors were included when several other similarly worthy people were not, and I wish that the authors had selected one of the many photos of Bott that display his characteristic warmth and vivacity instead of one that shows him as a tired old man.

The biographical sketches, including interesting personal details, are well done. The descriptions of the subjects' mathematical accomplishments, however, present a problem. The authors (or perhaps an editor?) made the questionable decision to try to write these descriptions not for a mathematically literate audience but for a general one. This made their job much harder than it would have been otherwise: How do you explain Whitney's characteristic classes, the Ahlfors finiteness theorem, or Zariski's "main theorem" to the man or woman in the street? I am not persuaded that their effort is successful, and certainly it makes the book less valuable for mathematicians, presumably its primary audience. Those who are familiar with the relevant concepts and results will not get any enjoyment from the watered-down accounts here, and those who are not will just end up hungry for an exposition that really tells them something.

7.3. Review by: Albert C Lewis.
Mathematical Reviews MR3100544.

As the title implies, mathematics is central to this history of the Harvard mathematics department. While most of the few departmental histories that exist for the United States tend to concentrate on the mathematicians, here at least as much attention is given to explaining the significant mathematical achievements of the department's renowned members. Though the achievements are viewed through very brief vignettes, the remarkable result is an account, at a consistently clear, non-specialist level, of a wide swath of modern mathematics. A sample of the many topics: Benjamin Peirce and his linear associative algebra; William Fogg Osgood and the Riemann mapping theorem; G. D. Birkhoff and dynamical systems; Marston Morse and Morse theory; Hassler Whitney and fiber bundles; Saunders Mac Lane and category theory; Lars Ahlfors and Nevanlinna theory; Andrew Gleason and Hilbert's fifth problem; Oscar Zariski and algebraic geometry; Richard Brauer and finite group theory; and Raoul Bott and his periodicity theorem. It is only in the Epilogue that the reader learns that there was no mathematics building through much of the department's history. John Tate is credited with helping to guide the design of the Science Center, completed in 1973, which now houses the department. Before then, he maintained, "professors had offices all over the place" (p. 208). We are given little insight into in what sense these individuals formed a "department" within the University, let alone into the establishment and filling of chairs. Still, it was not the authors' intention to write a complete institutional history.

Much new biographical information has been gleaned from archives and interviews, supplementing sources from previously published accounts.

7.4. Review by: Joseph W Dauben.
Isis 106 (2) (2015), 466-467.

This book, despite its title, is not, in sum, a history of the Harvard Department of Mathematics but focuses on thirteen of its brightest stars of the past two hundred years up until about 1975; here the authors bring their story to a conclusion, preferring not to write about those still alive and about history too close to the present to evaluate objectively. It is a warts-and-all story, beginning actually before 1825with the first mathematician on the Harvard faculty, the hapless Isaac Greenwood (1702-1745), the first Hollis Professor of "Mathematics and Natural Philosophy" whose love for the bottle eventually got the better of him.

Of the thirteen mathematicians featured in A History in Sum, three are given chapters to themselves - Benjamin Peirce (1809-1880), G D Birkhoff (1884-1944), and Lars Ahlfors (1907-1996). The others are grouped according to various shared features. W F Osgood (1864-1943) and Maxime Bôcher (1867-1918) are associated with the "Great Awakening in American Mathematics"; Marston Morse (1892-1977), Hassler Whitney (1907-1989), and Saunders Mac Lane (1909-2005) are credited for their work by which "Analysis and Algebra Meet Topology"; Andrew Gleason (1921-2008) and George Mackey (1916-2006) are linked by "The War and Its Aftermath," as well as by their shared contributions to the subject of Hilbert spaces; Oscar Zariski (1899-1986), Richard Brauer (1901-1977), and Raoul Bott (1923-2005) are presented simply as "The Europeans." Other well-known mathematicians, like Fields medalists David Mumford (1937-) and Heisuke Hironaka (1937-), also feature in the book, even if they are not mentioned in chapter titles.

