This list isn’t exhaustive. There’s so many stories out there, but I’ve chosen those that have helped me understand math culture and have helped me reflect on my own connection with mathematics. In particular, I would recommend reading:
On Proof and Progress in Mathematics (Bill Thurston). The powerhouse in the field of low-dimensional topology, Bill Thurston, gives insightful answers for the following questions in his essay above:
what is it that mathematicians accomplish?
how do people think and understand mathematics?
how do people communicate their understanding of mathematics?
Category Theory and Context: An Interview with Emily Riehl (by Beth Malmskog). The prolific mathematician Emily Riehl gave a very inspiring interview on how she got into math and what the math life looks like in her perspective. She has an intriguing reason for choosing her career, which I find a little funny and quite relatable! She also gives some cool insight on how she works on the job.
Wikipedia page for Emmy Noether, and Emmy Noether’s ‘Set Theoretic’ Topology: From Dedekind to the Rise of Functors (Colin Mclarty). Emmy Noether brought the powerful insight of structure into so many fields, such as commutative algebra, algebraic geometry, algebraic topology, representation theory, and even physics. The second article delves into her contributions during her second epoch. This was the time when she extracted the “set-theoretic” (structural, almost categorical-like) intuition for groups from Dedekind, which includes the isomorphism theorems. This was also the time when she collaborated with many of the influential topologists at the time (namely Brouwer, Alexandroff, and many others) to simplify and unify topology by using groups in the right way–a precursory thinking to the algebraic topology we know today!
Leray in Oflag XVIIA: The origins of sheaf theory, sheaf cohomology, and spectral sequences (Haynes Miller). This paper describes the genesis and evolution of the ideas of sheaf theory, sheaf cohomology, and spectral sequences in the 1940s, before it was adopted by algebraic geometers. You might be surprised to learn that these came out of a mathematician/prisoner-of-war Jean Leray during his captivity in a war camp at Austria from 1940 to 1945, who had to conceal his work on partial differential equations from the authorities to avoid being asked to work for them!
The Rising Sea: Grothendieck on simplicity and generality (Colin Mclarty). This wonderful paper describes how Alexander Grothendieck (as well as Jean-Pierre Serre, the Italian school of algebraic geometers, Wolfgang Krull, and many others) took some of the wild ideas from André Weil’s Conjectures and ran with it, to restructure algebraic geometry as we know today.
Jean Bourgain (Terence Tao). This is a wonderfully written obituary for Jean Bourgain by Terence Tao, which also delves into Terence Tao’s funny experience with reading one of his papers in graduate school. Also, if you’re curious about the kind of “toolkit” that Bourgain uses in his papers, see the complementary post: Exploring the toolkit of Jean Bourgain.
A database of the general examinations taken by graduate students in the Princeton University Math Department. This is a great repository for graduate students studying for a qualifying exam (pay attention to some of the questions posed by the examiners)! But also, there’s a load of interesting and funny stories in here, such as Terence Tao’s general exam and Manjul Bhargava’s general exam.