Boosting mathematical productivity

Forum for the GRE subject test in mathematics.
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Cyclicduck
Posts: 78
Joined: Sun Apr 14, 2019 9:55 pm

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.

ahhhhmeh
Posts: 54
Joined: Thu Mar 29, 2018 10:13 pm

Re: Boosting mathematical productivity

Post by ahhhhmeh » Sat Feb 01, 2020 5:50 am

For me the biggest issue is isolation. I try to get as much socializing done as possible. But it’s kinda hard to be social when you spend so much time on stuff that no one really cares or knows about.

I’ve heard group sports helped some people but I haven’t been able to find a group.

Cyclicduck
Posts: 78
Joined: Sun Apr 14, 2019 9:55 pm

Re: Boosting mathematical productivity

Post by Cyclicduck » Mon Feb 03, 2020 5:58 pm

ahhhhmeh wrote:
Sat Feb 01, 2020 5:50 am
For me the biggest issue is isolation. I try to get as much socializing done as possible. But it’s kinda hard to be social when you spend so much time on stuff that no one really cares or knows about.

I’ve heard group sports helped some people but I haven’t been able to find a group.
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Please see the following quote by the great Grothendieck.
Alexander Grothendieck wrote:In those critical years I learned how to be alone.[But even]this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation[1945-1948],when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law..By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member. or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc…it is in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

ahhhhmeh
Posts: 54
Joined: Thu Mar 29, 2018 10:13 pm

Re: Boosting mathematical productivity

Post by ahhhhmeh » Tue Feb 04, 2020 3:40 am

Cyclicduck wrote:
Mon Feb 03, 2020 5:58 pm
ahhhhmeh wrote:
Sat Feb 01, 2020 5:50 am
For me the biggest issue is isolation. I try to get as much socializing done as possible. But it’s kinda hard to be social when you spend so much time on stuff that no one really cares or knows about.

I’ve heard group sports helped some people but I haven’t been able to find a group.
????????????????????????
????????????????????????
????????????????????????

Please see the following quote by the great Grothendieck.
Alexander Grothendieck wrote:In those critical years I learned how to be alone.[But even]this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation[1945-1948],when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law..By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member. or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc…it is in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.
I don’t want to become a monk or a math-eunuch. Peers aren’t the same thing as friends.

Cyclicduck
Posts: 78
Joined: Sun Apr 14, 2019 9:55 pm

Re: Boosting mathematical productivity

Post by Cyclicduck » Tue Feb 04, 2020 4:56 am

What are you smoking? This is about not fearing solitude. And if you're trying to imply something about who Grothendieck was, man up and say it.



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