![]() ![]() ![]() It asks: If you look at billiards in a triangle, is there a periodic billiard path-one that traces the same path over and over again? This was known for acute triangles, but it wasn’t known for obtuse triangles. One problem I got very interested in was the triangular billiards problem. I like to just get my hands dirty and start right away. I don’t feel like spending six months reading the literature until I get to the point where I’m ready to attack this problem. If I hear about some conjecture in some fancy area of math, I’m lazy about it. I like things where I can just start working. The third thing, which maybe sounds a little silly, is that the simple problems I like don’t require much background to get into them. Now I have this feeling that almost everything is unknown about mathematics. When you’re young you have the impression that almost everything is known. Like, I’m just going to program the computer and run some experiments and see if I can turn up some hidden patterns that nobody else had seen just because they hadn’t yet done these experiments. The modern computer is a new tool, and I think of these simple things as excuses for data gathering. The second thing is that I like doing computer experiments, and so I feel that sometimes I have a chance of making progress. In other words, there’s something missing in human knowledge that prevents people from solving the problem. I feel if it’s a simple problem that hasn’t been solved, it probably has some kind of hidden depth to it. Finally, I like the beauty of pure mathematics, much in the same way someone might like a work of art. There’s kind of a mountain-climbing aspect to it. I like solving problems, trying to solve problems that people can’t solve. I’ve always had a love for these kinds of things for some primitive reason I can’t quite explain. ![]() I like that you can get to the bottom of questions, unlike, say, politics or religion where you can just talk for years with people and no one will change the other’s opinion. I like the fact that it’s procedural, and there’s a method to it so you can make progress. The first thing is I like the fact that it works, somehow. An edited and condensed version of those conversations follows. Quanta Magazine spoke with Schwartz about his taste for simple problems, what he called the “miracles” of mathematics, and his upcoming math book for kids about infinity. As he explains, computers complement human mathematical thought in several ways: They draw out patterns that provide hints which lead to proofs that might not have been apparent to the mind alone. ![]() Schwartz uses computer experiments in much of his work-he’s on the vanguard in that respect. In 2008 he proved that every triangle with angles all less than 100 degrees contains at least one periodic billiard path-a repeating path that a ball will trace and retrace forever. He is now a tenured professor at Brown University whose most important work has taken place in the field of dynamics, which studies the long-run behavior of iterative processes, like a billiard ball ricocheting on a frictionless table. He received his doctorate at Princeton University under the mentorship of Bill Thurston, one of the most important mathematicians of the last half-century. Yet none of this is to say that Schwartz is anything but a serious and accomplished mathematician. “I don’t think I have a mature attitude towards math,” he said. He likes problems he can read about today and start solving tomorrow-simple problems, fun problems, problems that have the aspect of a carnival game: Step right up and see what you can do with this one! It’s an unusual disposition among research mathematicians. Such questions don’t interest Richard Schwartz. If you want to attack the biggest problems, you’ll need to master a lot of highly technical material before you can even begin to say something new. Most of the important discoveries in mathematics take place after decades or centuries of effort. From Quanta Magazine ( find original story here ). ![]()
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