This is pitched at "readers with at most an undergraduate (ideally high-school) education", and the pieces often begin with some rapid prerequisites: explaining what a group is, how matrix multiplication works, or what a graph is, for example. And many results are genuinely accessible: "The Optimality of the Standard Double Bubble", for example, or "Arbitrarily Long Arithmetic Progressions of Prime Numbers" or some of the graph theory results. (I managed to explain at least the gist of some of these to a ten year old.) But others are more of a reach: it is hard to believe anyone who needs an explanation, or even a refresher, of what a derivative is will cope with the rapid segue through Banach spaces and Lebesgue measure in "On the Differentiability of Lipschitz Maps on Infinite-Dimensional Spaces". Despite the attempt to provide low level scaffolding, most of Theorems of the 21st Century seems unlikely to work, without support, for any except the most high-flying high school students.
The pieces are awkwardly ordered by publication date and the titles are often inscrutable even when the material is not ("Every Separable Infinite-Dimensional Banach Space Has Infinite Diameter"), which makes browsing difficult. It would have made more sense to group the theorems by area, with for example all the graph theory results together. This would also have allowed some of the repetitive explanatory material to be provided just once.
That said, I ended up reading through the whole of Theorems of the 21st Century and really enjoyed it. It is a nice survey of some core concepts in mathematics and conveys a feel for some current research topics.
January 2025
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