Ricci-flat or non-Kähler? Donut or sphere? Topology or geometry? Math or physics? There's no need to make a choice. Shing-Tung Yau loves them all, for the sake of their own beauty. With The Shape of Inner Space (2010) Yau creates a remarkable amalgam of personal history, advanced math presented at high school level and educated speculation about the deepest structure of reality. The book revolves around Calabi-Yau spaces. More precisely they are called manifolds, able to describe a proposed six-dimensional part of our large sized four-dimensional reality.
Yau tells the story of how he grew up poor in China and Hong Kong. Despite all the hardship he found his way to mathematics. He was fascinated by the Calabi conjecture (about specific properties of higher dimensional spaces or manifolds). He then paints the road to success via failure. As a 29 year old failing to show the Calabi conjecture wrong, he decides to assume it to be right and trying to prove it. In which he succeeds. The resulting manifolds are from then on called Calabi-Yau manifolds. Yau studies them because they are beautiful. Sometimes they are serious competition to his wife, Yau admits.
The importance of Calabi-Yau manifolds grows when the world of physics stumbles upon a concept called string theory. This is the world in which elementary particles are tiny, one-dimensional (open ended or closed) strings. Every vibration corresponds to a certain type of particle. But the vibrations have to occur in higher dimensional space. And a little symmetry would be nice too, thank you. Then suddenly Calabi-Yau manifolds seem to hold many promises for this ultra weird Lala-land. From then on Yau's story revolves more around the intensifying collaboration between string theorists and geometers like him.
With that ground firm established, Yau devotes the last part of the book to the description of the probabilities and possibilities of string theory. While the first part requires some knowledge of high school level mathematical language the second part only requires imagination. This makes it fun to read.
Yau (in collaboration with Steve Nadis) does great in describing concepts unhealthy for your every day perception of reality. The story often becomes abstract and hard to follow, but rewarding in the end. His personal history is fun to read, but the aforementioned amalgam makes it a book that won't cater everybody. Some might say: give me personal history or science talk, but not both. Others (like me) might say that the road traveled is as important and interesting as the results achieved. The only thing that bothered me was the elaborate description of scientists' then and now positions at institutes and universities. It slows down the story and makes it sometimes harder to keep up with the flow of the story. Like: The x,y-tensor, then described by John Doe from Alberta Institute of Mathematics, now at Princeton's Instutue of Advanced Studies, together with Paul Calcular, then a post-doc of mine at Standford, now at Santa Barbara sea aquarium, and Barbara Johnson, then at Caltech, now at Harvard institute of Mathematics, in what now is the DCJ-tensor equation, named after the three discoverers Doe, Calcular and Johnson, was imperative to.....
Well, you get the idea. Give credit where credit's due, but Yau could have done away with that in more footnotes, and it would have benefited the story.
In the end Yau makes a great observation when discussing the question if string theory is a proper description of reality, together with the six dimensional Calabi-Yau manifolds catering its children. Who knows, he muses. But whatever the physical fate of Calabi-Yau, the mathematical fate has been firmly established. In math they are true and will be true forever. You can't uninvent them. And ultimately, Yau thinks, anything mathematical will eventually pop up in our physical reality. I think he's dead right.