Hi! Welcome to my sixth BeyondResarch Insight!
Lately, I have been struggling to complete the weekly problem sets, and thus, in our last differential geometry session, we focused on tackling this issue. We spent a lot of time going through the problems that we failed to solve and discussing the right approach, which linked nicely to some advanced problem-solving techniques we discussed the previous week.
I found the Rising Sea method particularly interesting. It was used by Alexander Grothendieck, a prominent mathematician of the 20th century, who is behind modern abstract algebraic geometry. His ingenious problem-solving approach was to develop theory around a problem until it “dissolves,” i.e., becomes trivial. Grothendieck compared this to cracking open a nut by dissolving it in a body of water (the rising sea refers to building up the theory) as opposed to more traditional approaches to do so.
For example, a contrasting problem-solving technique is the Hammer and Chisel method, which was heavily used by Jean-Pierre Serre, Grothendieck’s contemporary working in algebraic geometry and algebraic number theory. Following the same metaphor, with this method, you try to solve a problem by using a chisel and a hammer to strike the shell of the nut at different places until you find the weak spot and it cracks.
The hammer and chisel is a much more direct, efficient, and, at least for me, conceivable approach. However, the rising sea method has a big advantage: it allows you to obtain an extremely deep understanding of the problem and its structure! Nonetheless, this problem-solving technique is very advanced. In fact, Grothendieck might be the only one who was able to use the rising sea method successfully!
In the session, we also discussed some simpler approaches I should be following to improve my problem-solving skills. First and foremost, I should start all problems by breaking them down in a mind map that outlines all the steps I need to take to reach the solution or to deliver the proof. Then, I should cautiously apply axioms and properties to execute the steps in the breakdown and not stray away from the problem at hand.
Lastly, there is one more thing I have been failing to do and that is taking advantage of the resources I have access to as a part of BeyondResearch and asking my fellow scholars and Dr. Bar for help when I am stuck on a problem. Unfortunately, this has most likely prevented me from taking the most away from the program and compromised my learning. Therefore, you can consider “asking questions” my New Year’s resolution!