The influence of each of these scholars is still quite palpable, and their legacies are inspiring. "In a half dozen or so separate fields - such as analysis, differential geometry and topology, algebraic geometry and algebraic topology, representation theory, group theory, and number theory - Harvard has led the way". And in bringing each of their chosen figures to life, and explaining the complexities of the mathematics they produced, the authors succeed admirably. However, this book is not written for a general audience; despite the authors' best efforts, appreciating this book fully requires at least a college-level understanding of the basics of abstract algebra as well as real and complex analysis.

As the authors explain, "Although lay readers will not be able to master these advanced subjects from our comparatively brief accounts, they can at least get a flavour of the work and perhaps get the gist of what it's about". Indeed, the book manages to convey the essence of many complex subjects in a few brief pages.

In crafting their narrative, the authors draw on published papers and biographies, interviews, and, to some extent, archival material available mostly at Harvard. ...

But there is much that a true history of the department would have included. The story is told here almost as if the department existed in a vacuum of its own. There is no discussion of the importance of the house system at Harvard and the role of tutors in undergraduate education. Given the number of its own undergraduates who found their way into the doctoral program and the faculty, this element of the story would have been worth telling. There is virtually no discussion of relations with other departments, let alone connections with mathematicians at MIT, for example. There is no mention of department meetings or hiring policies, and, above all, virtually no women appear apart from a few brief mentions, all after the period when the book nominally comes to a close - 1975. The senior author, Shing-Tung Yau, will need no introduction to mathematicians - he is a Fields medalist, a National Medal of Science awardee (1997), and, most recently, winner of the Wolf Prize (2010). He has done important work in differential geometry and string theory, the subject of The Shape of Inner Space (2010), another collaborative effort with Steve Nadis, a science writer and former MIT Knight Science Journalism Fellow.

In the final analysis, Yau and Nadis succeed admirably in making clear how Harvard has long served as a magnet, attracting some of the world's best mathematicians and thus making it an exciting place to be, a melting pot of mathematical activity.

8.1. From the Preface.

Having no prior experience in committing "the story of my life" to the printed page, I'll try to keep things simple - for my sake, if not for yours - and start at the beginning. I was born in China in the spring of 1949 in the midst of the Communist revolution. A few months later, my family moved to Hong Kong, where I lived until going to the United States for graduate school in 1969. In the nearly five decades that have elapsed since my first transpacific crossing, I have gone back and forth between America and Asia on countless occasions. At times, it is hard for me say which is my true home or whether it would be more accurate to say that I have two homes, neither of which I'm fully at home in.

To be sure, I have carved out a comfortable existence in America without ever feeling truly at one with the society around me. I also have strong emotional and familial ties with China that are deeply engrained and seemingly hardwired into my being. Nevertheless, after many decades away, my perspective on my native land has shifted as if I were always observing things from at least one or two steps removed. Whether I'm in America or in China, I feel as if I have both an insider's view and an outsider's view at the same time.

This sense has left me occupying a rather peculiar place that cannot be located on a conventional map - a place lying somewhere between two cultures and two countries that are separated from each other historically, geographically, and philosophically - and through rather profound differences in cuisine. I have a home in Cambridge, Massachusetts, not far from Harvard University, which I'm happy to say has been my employer since 1987. I also have an apartment in Beijing, which I'm delighted to make use of when I'm in town. But there is a third home I've had much longer, and that is mathematics - a field I have been fully ensconced in for almost a half century.

For me, mathematics has offered a kind of universal passport that has allowed me to move freely throughout the world at the same time I ply its formidable tools toward the task of making sense of that world. I've always found mathematics to be a fascinating subject with seemingly magical properties: It can bridge gaps of distance, language, and culture, almost instantly bringing onto the same page - and onto the same plane of understanding - people who know how to harness its power. Another thing that's magical about mathematics is that it doesn't take much, if any, money to do something significant in the field. For many problems, all you need is a piece of paper and a pencil, along with the ability to focus the mind. And sometimes you don't even need paper and pencil - you can do the most important work in your head.

