Episode #472 from 0:49
First hard problem
What was the first really difficult research-level math problem that you encountered, one that gave you pause maybe? Well, in your undergraduate education you learn about the really hard impossible problems like the Riemann Hypothesis, the Twin-Primes Conjecture. You can make problems arbitrarily difficult. That's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting are the problems just on the boundary between what we can do rather easily and what are hopeless, but what are problems where existing techniques can do 90% of the job and then you just need that remaining 10%. I think as a PhD student, the Kakeya Problem certainly caught my eye. And it just got solved actually. It's a problem I've worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese mathematician Soichi Kakeya in 1918 or so. So, the puzzle is that you have a needle on the plane or think like driving on a road something, and you want it to execute a U-turn, you want to turn the needle around, but you want to do it in as little space as possible. So, you want to use this little area in order to turn it around, but the needle is infinitely maneuverable. So, you can imagine just spinning it around. As the unit needle, you can spin it around its center, and I think that gives you a disc of area, I think pi over four. Or you can do a three-point U-turn, which is what we teach people in their driving schools to do. And that actually takes area of pi over eight, so it's a little bit more efficient than a rotation. And so for a while people thought that was the most efficient way to turn things around, but Besicovitch showed that in fact you could actually turn the needle around using as little area as you wanted. So, 0.01, there was some really fancy multi back and forth U-turn thing that you could do that you could turn a needle around and in so doing it would pass through every intermediate direction. Is
Why this moment matters
What was the first really difficult research-level math problem that you encountered, one that gave you pause maybe? Well, in your undergraduate education you learn about the really hard impossible problems like the Riemann Hypothesis, the Twin-Primes Conjecture. You can make problems arbitrarily difficult. That's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting are the problems just on the boundary between what we can do rather easily and what are hopeless, but what are problems where existing techniques can do 90% of the job and then you just need that remaining 10%. I think as a PhD student, the Kakeya Problem certainly caught my eye. And it just got solved actually. It's a problem I've worked on a lot in my early research. Historically, it came from a little puzzle by the Japanese mathematician Soichi Kakeya in 1918 or so. So, the puzzle is that you have a needle on the plane or think like driving on a road something, and you want it to execute a U-turn, you want to turn the needle around, but you want to do it in as little space as possible. So, you want to use this little area in order to turn it around, but the needle is infinitely maneuverable. So, you can imagine just spinning it around. As the unit needle, you can spin it around its center, and I think that gives you a disc of area, I think pi over four. Or you can do a three-point U-turn, which is what we teach people in their driving schools to do. And that actually takes area of pi over eight, so it's a little bit more efficient than a rotation. And so for a while people thought that was the most efficient way to turn things around, but Besicovitch showed that in fact you could actually turn the needle around using as little area as you wanted. So, 0.01, there was some really fancy multi back and forth U-turn thing that you could do that you could turn a needle around and in so doing it would pass through every intermediate direction. Is