First, let me tell you about the math, page by page.

• On page 47 there is a discussion of factorization of large numbers and the role that quantum computers can play. (However, since Kyle believes in the many universe interpretation of quantum physics, it is described in a ridiculous way: the computation is done by infinitely many computers in infinitely many different universes.)
• On page 79 we are given the symbolic expression of the right-hand side of the Drake equation for estimating the number of radio-broadcasting civilizations in the galaxy as well as a description of the terms.
• Page 81-84, back to this crazy sounding description of factorization on quantum computer. In particular, it is argued that since the number of computations involved in factoring the huge number he asked his computer to factor is larger than the number of atoms in the universe, it is proof that the computation is done in multiple universes. (Not really, it could just argue that a single atom or particle is capable of doing lots of traditional computations by virtue of its quantum nature!) But, it doesn't matter, because it only finds two factors (though he expects more) which (according to Kyle) could have been done using only the number of atoms in this universe.
• Page 92 has a diagram of a hypercube. This is a real mathematical object which is to a cube what a cube is to a square. In particular, it is what you get by taking a row of four cubes, sticking four more on the four visible sides of the second one and then identifying all of the remaining sides by `folding it' up. (This object can only be embedded into spaces of dimension greater than four so it is not something we can really see in our daily experience.
• On page 113 there is a table of the ratio of 2832 by its integer factors. It is apparently a clue that she is supposed to recognize in order to decode an alien message that 59 is the only prime factor of 2832 greater than 3.
• With the knowledge of the factorization of 2832 determined on page 113 she is able to recognize that the message is a finite set of tiles. On page 117 she mentions this to her physicist husband who makes a remark (a non-sequitur as far as I'm concerned) about how it is funny that the aliens would send a solvable, finite tiling problem when there are problems about tiling the plane with infinitely many tiles that are not clearly solvable. (This is a brief mention of some real math, like the question of whether local rules can allow one to tile the plane with Penrose tiles, but it is strange to mention it here and pretty far fetched to say that this `echoes one of the big debates in human consciousness.'
• One of the coolest mathematical discoveries of the late 20th Century (IMHO) is the existence of `fake R^4's', or more specifically the fact that there are no `fake R^n's' for any n other than 4. Let me put it like this: you can ask whether there is a space that is topologically equivalent to n-dimensional space (R^n) but not smoothly equivalent to it. The surprising result is that there are such spaces (topologically but not smoothly equivalent) only for n=4. This makes R^4 a very special sort of space indeed. I always thought that this should be worked into a science fiction story somehow. Sawyer touches on this on page 121. He even has a two paragraph quote from Ivars Peterson's Science News article on the subject. The way he works it in to the story is that our heroine is decoding a message from aliens and she figures there must be some assumption they are making, something they will assume we know as well as they do. She decides that they must be utilizing 4-dimensional space because it is so special, and she seems to be right because it all works out. However, I find this rather implausable. It is true that 4-dimensional space is special because it is the only one with the property described above, but other dimensions are also singled out as answers to certain questions! The unit-ball (the set of all points one unit away from a fixed point) is discrete only in R^1. Certain wave propogation phenomena (Huygens' Principle) apply only in R^2. Etc, etc.
• There is a reference to Alan Turing on page 132 when it is mentioned that one character in the book killed himself by eating a poisoned apple (the same method that Turing used).
• On page 172 there is a discussion of Roger Penrose's theory of the quantum mechanical nature of consciousness. The theory itself is not presented in a mathematical formulation, but it is mentioned that Penrose is a mathematician.
• Turing is mentioned again on page 173 in the context of the Turing test for artificial intelligence.
• When Heather enters the machine that the aliens have instructed her to build on page 192, we read that she was ``looking through a twisted lens, a Mobius microscope, a topological telescope. A hyperscope. And the hyperscope was allowing her to see the four-diemsional reality that surrounded her quotidian world, a reality she'd been no more aware of than A Square -- the hero of Abbott's Flatland -- had been aware of the three dimensional world surrounding him."
• We learn on page 218 that we learn of seven colors in the rainbow (ROYGBIV) because Isaac Newton liked the idea that there should be a prime number of colors. (Is that true?)
• On page 260 we learn that ``Time isn't the fourth dimension....The fourth dimension is a spatial direction, precisely perpendicular to the other three.''
• On page 264 the standard description of how a lower dimensional cross section of a connected object can have more than one connected component: a donut passing through a tabletop can appear as two separate disks. This is then applied to the overmind: all of our minds appear separate, even though they are all actually connected if you can travel into the fourth dimension. This is used, on page 296, to eliminate the paradox of the Schr"odinger's Cat thought experiment. (Well, if you're willing to accept a special role for human knowledge in quantum physics, which I still cannot bring myself to do.)
• Binary numbers appear on page 302 as part of an alien message. We are supposed to believe that merely recognizing that the message is made up of the binary numbers 6 and 14 would lead us to guess that it is refering to carbon and silicon by atomic number.

The contrived plot of this novel concerns a husband and wife, Heather and Kyle, who are learning to deal with the suicide of their daughter during a trial separation when their surviving daughter hits them with a bombshell: she accuses her father of having sexually abused her and her sister. Doesn't sound much like a science fiction novel so far? Well, Kyle is a physicist doing research in quantum computation and Heather is a psychologist whose area of expertise is the Jungian overmind. Also, there is a message from space which Heather is working to decode so that she can get tenure (in a psych department?). Also, Kyle is being approached by government agents who want to use his computer to decode (coincidentally) a previous message from space that happened (coincidentally) to have been intercepted by Heather's boyfriend who (coincidentally) also commited suicide. Coincidentally, the picture of a hypercube (an unfolded tesseract) that Kyle has hanging in his office is exactly the key Heather needs to decipher the alien message. The message turns out to be blue prints for a machine which (coincidentally) allows her to travel through the Jungian human overmind and (coincidentally) allows her both to resolve the sexual abuse charges and figure out what the original message from space said. I guess if you are a Jungian then all of these coincidences (synchronicities?) and the whole idea that all humans share a single mind might not be a problem for you. I'm afraid it was a bit much for me.

As with Sawyer's other novels (See Calculating God), there are many `inside joke' references to other science fiction, especially Star Trek. This and the reasonably facile presentation of real scientific ideas (even if they are extremely controversial ones) make this book an enjoyable read even though it appears to me in the end to collapse under its own weight.