I feel lucky that ever since finishing graduate school, and even before obtaining my PhD, I have never stopped pursuing research in my chosen field. Along the way, I've made some contributions to this discipline that I'm proud of. But a career in mathematics was by no means assured for me, despite a fascination with the subject that took hold of me during childhood. In fact, early in my life, the path I currently find myself on appeared to be well beyond reach.

I grew up poor in terms of the standard financial metrics but rich in the love my mother and father bestowed upon my siblings and me, and in the intellectual nourishment we received. Sadly, my father, Chen Ying Chiu, died when I was just fourteen years old, throwing our family into dire economic straits - with no "nest egg" to fall back on and mounting debts from all sides. My mother, Yeuk Lam Leung, was nonetheless determined for us to continue our education - a wish that was consonant with that of my father, who had always encouraged us toward scholarly pursuits. I became serious about my studies and found my calling in mathematics - a subject I was drawn to in middle school and high school in Hong Kong.

A big break came during my college years in Hong Kong upon meeting Stephen Salaff, a young mathematician from the University of California, Berkeley. Salaff arranged for me to pursue graduate studies at Berkeley, enlisting the services of a powerful member of the school's math department, Shiing-Shen Chern, who was then the world's foremost mathematician of Chinese descent.

I don't know whether I would have gotten far in my field had it not been for the fortuitous chain of events that brought me to California. But I am certain of one thing: I never would have been able to secure such a career had it not been for the sacrifices that my mother made for all of her children and for the love of learning that my father instilled in all of his progeny. I dedicate this book to my parents, who made it possible for me to live out the story told here. I also thank my wife Yu-Yun and my sons, Isaac and Michael, who have put up with me over the past several decades, and to all of my brothers and sisters.

I have spent innumerable hours indulging my obsession for shapes and numbers, as well as for curves, surfaces, and spaces of any dimension. But my work, as well as my life, has also been enriched, immeasurably so, by my relationships with people - family, friends, colleagues, professors, and students.

This is the story of my odyssey - between China, Hong Kong, and the United States. I have travelled the world in my pursuit of geometry - a field that is crucial to our attempts to map out the universe on both the largest and smallest scales. Conjectures have been made during these excursions, "open problems" raised, and various theorems proved. But work in mathematics is almost never done in isolation. We build upon history and are shaped by myriad interactions. These interactions can, on occasion, lead to misunderstandings and even fights, which I have, unfortunately, been caught up in from time to time. One of the things I've learned through these incidents is that the notion of "pure mathematics" can be hard to realise in practice. Personalities and politics can intrude in unexpected ways, sometimes obscuring the intrinsic beauty of this discipline.

Nonetheless, chance encounters with peers can also send us in unexpectedly fruitful trajectories that may last years or decades. In the final analysis, we are the products of our times and of our milieus, of whom we come from and where we come from. It now seems as if I come from many places - a fact that has made my life both richer and more complicated. In the account that follows, I hope to convey a sense of my upbringing, growth, and personal journey to any readers who might take an interest.

I take this opportunity to thank some of the many people who - if not contributing to this book directly - helped make the narrative arguably worth telling. For starters, I owe an incalculable debt to my parents, who supported my siblings and me as best they could, through hard times, while always trying to teach us good values. The main purpose of life, I learned, is not about making money - a lesson that enabled me to pursue a career in mathematics rather than in, say, business or banking. I was close to all of my siblings but am especially grateful to my older sister, Shing-Yue, who, up to the moment of her death, sacrificed so much - foregoing a professional career of her own - in order to help me and her other brothers and sisters.

I was also lucky to have fallen in love with, and eventually married, a woman who shared my view that there is more to life than seeking personal wealth, material possessions, and luxuries - that greater rewards can come from scholarly endeavours. I'm proud to see that our sons have also ventured far along academic paths.

I'm lucky to have lifelong friends, like Shiu-Yuen Cheng, Siu-Tat Chiu, and Bun Wong, whom I've known since my school days in Hong Kong. One grade school teacher, Miss Poon, stands out for the kindness she bestowed upon me when I was young and vulnerable. I got an early taste of mathematics from the lecturer H L Chow during my freshman year at Chung Chi College. And I was extraordinarily fortunate to have crossed paths during college with Stephen Salaff, who guided me to Berkeley with the help of Chern, Shoshichi Kobayashi, and Donald Sarason.

