Boosting mathematical productivity

Forum for the GRE subject test in mathematics.
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Boosting mathematical productivity

Post by Cyclicduck » Tue Jan 14, 2020 7:35 pm

Everyone knows that working at a desk all day long isn't good for your productivity. You need to find other things to let your mind take a break, allowing it to work on a problem in new ways in the background. For example, Yitang Zhang claimed that he achieved his famous breakthrough while on vacation on a walk. Of course going on vacations and walks are great, and sometimes they do lead to great success, but I think that as (future) mathematicians, we can do better. That is, we should be able to find activities that truly optimize the effectiveness of the downtime used for them. I think I've hit on a really good activity, but I'm also looking for other people's suggestions on what's good.

The activity I believe is best for mathematics, at least for me, is armwrestling. Unfortunately, there are many misconceptions about this sport that need to be corrected before we have a meaningful discussion about it. However I don't want to spam this forum with a bunch of irrelevant material, so I'll hold this information until necessary to address the concerns of others.

Armwrestling is a lot like math. For both, you have to train very specific muscles and ways of thinking. But the connection goes far deeper than that in two ways. First, the rhythm of armwrestling seems, subjectively, to be very similar to that of mathematics. In armwrestling, there is a subtle game you play with your opponent through the hand. Not being able to cover every weakness, you leave some gaps, which can either be real weaknesses or real traps. Doing math, you are always competing with someone, be it for a job, a prize, a paper in a journal, etc. While we are all friends in the end, we need to learn to expose some of our weaknesses and draw our opponents in, where they will succumb to pushing too far and falling into traps. Look at how, in armwrestling, Devon Larratt does this to Andrey Pushkar in this supermatch. I know it may not sound nice, but I've heard that math as a career is indeed very competitive and understanding these dynamics are important.

The second connection between math and armwrestling is more object-level. There are many themes in math that can be better understood through armwrestling. For example, my sparring partner and I were discussing the infamous Dave Chaffee-Devon Larratt match from last year. We were replaying it; we strapped up and I fully committed to a toproll with side pressure while he rolled down into a King's move to defend. I came close to the pinline; I measured his hand to be only a few inches off the pad. I went for those few inches, but every time I surged, he pulled back and contained my hand more when I let up. For every surge I made, he had a response that slowly grew more and more powerful. It was like he was sucking in the strength I was pouring in until he had all the hand control and finished me. Reflecting on this, I came to a better understanding of how different ways of studying finite phenomena which naturally occur works. Namely, I saw how in many instances, the goal is not to simply measure these phenomena, but control and match how they work. For instance, when studying ramification in number fields, it's like you've got everything except for a finite set of primes which are ramified - just like the couple of inches to the pinline. You can work and work to just study unramified extensions, but when you get back to the global picture you still have to deal with it. Thus, you control how ramification works through the discriminant. Then you study the ramification groups to see how it works step by step, until you reconstruct how every piece of ramification arises. Another example is through Galois connections. For instance, with covering spaces, you have this fundamental group that measures how far away your space is from being simply connected. But studying this properly doesn't just involve measuring how big it is, but by controlling it step by step through the appropriate bijection between covering spaces and subgroups of the fundamental group, just like my friend had a response for every surge I made.

I know this all sounds sort of crankish, and I'm not saying that you need to armwrestle to understand these notions, but armwerstling really helped me connect these ideas and better understand what I see to be a theme of math. As I armwrestle more and do more math I see these connections more and more. Thus, for me armwrestling pairs very well with math, and I think there are a lot of other connections between math and fields which at first glance have nothing to do with math.

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