This Quanta article A New Agenda for Low-Dimensional Topology highlighted an important part of doing math that I think is underrated: making lists of interesting unsolved problems. An early, impactful example of this was David Hilbert's list of 23 unsolved problems he introduced in 1900.

The first Hilbert problem was Cantor's continuum hypothesis, which asks if there is a size of infinity strictly between \( \aleph_0 \) (the number of integers) and \( c\), where \( c = |\mathbb{R}| \) (the number of real numbers).

The second Hilbert problem, The compatibility of the arithmetical axioms, was what Bertrand Russel worked on in his Principia Mathematica, which a young austrian genius named Kurt Gödel used as a starting point for his work on his Incompleteness Theorem. It cannot be stated how much this changed our philosophical understanding of mathematics. Inspired by Gödel, a young Brit named Alan Turing applied Gödel's ideas in his work on computable and uncomputable functions, and Turing's work became the basis for the development of computers.

Hilbert's list sparked a revolution in mathematics, philosophy and computer science.

The quanta article points at a similar list, but for the niche field of low-dimensional topology, which includes knot theory and 3-manifolds.

The analogous list is from a mathematician named Rob Kirby:

The first Hilbert problem was Cantor's continuum hypothesis, which asks if there is a size of infinity strictly between \( \aleph_0 \) (the number of integers) and \( c\), where \( c = |\mathbb{R}| \) (the number of real numbers).

The second Hilbert problem, The compatibility of the arithmetical axioms, was what Bertrand Russel worked on in his Principia Mathematica, which a young austrian genius named Kurt Gödel used as a starting point for his work on his Incompleteness Theorem. It cannot be stated how much this changed our philosophical understanding of mathematics. Inspired by Gödel, a young Brit named Alan Turing applied Gödel's ideas in his work on computable and uncomputable functions, and Turing's work became the basis for the development of computers.

Hilbert's list sparked a revolution in mathematics, philosophy and computer science.

The quanta article points at a similar list, but for the niche field of low-dimensional topology, which includes knot theory and 3-manifolds.

The analogous list is from a mathematician named Rob Kirby:

Kirby attributes this early-career success in part to the existence of the Milnor list, which provided him with a greater variety of projects to choose from than he would have received from the people immediately around him in graduate school

“If you’re writing a letter of recommendation for someone and they’ve solved a Kirby problem, you mention that in your letter,” said John Baldwin, a mathematician at Boston College who participated in the workshop and is helping to edit the list.

The lists serve a social function, naming relevant problems and making them legible to the wider community. This is an essential way to transfer status to budding young mathematicians. It also helps the young mathematicians by presenting them with a mathematical frontier they can explore that has been mapped out, but not conquered, by the previous generation. This succession process is incredibly important to the long-term success of mathematics, and it should be more widely appreciated.