So, everyone in my group has met with our professor several times, and we’ve all nominally started working on our problems–figuring out what we want to figure out about them, making conjectures and figuring out they are false after several hours of calculation, writing computer programs to search efficiently through ~2^49 matrices for counterexamples, etc. It’s a good day when I discover and can prove a formula for one case of the objects I’m counting.
There’s a lot of literature review, too. I find the process comparatively relaxing–when I’m reading papers on what has already been done in the area surrounding my problems, I often feel like I’m building a mathematical toolbox and intuition that will help me approach the problems I want to work on. Diving in head first and tackling the problems naively can be fun, but it’s also daunting, emphasizing how much one doesn’t know when one starts work on a problem.
Right now, I don’t feel like I have any traction on any of the problems I’m working on–so I’m mucking around: testing small cases in the rare event that they exist (the numbers associated to everything I’m working with grow super-exponentially! Unfortunate for my concrete-cases-loving, pattern-seeking brain), looking for recurrences (and then discovering that they do worse than the naivest of upper bounds I can establish), and brainstorming other angles I can take on the problems. All this is towards the goal of actually defining what those problems are, of course! I’m not in a comfortable place where I have a list of calculations or concrete angles I want to try out–still casting around. Which is sometimes one of the best parts of research, but also, for me, the hardest research skill to develop.