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# Ellenberg Inspires Teens at 2024 Arnold Ross Lecture

“It warms my heart to see so many people dedicating their summer to number theory!” Ellenberg told the audience at the 2024 Arnold Ross Lecture, which he delivered at the Program in Mathematics for Young Scientists (PROMYS) math camp.

The near-full room held 80 high-school campers and 25 undergraduate counselors, plus PROMYS faculty, PROMYS for Teachers participants, and Boston University (BU) faculty members.

“How many of you know the card game Set?” he asked. Hands waved.

Held at BU, Ellenberg’s talk was part of an AMS tradition: a lecture series for talented high-school mathematics students, dedicated in 1993 to Arnold Ross for his many contributions to the development of mathematical talent, which includes an Ohio State University summer program that survives to this day.

In 1957, Ross launched a multilevel summer program for gifted high-school students as chair of the mathematics department at the University of Notre Dame. When Ohio State hired him in 1963, Ross brought the youth program with him and ran it every summer until 2000, giving the number theory lecture each morning.

Number theory is the bread and butter of Ellenberg, who is John D. MacArthur Professor of Mathematics at the University of Wisconsin-Madison, a Fellow of the AMS, and the author of two best-selling nonfiction books, *Shape* and *How Not to Be Wrong. *

“I really like talking to teenagers because they’re very willing to engage with you,” Ellenberg said. “If they disagree with something you say, they’ll tell you! And that automatically raises the energy level in the room.”

“Certainly, I pitch it differently than I do to peers,” he said. “For one thing, I don’t have to establish that the problem is connected to some existing body of research they already know is important or valued, because teenagers are *not* already in the field, and they don’t have prebaked ideas about what’s important, you just have to convince them directly that the thing you’re going to talk about is interesting!”

In a vigorous hour, Ellenberg introduced the “cap set problem,” a popular question in combinatorics which is related to large subsets of integers containing no three terms in arithmetic progression, and to the card game Set.

He engaged the audience, asking them to work together on a simpler example so they could get a feel for the problem. Most everyone called out the same answer—“4!”—to Ellenberg’s delight.

In closing, Ellenberg described a project with scientists at Google DeepMind where his team tried, with some success, to coax a computer into producing interesting examples of cap sets: hypothesis generation by machine. This was his springboard to discuss the near-future role of machine learning in mathematical practice.

Students volunteered their experience asking math questions of the program ChatGPT. “BAD,” one emphasized with a grimace.

Ellenberg noted that machine learning has its uses, though some tasks may be “parochial.”

“The goal is for me to get a new idea about the problem,” he told the audience. “Any mathematical way that we can get ideas is a great thing. There’s a very long tradition of computers assisting mathematicians, and I don’t find that the human mathematicians have lost intuition.”

“I found it fascinating how Jordan Ellenberg connected the game Set, which I’ve enjoyed since childhood, to number theory conjectures,” said Emily, a 16-year-old PROMYS student from New Jersey, after the lecture. “As a STEM enthusiast with a passion for both math and computer science, I was particularly intrigued by the intersection of these subjects.”

She added, “Ellenberg’s use of machine learning, which is not currently highly effective at solving math problems, to tackle mathematical conjectures was especially cool to see!”

Luna Dole, an 18-year-old PROMYS student from Texas, said, “I especially enjoyed the part where he described the technique for generating solutions using AI. It’s one of the most hilariously absurd concepts I have ever heard of and the fact that it turned out to actually work is delightful to me.”

PROMYS Executive Director Li-Mei Lim praised the lecture. “I think it’s great for students to see how seemingly simple things can actually lead to interesting questions, whose answers are still unknown,” she said.

Sixteen-year-old Lisa from California summed it up. “I thought the way Jordan related a simple yet well-known game of Set to a fundamental problem in number theory reflected a core value of PROMYS: Think deeply about simple things.”

Not coincidentally, “To think deeply about simple things” was a motto of Arnold Ross.

*Watch Jordan Ellenberg’s 2024 Arnold Ross Lecture on the AMS YouTube channel at https://www.youtube.com/watch?v=09TUwrgnKZs.*

## Credits

Figure 1 is courtesy of Elaine Beebe.