Po-Shen Loh has resurrected the United States International Mathematical Olympiad team, leading it to four first-place rankings in the last six years as the team’s head coach.
But in 2002, when a friend suggested Loh apply for an open position as a grader with the team, he hesitated. “I had never thought to apply before,” Loh said. “Not because I didn’t want to. But because I thought there are better people out there.”
He eventually agreed, and by the end of the team’s June 2002 training program, he’d made an impression. “Somehow I got voted best lecturer,” he said. In 2013 the Mathematical Association of America, which coordinates the team, asked Loh to become the head coach. He accepted, and two years later the U.S. achieved a top ranking in the IMO for the first time in 21 years.
Math has always been a part of Loh’s life. He grew up in Madison, Wisconsin. His father was a statistics professor, and his mother had taught math in Singapore. A local news story from 2015 dubbed them the capital city’s “first family of math.” It’s in this context that Loh learned early in life that tackling hard math problems requires persistence and, often, unorthodox thinking.
Loh brings this perspective to his work as IMO coach. Every June, the mathematical association invites 60 high schoolers to a national training camp in Pittsburgh. Based on a battery of exams given the preceding year, six of them have already been tapped to represent the U.S. at the IMO in July. One of Loh’s main innovations at the camp has been to invite Olympiad teams from many different countries to practice alongside the 60-person U.S. contingent. “At first people were surprised, because we paid for all expenses, with no catch,” he said.
Loh also works hard to expose kids from all backgrounds to mathematical ideas. Before the pandemic he toured the country giving math talks like a “traveling salesman of mathematics,” he said. In 2014 he launched an app called Expii, which uses interactive puzzles to teach basic math concepts. He also made a YouTube video in which he helped coach a young cheerleading team in New Jersey by explaining the basic math behind their choreography.
When he’s not filming videos or hosting international training sessions, Loh works as a mathematician at Carnegie Mellon University in Pittsburgh. He studies extremal combinatorics, meaning objects like very large graphs or networks. In particular, he studies how overarching characteristics of those networks affect their small-scale features. One of his results quantifies how the total number of nodes and edges in a certain class of large networks constrains the number of nodes that can be disconnected from the others.
Quanta Magazine recently spoke with Loh about his approach to coaching, why he enrolled in improv comedy classes and the reason he often runs from place to place. The conversation has been condensed and edited for clarity.
What was it like growing up in such a math-oriented house?
When I was around fifth grade or so, my father found a book of interesting math puzzles. These were not like “Do 100 arithmetic problems as fast as possible.” They were problems that made you think outside the box, like, you have six toothpicks, can you line up the six toothpicks so that there are four triangles made? The solution is that you make a tetrahedron. Which is not what you expect, right?
That introduced me to the practice of thinking about something for a while and then finding out, “Oh, my gosh, you do it that way.”
You started participating in math competitions in middle school and you’re still involved with them today. What do you like about them?
I found out that you could meet interesting people, and I found out that the problems were insanely hard. That actually appealed to a certain element of my personality. I really love people. I love doing things to help people. And I like something that’s supposed to be ridiculously hard and feels good if you solve it. So that’s actually what drove me onwards.
What about the competitive part? Are you drawn to that?
I’m not competitive in the sense of trying to beat other people, but I’m very competitive with myself. Even crazy things like, if Google Maps tells me it’ll take me this amount of time to walk from here to there, I might challenge that just because I feel like it.
When the Mathematical Association of America was thinking of naming you head coach, you warned their executive director that the team would probably do worse with you in charge. Why did you think that?
The young people on the IMO team are very interesting people that we have a chance to touch. They are people who could lead science, technology and innovation in this country in the next few decades. My goal as coach wasn’t going to be to win, it was going to be to maximize the number of them that I read about in The New York Times in 20 years.
Why is nurturing future leaders at odds with training to win the IMO?
Because if you put your mind on a finish line, and the finish line is going to happen within two months, you’re going to calibrate for that final sprint. For example, if I have two months left before the International Math Olympiad, what should we really be doing? We should spend all day and all night helping the students know competition tricks inside and out, so that when they get to the actual exam, they’ll just tear it apart like robots.
So if you don’t train the team to tear apart the test, what do you do?
We use our time together to also expose them to what people do with all of these math skills. I organize evening seminars where people talk about the kinds of math or applications they’re doing now. You might have somebody talking about quantum algorithms for factoring numbers even though there’s not going to be anything about quantum mechanics on the International Math Olympiad.
How else does that different mindset manifest in your training?
