Mathematicians and computer scientists had an exciting year of breakthroughs in set theory, topology and artificial intelligence, in addition to preserving fading knowledge and revisiting old questions. They made new progress on fundamental questions in the field, celebrated connections spanning distant areas of mathematics, and saw the links between mathematics and other disciplines grow. But many results were only partial answers, and some promising avenues of exploration turned out to be dead ends, leaving work for future (and current) generations.
Topologists, who had already had a busy year, saw the release of a book this fall that finally presents, comprehensively, a major 40-year-old work that was in danger of being lost. A geometric tool created 11 years ago gained new life in a different mathematical context, bridging disparate areas of research. And new work in set theory brought mathematicians closer to understanding the nature of infinity and how many real numbers there really are. This was just one of many decades-old questions in math that received answers — of some sort — this year.
But math doesn’t exist in a vacuum. This summer, Quanta covered the growing need for a mathematical understanding of quantum field theory, one of the most successful concepts in physics. Similarly, computers are becoming increasingly indispensable tools for mathematicians, who use them not just to carry out calculations but to solve otherwise impossible problems and even verify complicated proofs. And as machines become better at solving problems, this year has also seen new progress in understanding just how they got so good at it.