I'm grateful to the American educational system for providing, since the moment of my arrival, a wonderful environment for pursuing mathematical research. A great feature of this system is that it recognises and fosters a person's talent, regardless of his or her race, background, or accent. I should single out Harvard in this regard, which has served as a convivial home for me over the past thirty-plus years. I've had many terrific colleagues in the Harvard Mathematics Department - too many in that time, unfortunately, to list here.

My career has been aided immeasurably by somewhat older and more established mathematicians who've gone out of their ways to help me. First and foremost is my former advisor and mentor S S Chern. But many others have been of tremendous help, including Armand Borel, Raoul Bott, Eugenio Calabi, Heisuke Hironaka, Friedrich Hirzebruch, Barry Mazur, John Milnor, Charles Morrey, Jürgen Moser, David Mumford, Louis Nirenberg, Robert Osserman, Jim Simons, Isadore Singer, and Shlomo Sternberg.

Some mathematicians prefer to work alone, but I do best in the company of friends and colleagues. I am pleased to have had some great ones over the years, among them S Y Cheng, John Coates, Robert Greene, Dick Gross, Richard Hamilton, Bill Helton, Blaine Lawson, Peter Li, Bill Meeks, Duong Phong, Wilfried Schmid, Rick Schoen, Leon Simon, Cliff Taubes, Karen Uhlenbeck, Hung-Hsi Wu, Horng-Tzer Yau, and my brother Stephen Yau. I've collaborated closely, in particular, with Rick Schoen for about forty-five years and have done some of my best work with him. Although he started out as my student, I'm sure I've learned as much from him as he has from me. I truly value his friendship.

I continue to collaborate with other former students and postdocs - such as Huai-Dong Cao, Conan Leung, Jun Li, Bong Lian, Kefeng Liu, Melissa Liu, and Mu-Tao Wang. I've got some outstanding maths colleagues in China and Hong Kong: Yang Lo, Zhouping Xin, and many others. I've also had close ties with physicists for most of my career, enjoying my interactions with people like Philip Candelas, Brian Greene, David Gross, Stephen Hawking, Gary Horowitz, Andy Strominger, Henry Tye, Cumrun Vafa, and Edward Witten. My work in mathematics has definitely profited from these associations, and I'd like to think that some benefits have trickled down to physics as well.

All told, it's been an exciting journey so far, and I hope (and expect) there will be a few pleasant surprises on the road ahead.

Shing-Tung Yau, Cambridge, 2018.

8.2. Review by: Peter Giblin.
The Mathematical Gazette 105 (564) (2021), 564-566.

Shing-Tung Yau (whose family name, he says, should have been transliterated as Chiu) is one of the great geometers of our time, a mathematician of exceptional power and vision whose ideas, particularly the systematic application of 'geometric analysis' including partial differential equations such as 'Monge-Ampère equations', to geometry and topology, have led to solutions of a succession of very hard problems. This includes his early success on the 'Calabi conjecture', first mistakenly disproved in 1973 and then successfully proved in 1976 at the age of 27, shortly after his wedding to the physicist Yu-yun Kuo. This conjecture concerns the differential geometry of certain complex manifolds which have a special metric called a 'Kähler metric'. It is connected to many other areas of mathematics and physics, including string theory and (via the 'Ricci curvature') the Poincaré conjecture.