The most obvious change is we invite people from other countries’ teams to train with the United States. We teach them all the secrets. We treat them as our own. And we just have fun for three weeks learning all kinds of crazy mathematics.
That’s unusual, to share your training secrets with the competition. Why do you do it?
If you let 60 Americans see and work with and build relationships with their peers who are going to be leaders from other countries, this is extremely valuable. If I was a student, would I want this? Of course. Because if I was a student, one of those 60 people, I will be pretty sure I’ll never make the International Math Olympiad team — what’s my chance of being one of the top six in the U.S.? Instead of sending six people to the IMO, we bring the IMO to 60 people.
When you first coached the winning team, all the participants were male and you acknowledged the gender gap. Your winning team from 2019 was also all male. What could you be doing to achieve better gender balance?
You need to make sure there are enough people who are trying to pick up these very unusual skills that are in the math Olympiads. When I think about the issues of diversity, I think about what is involved in getting people interested. So when I give a talk, I can tell who feels comfortable and who doesn’t feel comfortable. And actually one of my goals is to go and try to help the people who look like they don’t think they belong and to help them feel like they can.
Besides coaching the IMO team you’re also a successful mathematician focusing on extremal combinatorics. What drew you to that specific field?
I think I like retrospectively obvious observations. I just thought it was beautiful to be able to bring in some different angle, and suddenly, because of that different perspective, you’re cutting through it. It’s like if you try to cut wood and go against the grain, it’s a totally different experience from going with the grain, and pop, the whole thing falls apart.
So how does extremal combinatorics do that?
Say you’re trying to make some kind of a network that has some complicated properties, and we just can’t imagine what it would look like. Sometimes it turns out that you can’t make it, but you can prove that it exists by using probability. I’ll show you how. Give me a decimal between zero and one.
How about 0.2?
So 0.2 is what’s called a rational number. It’s two over 10, or one over five, we can write it as a whole number over a whole number. But it turns out that if you pick a random number between zero and one, the probability it’s rational is zero. This is because there are way more irrational numbers than rational numbers. So what’s kind of funny is that you didn’t provide me a number — an irrational number — that is actually the most common kind of number there is.
In the context of extremal combinatorics, you can use probability to show that networks with certain properties actually exist, even if you might have a hard time coming up with an example of one yourself.
I’ve read that you take improv classes to improve as an educator. Does that training help with actual mathematics?
In improv the principle is if you start doing something, you’re not supposed to suddenly go and say, “Oh, no, sorry, that was wrong.” Solving a math problem, it’s also the same. You can’t just sit there and say: “I don’t know if this idea will work. I don’t know if that idea will work. I’m not going to try any idea.” No, you’ve got to dive in. You have to already have the attitude that “I don’t know where this idea is taking me, but I’m going to push it all the way through.”
So you’d never drop a problem?
If it’s a math problem, it’s hard to get something that would make me drop it, unless somehow something was proved that said that it is not possible. To me, “drop” is a really strong term. Because what if in 10 years a new technique developed? It’s a new weapon, you should try it.
You also spend a lot of time teaching math to young kids and working on other outreach projects to get more people interested in math. What motivates you to do this?
I would say that when it comes to teaching, there’s a big problem that we haven’t solved yet, which is how to solve the issue of mathematics not necessarily being something that everyone thinks is great and everyone wants to do. That’s a big problem.
Mathematics is the heart of what helps people think. If you want to live in as reasonable a society as possible, it helps if everyone’s comfortable reasoning. And I’m not talking about mathematics from the point of view of sines and cosines. I’m just referring to mathematics from the point of view of logic. But it’s hard to learn logic in a vacuum.
Does your devotion to outreach, as opposed to pure mathematics, affect your reputation in academic circles?
I would not be surprised if that affects how I’m perceived. But at the same time, that is not necessarily the thing that I would use to decide what I would do. I decide what to do based on what I think is going to have the most impact. And I don’t mean this in any disparaging way to people who don’t choose to do these things.
I’m actually lucky that in my field, we’re quite close-knit. I think that virtually everyone in my field knows what I’ve been up to. And we also have mutual respect, in the sense that I have tons of respect for people who spend their time primarily proving theorems. I think that’s great.
Finally, I’ve heard that you often run from place to place. Is that because you’re always racing Google Maps?
It’s because I’m always trying to milk every second to do something. And then, by the time you finish, you realize that there’s no time left to work. So you run. I always think: just one more task. And by the time you’re at that point, you say, “Oh, no, now I need to hurry.”