Open Problems
First, I want to give a quick advertisement for a Floer homology open problem website run by John Baldwin ( https://floerhomologyproblems.blogspot.com/ ). This has a mix of problems, some of which seem very hard, and some of which might be a bit more manageable. If I was more thoughtful, I would collect a bunch of different open problem lists here for reference, but I'm not that thoughtful. Speaking of which, if you're a younger grad student or someone adjacent but not in low-dimensional topology, you may not be familiar with the famous Kirby problem list from the 1990s. (That is an update of an initial list made by Kirby many years earlier.) Many of these problems have been solved (e.g. the 3D Poincare Conjecture), but many of the outstanding ones are the biggest problems in the field (e.g. the 4D Smooth Poincare Conjecture, additivity of unknotting number under connected sum, etc) and have guided a lot of research directions. If you solve a Kirby problem, it definitely