Readers of the book under review will naturally not expect a technical discussion of the conjecture or its consequences. Though the book does contain some figures (Newton-Raphson approximation, Euler characteristic, loops on 2-dimensional surfaces for example) and some general descriptions of the kind of mathematics Yau was working on, the main interest must be in what Yau has to say about his life and about the international community of scientists of which he is a member. A significant part of this book is in fact on the theme of 'setting the record straight' where he feels that a commonly held version of events is distorted. Yau is no stranger to controversy: for example there is a New Yorker article 'Manifold Destiny' (28 August 2006) which is critical of Yau's record during the checking of the Hamilton/Perelman proof of the Poincaré conjecture. Many points in this article are rebuffed in a subsequent New York Times article 'The Emperor of Math' (17 October 2006). (At the time of writing, both these articles are available on the internet, and they give additional background to their stories.) In fact Yau faced many obstructions and troubles in his professional and administrative career, often, he says, caused by his outspokenness, strict honesty, impatience with those in power and adherence to doing what was right, as well as 'getting tangled up in situations which don't concern me'.

Yau's childhood (with the nickname 'little mushroom') and rise to prominence are astonishing. One of eight children, his 'wonderful, amazing father - a noble scholar who placed learning and honour above everything else' died when Yau was 14, plunging the family into a crisis which only his indomitable mother, helped by a certain amount of charity and Yau's initiative in tutoring other young people, brought them through. The family is remarkable by any standards - for example Yau's younger brother Shing-Toung (Stephen) is also a well-known mathematician. Just how sheer hard work, a fair dose of luck and the self-sacrifice of Yau's mother took the teenager from poverty in Hong Kong to graduate studies at the University of California, Berkeley, makes pretty compelling reading. On arrival in California he started to work with the great mathematician Shiing-Shen Chern who suggested the Riemann Hypothesis as a thesis topic; however Yau seems to have had a strong instinct for a very hard but potentially tractable problem, and he focussed his interests elsewhere. Yau's relationship with Chern remained difficult until Chern's death in 2004.

Yau has worked collaboratively for most of his professional life, and he has used his phenomenal energy to supervise many graduate students (he says about a dozen followed him from San Diego to Harvard in 1987), organise conferences and workshops, edit major mathematical journals and found mathematical research centres in the United States and especially in his native China (he was born in mainland China but his family fled to Hong Kong when he was a baby). All this is related in detail in the book, including the disagreements and rows which his actions provoked. The picture he presents of academia and education in China is rather negative: oldest people hold the most power; academicians can exert a corrupt influence; '[Academicians] are treated like royalty, without necessarily having done much to earn their lofty titles (and in that respect they might be like royalty too)'; 'It is almost impossible to contest a statement made by a Chinese leader'; 'Chinese education can suck the life out of a subject'. But more optimistically: '[The Chinese High School Mathematics Award] is part of a broader attempt on my part to counteract years of education in a rigid system in which Chinese students are trained to memorize things - to be passive receptacles that do whatever their teacher says.'

Many famous mathematicians make an appearance in the book, of course, and Yau is forthright in his commentary on them and on the tensions and feuds which he has seen, and in some cases been part of, during his career. This, and the personal story of his family and his determination to keep his children alive to their Chinese heritage, turn what could have been, after the spellbinding start of the book, a slightly dull catalogue of amazing achievements into a real story of a remarkable mathematician and of contemporary mathematics, written with passion by one of the key players.

8.3. Review by: Casey Sherman.
Math Horizons 27 (2) (2019), 29.

To paraphrase a quote attributed to Confucius, "By three methods may we learn wisdom: first, by reflection, which is the noblest; second, by imitation, which is the easiest; and third by experience, which is the bitterest."

In The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry (Yale University Press, 2019), authors Shing-Tung Yau and Steve Nadis reflect deeply on the life of Shing-Tung Yau. The book is a striking presentation of contrasts: East and West, harmonious research with peers and messy political fallout, the purity of mathematics, and the practicality of physics. Perhaps what it does best, however, is blend these dualities beautifully in a single narrative throughout his journey from growing up in abject poverty to receiving the Fields Medal. In my view, that blending is the whole point of the book.

The book follows Yau's life from his childhood to the present day. It begins with his early difficulties due to poverty; his love for a father who valued study, education, and poetry; and the budding academic wonder of discovering beauty in mathematics. Next, it takes us to California for Yau's graduate studies, where we witness his early relationships with other mathematicians, see his desire to bridge mathematics and physics, and are introduced to the Calabi conjecture, a focal point for his career. Later, we follow Yau through positions at the Institute for Advanced Study, Stony Brook University, Stanford University, and eventually Harvard University, doing work with a host of famous mathematicians.

