### Review

**AUTHORS' STATEMENT by SHING-TUNG YAU and STEVE NADIS**

There is a certain irony running through this book that one of the smallest things you can possibly imagine--six-dimensional geometric spaces that may be more than a trillion times smaller than an electron--could, nevertheless, be one of the defining features of our universe, exerting a profound influence that extends to every single point in the cosmos. This book is, in many ways, the story of those spaces, which physicists have dubbed "Calabi-Yau manifolds." It tells how one of us, Yau, managed to prove the existence, mathematically, of those spaces, despite the fact that he had originally set out to prove that such spaces could not possibly exist. It then goes on to explain how this mathematical proof, which had initially been ignored by physicists (partly because it was steeped in difficult, nonlinear arguments), nevertheless made its way into the center of string theory, which now stands as the leading theory of the universe and our best hope yet of unifying all the particles and forces observed--and yet to be observed--in nature.

Of course, none of this could have been foretold more than a half century ago when a man named Eugenio Calabi--the first half of the Calabi-Yau duo--proposed that there could be multidimensional spaces with properties so special that many mathematicians, including one of this book's authors, considered them "too good to be true." Calabi had not been thinking about physics at the time, in the early 1950s, when he advanced the famous conjecture named after him. Following the proof of the Calabi conjecture, we have learned many new and wonderful things in both physics and mathematics--all of which suggest that Calabi-Yau spaces are not only too good to be true, as the skeptics used to say, but that they may be even better.

**BLURBS**

**Brian Greene, Professor of Mathematics & Physics, Columbia University; author of ***The Fabric of the Cosmos* and **The Elegant Universe**

“*The Shape of Inner Space* provides a vibrant tour through the strange and wondrous possibility that the three spatial dimensions we see may not be the only ones that exist. Told by one of the masters of the subject, the book gives an in-depth account of one of the most exciting and controversial developments in modern theoretical physics.”

**Joe Polchinski, Professor of Physics, University of California - Santa Barbara; author of ***String Theory***, Vols. 1 & 2**

“Einstein’s vision of physical laws emerging from the shape of space has been expanded by the higher dimensions of string theory. This vision has transformed not only modern physics, but also modern mathematics. Shing-Tung Yau has been at the center of these developments. In this ambitious book, written

**Newsletter of the European Mathematical Society**

"An interested reader, even one with little background in mathematics, will be able to gather much new knowledge of, and appreciation for, both mathematics and physics from the elegant analogies and beautiful illustrations in this book... The book gives insight into the mind of one of the world’s greatest mathematicians and will provide intellectual stimulation to interested readers with any kind of background."

**Simon Donaldson, Royal Society Research Professor in Pure Mathematics and President of the Institute for Mathematical Science, ****Imperial****College****London**

“*The Shape of Inner Space* has a distinctive style: in part autobiography, in part an account of developments in geometric analysis and string theory over the past 40 years, and comments on future directions. It gives a unique insight into the thoughts of one of the most important and influential mathematicians of our times.”

**Edward Witten, Professor, Institute for Advanced Study**

“Shing-Tung Yau and Steve Nadis take the reader on a fascinating tour of many contemporary topics in geometry and physics. Readers will find many challenging ideas to explore in this book, and even specialists will enjoy Yau’s reminiscences about his education and work.”

**Steven Strogatz, ***New York Times* Opinionator contributor and professor of mathematics, Cornell University

“A fascinating first-hand account of how the math underlying string theory was discovered. Fields medalist Yau and ace science writer Nadis have teamed up to show the rest of us the deep geometry that just might lie at the heart of the universe. It’ll twist you into knots of pleasure!”

**John Coates, Sadleirian Professor of Pure Mathematics, University of ****Cambridge**

"This extraordinary book by Shing-Tung Yau and Steve Nadis gives the layman a remarkable glimpse into the mysterious inner world of one of the most beautiful and important parts of mathematics."

**Andrew Strominger, Professor of Physics, ****Harvard ****University**

“This book tells an inspiring story about how progress in science is made by breaking traditional boundaries in disciplines. It's really the only book of its kind—and, of course, written by someone who not only witnessed but also inspired and produced many of the major developments in this field over an exhilarating period of four decades.”

**David Gross, Frederick W. Gluck Professor of Theoretical Physics, University of California – Santa Barbara; Nobel Prize-winning physicist**

“*The Shape of Inner Space* takes one on a marvelous journey that explores many beautiful areas of modern geometry and physics, and the people behind recent discoveries. It is a journey that I highly recommend to the intellectually curious.”

**Michael D. Smith, Dean of the Faculty of Arts and Sciences, John H. Finley, Jr. Professor of Engineering and Applied Sciences, Harvard University**

“Though this wonderful new book helped me to better understand the discoveries underpinning string theory, what I enjoyed most was what it reveals about the beauty of mathematical inquiry. This book shares a very human process of thought, discussion, and wonder that is enormously appealing, in addition to being quite obviously fertile ground for discovery. Words from Yau’s poem from the front pages—‘Inexhaustible, lovely in every detail’—provide an apt description of the book itself. Well done!”

**REVIEWS**

*Publishers Weekly*

“With the help of *Astronomy* magazine contributing editor Nadis, Yau relates the saga of [his] groundbreaking work which provided the foundations of string theory. Yau confidently draws readers into a realm of abstract concepts, from multiple dimensions to the exotic spaces called ‘manifolds,’ or Calabi-Yau spaces, whose curvature gives space its shape. From here it’s a hop, skip, and a jump to the geometry of space around the Big Bang, black holes, and the end of the universe.”

