The short stories in the anthology Mirror Shards all focus on augmented reality (AR), the idea that our perception of the world around us will be fundamentally changed by the use of advanced technology. Ken Liu's contribution, Music of the Spheres considers the effect that it might have on math research.
Central to the story is the evolving relationship between a student of mathematics and his younger sister, destined to become a biologist. As the story opens, he is helping her with her geometry homework. (Side Note: I really hope that children learn Euclidean geometry in school in the future, as they apparently do in Liu's imagination. In the reality in which I live, this important part of cultural and mathematical history has been all but wiped out of the K12 curriculum!) However, by the time he is studying math in grad school, the world is changing too quickly, and it is her knowledge of evolutionary biology helps him to be able to deal with the situation.
The story presents an interesting viewpoint on mathematical aesthetics and on differing mathematical abilities between individuals.
Moreover, aside from the Euclidean geometry mentioned above, there is also a bit about coordinate geometry and transformations and a good discussion of the role of computers in proving the Four Color Theorem. There is no doubt that the relationship between the two protagonists is emotionally potent, but it seems to me, that what this story is really about what it takes to prove interesting theorems today, and how that might change as a consequence of AR.
