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John Baez (Nov 14 2024 at 17:18):On Mathstodon Terry Tao responded to some recent gossip, and explained some of his early steps in learning gauge theory:
An anecdote that I shared about rolling around on the floor back in 2000 to solve a math problem, both in my #Masterclass at https://www.masterclass.com/classes/terence-tao-teaches-mathematical-thinking/chapters/transforming-problems , and on #MathOverflow at https://mathoverflow.net/a/38882/766 , as well as the #NewYorkTimes https://www.nytimes.com/2015/07/26/magazine/the-singular-mind-of-terry-tao.html , has for some reason recently gone viral on various social media. Just for the record, I wanted to add some mathematical background behind the story, which eventually led to my paper https://arxiv.org/abs/math/0010068 . At the time, I was trying to construct solutions to an equation known as the wave maps equation on the sphere: the solution was like a solution to the wave equation, except being forced to take values in a sphere rather than in a vector space.
I was trying to solve the equation iteratively, breaking up the solution to a low frequency base solution and a high frequency correction. As a first approximation, the low frequency base could also be assumed to stay on the sphere and solve the wave maps equation, so the main problem was to work out what the high frequency correction was doing.
Because the high frequency correction also had to keep the solution on the sphere, one could assume as a first approximation that the high frequency correction was tangent to the low frequency base. So, at any given point in space and time, the low frequency base solution was located on some point on the sphere, and the high frequency correction basically lived on the tangent plane to the sphere at that point. But because the base solution evolved (slowly) in space and time, this tangent space kept rotating around the sphere.
I had the idea to try to perform a change of variables to rotate these tangent planes to line up and become parallel, in the hope of simplifying the equation. At the time, I did not have the required background in differential geometry to recognize that I was actually trying to perform a gauge transformation; but in fact this project finally set me on the path to learn differential geometry properly (I had not fared well with it as an undergraduate, and was slightly intimidated by it as a graduate student), which has definitely helped my mathematical career since.
The problem was that the coordinate changes I wanted to implement were not unique. If I wanted to rotate a tangent plane on the sphere to be tangent to the north pole, say, I could easily visualize rotating the sphere so that the point of tangency was on the pole, but then I could also twist the plane around that pole arbitrarily. So, for each point in space and time, there was an additional twist I could perform, and this transformed the equation that I was trying to solve in a way that could either make the equation more like the free wave equation (and thus easier to solve), or less like the wave equation.
If I focused only on how these twists evolved in a single spatial direction - such as the x-axis, it was like trying to smooth out a chain of disks to all be parallel, and I could see how to perform a change of variables to do that. (Now that I know differential geometry, I know that I was integrating a connection along this direction.) But then this operation made things curve in a funny way in the y-axis. I couldn't visualize all this in my head.
So, somewhat in desperation, I lay on the floor, closed my eyes and tried to imagine _being_ the plane being rotated. I imagined the base point being moved in the x-direction, and rolled accordingly; then in the y-direction. With this, I was able to figure out that if I could find a rotation matrix depending on both time and space whose derivatives behaved in a certain way, then I could find my good change of variables and solve the problem. (In retrospect, I was trying to flatten a connection by choosing an appropriate gauge.)
Miraculously, the algebraic form of the equation was such that a natural choice of gauge that did nearly flatten everything was evident. (Again, in retrospect: the connection attached to this problem happened to have a small curvature.) After some frenzied calculations with pen and paper, I was able to get a plan to solve my problem, and then write the paper. (Which actually ended up earning me a Bocher prize.)
I was visiting my aunt in Australia at the time, and she managed to walk in on me while I was rolling around on the floor with my eyes closed. In the grand scheme of things, not the most embarrassing situation to be in, but I am not sure how satisfied she was by the explanation that I was "thinking about math". In a strange way, though, my profession can occasionally benefit from our reputation for eccentricity; she did not ask any further questions.
Jean-Baptiste Vienney (Nov 14 2024 at 22:16):That's really encouraging to read him say that he had trouble with differential geometry when he was a student while he is part of the strongest people in math out there! Honnestly, I didn't think he would have struggled with any topic while being a student. But it looks like he is much more a normal human being than I would have thought.
Todd Trimble (Nov 15 2024 at 01:19):The rolling around on the floor reminds me of remarks of Thurston on the relatively underappreciated role of the kinesthetic sense in mathematics (compared to the visual and auditory senses). This is from the collection of interviews in More Mathematical People; Thurston starts off discussing the case of the French topologist Bernard Morin, who lost his sight at the age of 6 from glaucoma, but who was instrumental in the effort of making sphere eversion explicit:
[sphere eversion is] something most people have a great deal of trouble visualizing. In fact, I think that vision is somehow distracting to the spatial sense, because we have a spatial sense that is more than just vision. People associate it with vision, but it's not the same. If I close my eyes and imagine what this room is like, I will have a sense in my mind that there's a table here and something here and there. It will be a sense of the room that doesn't have much to do with perspective. It's hard -- that is, it takes a lot of training -- to go from a spatial image to a picture on paper. So these things are not necessarily stored in our minds in a visual sort of way. We translate what we see into a sense of space. If you think of it, you realize that if you imagine a table with four chairs around it, it doesn't matter whether you can see the seats of the chairs. You just know that they are there. It's kinesthetic as well as visual.
Mike Shulman (Nov 15 2024 at 01:57):I wonder whether that is equally true for everyone. I certainly agree with it: when I imagine a table I don't "see" the seats of the chairs. But when I learned about aphantasia it sounded as though there are people who really do "see" the seats of the chairs when they imagine a table.
John Baez (Nov 15 2024 at 02:29):Jean-Baptiste Vienney said:
That's really encouraging to read him say that he had trouble with differential geometry when he was a student while he is part of the strongest people in math out there! Honestly, I didn't think he would have struggled with any topic while being a student.
Of course we should remember that "struggling" for a world-class mathematician may mean something a bit different than "struggling" for an ordinary student. Maybe this just means he didn't excel in differential geometry as he did in other fields, so he focused on those other fields.
But I don't know!