R. F. Kuang is one of my favorite authors, so I was thrilled when the first chapter of her newest novel Katabasis used “Ramanujan summation” in a magical spell! The rest of the story exceeded my wildest dreams for mathematics in mainstream fiction, and I think its maths would be advertised much more if it wasn’t so important for the plot (spoiler warning). The two main characters are graduate students at Cambridge’s Department of Analytic Magick, which uses paradoxes to change how reality works. Alice excels in linguistic paradoxes, and Peter most frequently uses logic, but my favorite part was their exploration of the non-euclidean world of Hell because I constantly imagine stories in the game HyperRogue.

The geometry of Hell was arguably the biggest mystery until the very end, where [SPOILERS] it is revealed in maps to be a hyperbolic surface that is described in detail as non-orientable like the real projective plane. I think that makes sense after reading Wikipedia for a deeper understanding, and the characters explained every plot point just enough for an amateur like me to understand and become curious for more. Curry’s paradox is my new favorite, but Peter also explains Gabriel’s Horn, the Unexpected Hanging paradox, and Godel’s Incompleteness Theorem [sic] as a taste of what to expect. Alice also points out Escher’s Nil geometry, Bach, and a “golden braid,” which seems like an allusion to me...

[no more spoilers]

However, I think you will agree that the characters are unfortunate stereotypes about mathematicians (even though the characters insist “mathematicians hate magicians” like themselves). As a parody of leaving academia, Alice has a long-held dislike of math, and Peter has always been an erratic genius, so other tropes and traits are subverted instead like disabilities. Studying is used to torture souls in Hell on a few occasions, and the pursuit of knowledge for its own sake is frequently portrayed as misguided. The first half of the book is a rare example of mathematics being both fun magic AND being cold/boring/useless… but I think the ending is worth it for closure as we find reasons to survive when the world is upside down.

She had to look away; the looping made her head hurt. And yet this made perfect sense. Lewis Carroll had theorized this -- how else did you conceptualize life and death, the membrane of passage, except as continuity? -- but no one believed him. Take a strip of paper, twist it in the middle, and connect the ends. Very good. Now you have a ring, a three-dimensional object you can hold in your hand. But it only has one side. The inside is continuous with the outside. Now do the same thing with a four-sided handkerchief. Twist the edges, line them up, and stitch it all together so that the inside is continuous with the outside. All is external to the bag, which means all is also internal to the bag, and so the bag holds the world.

Impossible to draw, impossible to even conceive. But here Alice was looking right at it.

"The projective plane," Elspeth murmured. "Astonishing."

The black sands were so close now. All Alice could see on that shore was a single golden braid of light, stretching from the bank to the unknown beyond.

I have to disagree a bit with the preexisting review on this site. I think this book mainly uses maths as buzzwords to make the characters seem smart. If googling the paradoxes that are mentioned throughout the book is your thing then you will definitely learn something. While there are some reasonably good maths discussions there are also quotes like:

I think anyone with basic knowledge of probability theory would know that you cannot equate a probability of zero with impossibility. To be fair, Huemer is an actual philosopher who seems to have written a paper about this exact argument.

Another line in the book that stood out to me was:

Which, since this book takes place at least after the 1960s as the previous quote demonstrates, is clearly not true.

I agree with the OP, Aidan. There is really a lot of math in this modern twist on Dante's Inferno. It runs through the whole book as a theme. There is real math (like hyperbolic geometry and Goedel's Theorems) that is important to the plot. There is also a mathematical approach to magic that is made up but feels real.

I liked the book a lot at the start. It is like a metaphor for the challenges of grad school. But, towards the end, it got a bit too negative for me. Academia isn't perfect, but this book makes it seem worse than it is. For the last quarter of the book I just wanted it to be over already. Overall, I liked it and recommend it to other math fiction fans.