Throughout, we have the treat of sharing in Yau's insights and motivations. His belief in the beauty of mathematics is contagious and he demonstrates it for us often. We watch as he delves into curvature, minimal surfaces, and nonlinear differential equations. To be fair, we see them from afar, but the authors provide simple analogues for each mathematical concept they introduce so that we can glimpse the importance of the ideas.

We witness the emergence of geometric analysis as a popular field. Yau brings us along as he experiences the elation of proving the Calabi conjecture to be false, the anguish of finding an error, and the long systematic journey he took back up Mount Calabi to proving it true. For me, the most rewarding portion of the book was seeing his desire to link mathematics and physics bear fruit when his work leads him deep into the heart of string theory, where Calabi-Yau manifolds are hidden.

Yau prefers work with other researchers to the realm of solitude, and there is no shortage of names that readers will (or soon will) recognise. Some of these mathematicians helped build a foundation for geometric analysis, some we simply hear about (such as those working on the Poincaré conjecture), and with some Yau includes to show the trials that can happen when research becomes political. This last group is inevitable, as the subject of the differences between East and West are another focal point of the book. Yau provides observations on the differences between academia in the East (specifically China), where politics (sometimes departmental and often governmental) can trump skill, and the West, where he contends that academic prowess holds more sway.

While this book is good for anyone, it is particularly relevant for younger mathematicians. It provides a unique perspective of the rise of a field of mathematics from the point of view of someone who follows the example of the scientists of ancient Greece and many great mathematicians - that mathematics is beautiful and profoundly connects to the sciences - and who further demonstrates how it is possible to reach across fields through healthy collaboration.

In Yau's noble reflection, perhaps we can gain wisdom from some of his experiences without the bitterness that Confucius warns us of.

8.4. Review by: Thomas William Murphy.
Mathematical Reviews MR3930611.

This is a remarkable text offering insight into the achievements and life of Fields medalist Shing-Tung Yau, undoubtedly a leading mathematician of our time. In direct and uncluttered prose, the reader is treated to a frank account of Professor Yau's life. It will reach a large audience and is a valuable read for those interested in the development of geometric analysis and string theory, topics which are bound up in his remarkable career.

The book is written in an idiosyncratic style, beginning as an autobiography before gradually blending the tale of Yau's life with explanation of his main contributions to mathematics and discussions concerning the politics of modern research mathematics. This works well, as the reader comes to learn just how central Yau has been to the development of various areas of mathematics and his role as an educator and administrator.

The first few chapters concern Professor Yau's early life until he arrives at U.C. Berkeley to undertake his Ph.D. in mathematics under the supervision of S. S. Chern in 1969. Yau's passion and dedication to mathematics is evident from a young age. His rise to the top of mathematics from humble beginnings yields an inspiring tale, simply and effectively presented.

One of the book's strengths is the account of how geometric analysis developed from one of the main instigators of the field. The early impact of C B Morrey, B Lawson, and Chern on Yau's intellectual development and the account of his quest to prove the Calabi conjecture via estimates for a complex Monge-Ampère equation make for fascinating reading. Towards the end of the book a lucid exposition of the key ideas underlying mirror symmetry and string theory is developed in tandem with a discussion of Yau's contributions. Taken together with the remarkable tale of his early life, it is clear there is much content to inspire younger generations.

Large parts of the latter half of the text concern Yau's efforts to strengthen the mathematical community, with a particular focus on his work in China developing several research institutes. While there is little doubt of Yau's enormous positive impact on mathematics, this necessarily involves discussion of various disputes and arguments from his perspective. In places this is understandable (such as his desire to explain his position regarding the controversies over Ricci flow) but at other times it is too much and the overall narrative is somewhat marred.

The shape of a life affords the reader a readable and candid encounter with Professor Yau and his remarkable life. The reviewer has little doubt it will attract a large readership who will be rewarded with a glimpse of the measure of a great geometer.