All of their lives become intertwined in this debut novel filled with emotion, tragedy, and the big questions of human existence. The author takes care to dole out information slowly and out of chronological order. I will minimize my discussion of any of the non-mathematical aspects here so as to avoid ruining it for any of you who might read the book in the future.

As for the math, there is a lot I can say about it without revealing too much. For instance:

• A key plot point is that since the accident, Marissa has been completely absorbed by her math research which she claims will allow her to communicate with the dead. Though her husband thinks this is simply a neurosis growing out of her guilt, other experts in mathematics and physics respond positively to her partial results, suggesting it is not only correct but ground breaking and important. (More on this below after the "spoiler alert".)
• Through much of the novel, Marissa is portrayed as a stereotypically cold, introverted mathematician. Like many mathematician characters in fiction, she seemingly hides from the real world by focusing on mathematics. Her husband is bothered by the fact that she is unaware of and unconcerned about his own feelings and actions. However, near the end of the novel, Marissa acknowledges to her husband that she was essentially taking advantage of this stereotype as she dealt with mourning and guilt:

  (quoted from The Intangible) People, especially you, cut me slack because I'm a mathematician. I get away with things because everyone assumes I don't know any better. But, I knew that you were hurting. And I let you hurt alone...

• Another stereotype that she represents is the "bad math teacher":

  (quoted from The Intangible) Marissa never thought of herself as a teacher. If a student was anything less than brilliant, there wasn't a lot Marissa could or would do for her.

• It is not clear to me in what area of research Marissa specializes. When she is young she talks a lot about Fermat's Last Theorem and people say she "has a good grasp of number theory". Later, she gives a talk on low-dimensional topology. And, her work on communicating with the dead seems to be in the area of mathematical physics.
• The accident is not the only thing Marissa is guilty about. While she is a student at Princeton she is portrayed as being extremely ambitious. She thinks of the famous mathematicians who were at that institution, of the mathematicians who did great work when they were young, and she desperately wants to be one of them. Because of that ambition, she opts to stay over the summer to continue working on her mathematics rather than going home to be with her sister. Marissa believes that decision contributed to her sister's descent into drug addiction.
• Even though Amanda's job title is "data scientist", suggesting to most people that she works with computers, the book definitely emphasizes that this also requires her to be a statistician and mathematician in addition to being a programmer. One of the anecdotes on a page containing several that a were supposed to convey her strength and determination was this one:

  (quoted from The Intangible) When she fell in love with computer science during her first year in college, math phobia stood between her and the degree. She bought the textbook for every required math course and spent her summers working through them. She was shocked when her combinatorics professor told her she had a gift.

• While being quizzed by her employer Amanda correctly applies Bayes' Theorem to determine the probability that someone actually has a rare disease after receiving a positive result on a test with high (but not perfect) specificity.
• It may be of interest to some readers of this website to know that Amanda is black. (FWIW The author also is a black data scientist living in Atlanta, GA.) I am not sure whether there was any indication of Marissa's racial background.

Marissa's research project combines string theory and the Everett Many-Worlds interpretation of quantum physics. Mathematical words are tossed around in a way that I did not find particularly coherent or meaningful:

(Note: The frequencies of "vibration" in string theory are restricted to a discrete set of values. This is what makes them quantized. So, it would not be possible to make an arbitrarily small change to them.)

It becomes clear that she had discovered a way to do something with these other universes. I wasn't really clear, however, what it was. It was described as "melding" her with the version of her in another universe, but she would supposedly not retain the memories from her universe so I wasn't sure what that meant or why it mattered. In any case, it is clear that it was the sort of ground breaking discovery she was hoping for, but she soon regrets having made it.

Pieces of the whole idea were released to other people before it was complete: the transformation found application in data science and physicists seem to like the parts they heard. But, at a point when she was the only one who fully understood the implications for some sort of interaction with other "dimensions" (i.e. universes where the things were slightly different because some quantum collapse had turned out differently), Marissa realizes that the potential for misuse of this power is too great and decides instead to destroy her work. This raises the question of whether destroying a discovery before making it public can be ethical, a question the book explicitly but briefly addresses.

This all also raises the question of whether I should be classifying this as "science fiction". I have decided not to do so, for several reasons. Although Marissa does develop this scientific theory (and even the part about Amanda's false pregnancy has a scientific component to it), we do not see any actual consequences of the novel science that go beyond what we already know. Moreover, the book does not read like science fiction. I think many people who do not enjoy science fiction would like this book and I do not want discourage them from trying it.