Although I describe the work as "creative", much of it sounds derivative. The idea that humans would use time-travel to save the world from a future ecological apocalypse is, by now, nearly a cliche. And, the method utilized here (sending the mind of a person from the future into the body of a person in the past) is also not particularly original (see Travelers). However, Reynolds adds more than enough new twists to these familiar ideas to compensate.

Three characters are presented as being mathematicians:

• Luba Lidova was a mathematical physicist studying a (fictional) sort of quantum entanglement which allows for manipulation of past events.
• The main protagonist, Valentina Lidova, is Luba's daughter. She is 70 years-old in the book's "present" and teaching math to children, even though she has every reason to believe that the human race will soon be extinct.
• Margaret Arbetsumian is the mathematical physicist whose job it is to practically apply Lidova's abstract theory for the time-travel project.

The most intriguing aspect of Lidova's research is how the paradoxes are handled. According to Lidova's theory, a paradox would ripple through spacetime just as a phonon travels through a crystal, eventually disappearing due to a sort of dispersion. In practice, this means that if you received a message sent from your future self and then decide that you will not send yourself that message, you would at first be shocked at the fact that there was an apparent contradiction -- if you didn't send the message to yourself how did you receive it? However, soon you would start doubting that you had received the message as that reality was replaced with another in which you didn't, and eventually you would not remember anything about the message being received in the first place. The paradox is resolved because it never happened; the "crystal lattice" of spacetime has returned to a stable and contradiction-free state.

(Ironically, I thought about this whenever there was something in the book that I had trouble believing. Of course, one often has to suspend disbelief to some extent when reading science fiction. Here I had a lot of trouble believing in the supposed plan to save the world. [Really, if you had the ability to manipulate the past, is this really the best way to save the world from disaster?] There's also a bit with a dog that didn't quite work for me. And every time a "Luba Pair" was mentioned I thought about how odd it was that Lidova would use her own first name that way. But, because I was enjoying the book and wanted to go on with it, I pushed these concerns to the back of my mind until they were forgotten, and I had the strange feeling that I had personally experienced this phonon-like paradox resolution ; )

Here is another passage from the book that discusses math, but the focus is more on the Lidovas and their relationship than on the mathematics:

I do find it interesting (and encouraging) that all three mathematician characters in this book just happen to be female, an apparently unimportant coincidence that doesn't merit any discussion. And, it is always nice to see math used in fiction for a good cause (which I presume we could agree that saving the world would be). However, aside from these points this book does not really use math or say much about the field of mathematics. So, the fact that it is mathematical fiction may not be much of an incentive to read it (unless mathematical fiction is your hobby, as it is mine).

The reason you should read this book, if you like mind-bending science fiction, is for the climactic scene which is original, beautiful, and worth the wait.