Disclaimer: I don't think I was the target audience for this book. I'm a physicist. I use math a lot, and many of the concepts in the book were either already familiar to me or at least on the periphery of my awareness. I got a lot out of the math explanations here. I really appreciated that, but I think that maybe Frenkel fell short of his goal of making these concepts accessible to the average reader.
Maybe the goal, though, isn't for everyone to understand the concepts, but appreciate his passion for them. In that, he definitely succeeds. His life story is captivating, and the conviction with which he tells us how interesting all these topics are is overwhelming. I generally don't like math -- that's what I have in common with the average reader, I guess. I do lots of math -- I use it, I need it, and I'm pretty good at it -- but pure math has never interested me. I'm interested in its applications, which is why I'm a physicist and not a mathematician. My vision generally starts clouding over whenever branes are mentioned. I found myself being drawn into his explanations of riemann surfaces and Lie algebras, though. He actually got me interested in some pure math. And so yes, he succeeded. He made someone who isn't crazy about math interested in what he's working on. The Langlands program actually is fascinating. Isn't that amazing? Connections between big ideas are startling and exciting.
Ultimately, I can't analyze this book from anyone's perspective but my own. So I'll just say that I though that was most assuredly a 5-star amazing book. I'll put all the usual caveats on there that people are different and maybe others won't like it as much, but that's true of any book I like.
The writing of this book was a bold, ambitious move. I applaud Frenkel for the effort and thoroughly enjoyed reading it.