**New Scientist**

“It is a testimony to [Yau’s] careful prose (and no doubt to the skills of co-author Steve Nadis) that this book so compellingly captures the essence of what pushes string theorists forward in the face of formidable obstacles. It gives us a rare glimpse into a world as alien as the moons of Jupiter, and just as fascinating…. Yau and Nadis have produced a strangely mesmerizing account of geometry’s role in the universe.”

**Nature**

“Physicists investigate one cosmos, but mathematicians can explore all possible worlds. So marvels Fields medalist Shing-Tung Yau…. Relating how he solved a major theoretical problem in string theory in the 1970s, Yau explains how the geometries of the vibrating multidimensional strings that may characterize the Universe have implications across physics.”

**Science Books & Films**

“Concepts are introduced in a clear way, preceding more detailed discussions. The subjects examined include topology, geometries, general relativity, quantum physics, the standard model of particles, and other topics relevant to the pursuit of the understanding of extra dimensions in our universe. Among the volume’s especially interesting discussions are the possible experimental tests of the theory, the potential semistability of our universe, the five candidate string theories, and black-hole entropy.”

**David Eicher, Astronomy.com**

“[A] masterwork on its subject…. The book is an entertaining read, written with the absorbing style that characterizes Nadis’ feature stories in *Astronomy*…. Those interested in cosmology and nature as a whole will be delighted with this new work!”

**New York Journal of Books**

“*The Shape of Inner Space* provides the opportunity to look over the shoulder of a giant in mathematics.”

**MAA Reviews**

“This fascinating book may well have a similar impact to Stephen Hawking’s *A Brief History of Time*…. I found this introduction to string theory totally absorbing, and well worth re-reading.”

**Nature Physics**

“An engaging exposition of elegant relations between geometry, topology, fields and strings, the book is also part memoir and part speculation about connections to physics…. Written with an easygoing sense of humour, and conscious of the distance between its subject and the daily concerns of the ‘average citizen’, the book in the end offers cautious optimism about the future of this ambitious programme of theoretical research. Altogether, Yau and Nadis’s effort covers some central developments in mathematical physics, and is well worth perusal by widely interested scientists as well as lay readers.”

**The London Mathematical Society Newsletter** “This book tells the fascinating story of strange geometric objects that have achieved some fame outside of [mathematics] called Calabi-Yau manifolds... The collaboration between a mathematician and a science writer has worked wonders in this book. It's crowded with beautiful metaphors that clarify complex ideas and provide a peek into higher-dimensional worlds... One thing that comes through on every page of this book is the beauty of the [mathematics] and its power to shed light on the secrets of our Universe. If this is the kind of thing that fascinates you, then this is a great book to while away those dark winter evenings.”

**PhysicsWorld.com**

"It is fascinating to see the story of string theory told from a mathematician's point of view rather than that of a physicist... By bravely attempting to explain areas of mathematics that no one has ever tried to relate to the public before, *The Shape of Inner Space* takes a huge step forward... It will undoubtedly influence how string theory is taught and written about in the future."

**Times Higher Education Supplement**

"A very well-written book, and one that scientifically minded laymen will find easy to follow… It is strongly recommended to those seeking a first-hand, simply explained account of one of the most fascinating evolutions in modern science, whose impact in mathematics is significant and enduring, and whose impact in physics may be forthcoming.”

*Choice*

"This book provides an excellent insight into the current ideas about string theory."

**Philip J. Davis, ***SIAM News*

* *"My experience in reading this book may be akin to that of a kibitzer in the presence of some moment of high creativity—perhaps of an onlooker in the atelier of Titian, watching how he painted the famous equestrian portrait of Charles V."

*American Journal of Physics*

“*The Shape of Inner Space *is a portrait of a beautiful branch of geometric analysis as seen through the eyes of one of its pioneers, Fields medal winner Shing-Tung Yau… After describing the sequence of events that led him to the United States and to his enamoration with geometry, Yau explains as only a master could the conjecture by Calabi and the subsequent discovery of Calabi-Yau manifolds that are the centerpiece of this book. The reader is thrown into a world of complex manifolds, geometric analysis, and differential equations, yet the book is written so that the persistent layperson could follow all of the main ideas.”

*Notes of the Canadian Mathematical Society*

“In the fascinating book, *The Shape of Inner Space*… Shing-Tung Yau, along with coauthor Steve Nadis, describes the exciting development of the theory of what are now called Calabi-Yau manifolds and their relationship to the structure of the universe.”

*Philippine Daily Inquirer*

“A journey into the mind of a brilliant mathematician, *The Shape of Inner Space* will delight readers who are not afraid to use their minds.”

*College Mathematics Journal*

“A worthy successor to *The Elegant Universe*.”

*Philippine Daily Inquirer*“A journey into the mind of a brilliant mathematician,

*The Shape of Inner Space* will delight readers who are not afraid to use their minds.”

*The Mathematical Intelligencer*

“What makes this book unique is that Yau has a deep insight not only into the mathematics but also into the physics governing our universe, and he uses this knowledge to build a bridge between both worlds.”

### About the Author

**Shing-Tung Yau** has won many awards, including the Fields Medal. He is a professor of mathematics at Harvard University and lives in Cambridge, Massachusetts.

**Steve Nadis** is a Contributing Editor to *Astronomy Magazine*. He has published articles in *Nature*, *Science*, *Scientific American*, *New Scientist*, *Sky&Telescope*, *The Atlantic Monthly*, and other journals. He has written or contributed to more than two dozen books. A former staff researcher for the Union of Concerned Scientists, Nadis has also been a research fellow at MIT and a consultant to the World Resources Institute, the Woods Hole Oceanographic Institution, and WGBH/NOVA. He lives in Cambridge, Massachusetts.