For mathematicians and computer scientists, this was often a year of double takes and closer looks. Some reexamined foundational principles, while others found shockingly simple proofs, new techniques or unexpected insights in long-standing problems. Some of these advances have broad applications in physics and other scientific disciplines. Others are purely for the sake of gaining new knowledge (or just having fun), with little to no known practical use at this time.

Quanta covered the decade-long effort to rid mathematics of the rigid equal sign and replace it with the more flexible concept of “equivalence.” We also wrote about emerging ideas for a general theory of neural networks, which could give computer scientists a coveted theoretical basis to understand why deep learning algorithms have been so wildly successful.

Meanwhile, ordinary mathematical objects like matrices and networks yielded unexpected new insights in short, elegant proofs, and decades-old problems in number theory suddenly gave way to new solutions. Mathematicians also learned more about how regularity and order arise from chaotic systems, random numbers and other seemingly messy arenas. And, like a steady drumbeat, machine learning continued to grow more powerful, altering the approach and scope of scientific research, while quantum computers (probably) hit a critical milestone.