As some background, I studied math, not that long ago, but its been around 10 yrs since I've been in a class room. That being said, when i was taking classes, I took them up to a reasonable level and done coursework at entry graduate level. This book is an overview of string theory and the evolution of some of the math behind it. In particular it is an attempt to communicate to the lay person what the author has been working on through his illustrious career in math. The author does succeed in some parts, but fails in others. The book is focused on the author's contributions rather than on trying to really explain the theory in the same vein as Brian Green's books. As such he gets into some very specific ideas that honestly, I have no idea how someone without a PhD or doing a PhD in the subjects he discusses could possibly understand.
The beginning of this book I found fantastic. It described some classical mathematical ideas in ways I hadnt thought about that really helped clear up some of the intention of some subjects I had difficulty with. The description of the ways in which mathematicians like Gauss and Riemann came to conclusions about something based on something that was seemingly distinct was often very illuminating. But soon, the cozy introduction started turning into stretched analogy and countless definition of ideas that the unfamiliar cant possibly pick up in a trivialized manner.
The book remains accessible up to around 1/3 of it and then soon becomes incomprehensible. Let me give an example - "Our four-dimensional example with K3 surfaces are topologically equivalent. The six dimensional example involving Calabi-Yau threefolds is more interesting. The components of this manifold include three-dimensional tori. Applying T duality will invert the radii of those tori. For a non singular torus, this radius change will not change the topology." The book is pretty much entirely written like this. This is not for a non-mathematician and all those who claim they understood this without a serious math background are kidding themselves.
The author is a serious mathematician and of a calibre few professional mathematicians can hope to match. Yet much of this book is to champion what his discoveries are to people who cant possibly understand them in a style that has the hint of self promotion. There are instances of bickering over credit of some theorem, the use of I proved this is very prolific (Yau was the leading mathematician who claimed that he and others proved the poincare conjecture and not Perelman, in this credit is given to Perelman, but even then it sounds a bit begrudging). The author doesnt need to prove himself so i'm not sure why its embedded in the writing.
All in all the book starts out really well, but gets the reader in over their head very quickly. In a certain sense it reminds me of the Road to Reality, by Penrose. The subjects the author discusses has taken him his career and would take most a lifetime if it is even possible. The subject matter cant be dumbed down to talking about the similiarities to y = x^2 once a page. I think the contents of this book should be read by graduate students and researchers who are curious about the history of calabi-yau spaces and the authors accomplishments and how they all fit together, but aside from the well informed few, this book largely remains out of the grasp of the most of us.