I really enjoyed this book. It was a bit of an impulse buy - I was dimly aware of the Poincare conjecture but hadn't heard of this author before, and my expectations weren't especially high. But I found the author's style to be very engaging indeed. She writes with great clarity and precision; her descriptions of Perelman's early years are very vivid and the different personalities in this story really leap off the page. Best of all, Ms Gessen doesn't condescend to the reader - it's really refreshing to read a 'popular science' book that treats its readers as intelligent adults.
My only criticism (and it's a very slight one), is that the book is relatively light on maths! I'd like to have seen a little more specifics about the conjecture and the various attempted solutions. I think it's fair to say that this book is intended to be more about the personalities involved rather than the mathematics itself - although there are plenty of good references at the end for those who wish to go a little deeper.
Anyway, in case I haven't made it clear - I really enjoyed this book, and i'll definitely be looking out for more of Ms Gessen's work from now on.