The book starts with conventional physics and cosmology but the author quickly delve into the premises of ｍultiverse and then the Mathematical Universe. The last chapter even touches the meaning of life (!), as well as discussing the existential risk due to various sources including self-inflicted ones such as nuclear war and unfriendly AI.
In this book, the author has categorized 4 levels of multiverse. I try to summarize my understanding (from this book) here. Level I is simply the space outside our visible Universe, due to eternal inflation. Level II is like many unconnected Level I multiverses as the eternal inflation could initially happen in several adjacent volumes which could further split in a treelike structure; each of these Level 1 multiverses may have different physics constants or solutions of the same law of physics (landscape). Level III multiverse is the collection of parallel universes as a result of the existence of various wavefunctions (of Schrödinger equation) in the Hilbert space. Level IV multiverse contains all mathematical structures corresponding to different fundamental laws of physics, which is implied by the author's Mathematical Universe Hypothesis. ( Table 6.1 in p.139 has a good summary.)
From the 1st paragraph on p.202, we learn that the author's original family name (his father's) was "Shapiro". But he changed to use his mother's family name "Tegmark" (while he was contemplating his publishing debut) because it's more unique than "Shapiro" :-) The author has also told us (p.243-246) that he has kind of hidden or not made it too obvious about his works/papers in the multiverse or the fundamentals of quantum mechanics until now (or after he was tenured at MIT ~ 2004 --- the 5th line from the bottom line of p.85) ! The author has certainly executed his academic strategy smartly which has allowed him to avoid the unfortunate fate of Hugh Everett III.
There may be quite a few crazy subjects in this book and the one that I like the most is "Quantum Suicide" (p.216-220). It invovles putting your head in front of a quantum machine-gun (which fires or not depending on the outcome of quantum measurement) and as long as the time scale to kill you is shorter than the time for you to become aware of the outcome of the measurement and it'd certain to kill you when it fires, you'd be/feel alive no matter how long your head stay there, if Hugh Everett is right.
Extremely trivial : I've noticed that the "Index" at the back of the book only shows pages 51 and 63 for "Alpher, Ralph" and "Herman, Robert". But "Alpher and Herman" also appears on p.61 (22nd line) --- almost in the same way as on p.63. Another example that I have noticed is qualia which has also appeared on p.288 (23rd line) and p.318 (12th line from the bottom) but the index of qualia has only mentioned p.234 and 290. I've always wondered how and/or how well indexing has been made in a book and my two examples here may show that it has not been done very well in this book.
The book has a lot of unbelievable ideas which are interesting and entertaining. However, I don't feel that I have been really convinced by what the author has written. I don't feel that the author has really proved anything to us in any substantial way and it just looks like that he's promoting the speculative ideas of multiverse and Mathematical Universe. Overall, I don't think this book has achieved something really great, but just that the author has enjoyed himself in writing a book with the subjects that he may be most interested in.
p.90 (4th line from the bottom): "our" in "...figure our where sound waves ..." should be "out"....Continua