Despite its name, "Our Mathematical Universe" is not a math book. It is an exploration of the nature of our physical reality according to the author's own Mathematical Universe Hypothesis (MUH). However, the MUH has been noted by math fans. For instance, it is the last subject covered in The Math Book
by Clifford Pickover.
At the end of the first chapter, Tegmark suggests that if you are a physicist you can skip ahead. I do not recommend this. Tegmark tells his story in a lively manner punctuated by illustrations and personal anecdotes. It is all a good read.
Along the way, we learn that the author has conducted a survey of physicists, repeated over time, about the quantum wave function. Early on, sentiment favors the Copenhagen interpretation. Later, Many Worlds is favored. I fall into that latter group. Tegmark proposes a life-or-death quantum machine-gun test of Many Worlds. I do not think that his test is necessary. The improbable victory of the Mets in game six of the 1986 World Series is sufficient proof for me.
Tegmark says that it is not enough to say that mathematics describes physical reality, but that our physical reality is mathematics. Our conservation laws are expressions of symmetries of the mathematical object that is our (multi)universe. Also, time is just another coordinate in space-time. Its passage is an illusion. I have read that Tegmark sends e-mails to his future selves.
Like many physicists, I believe that the Second Law is perhaps our most important concept. I think that Tegmark should have said more about how the MUH treats entropy.
Finally, Tegmark presents a way to test the MUH. If the universe is not a mathematical object, then physics will reach a dead end in which we can no longer describe reality by mathematics. If the MUH is correct, then we will continue to find mathematical descriptions.
Tegmark is an excellent storyteller. This work is well worth reading and thinking about.