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A Number Theorist Who Connects Math to Other Creative Pursuits

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In Quanta Steve Nadis interviews Jordan Ellenberg:

here are many different pathways into mathematics,” said Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison. “There is the stereotype that interest in math displays itself early. That is definitely not true in general. It’s not the universal story — but it is my story.”

That account was backed up by a biostatistician at the University of Pennsylvania — his mother, Susan Ellenberg. “Jordan recognized numbers before he could walk,” she said. “We’d be going someplace with him, and he’d start to call out numbers, and his father and I would have to figure out where he was seeing them. Each night, he’d ask me to teach him something new about math.” When he was in second grade, a local teacher began taking him through the high school math curriculum. Ever since, he’s been preoccupied with mathematics — though not exclusively so.

After graduating from Harvard University in 1993, Ellenberg completed a one-year master’s program in fiction writing at Johns Hopkins University, where he wrote a novel that was published a decade later, titled The Grasshopper King. But he always felt that he would eventually return to mathematics, and in 1994 he entered a doctoral program back at Harvard, pursuing research under the supervision of Barry Mazur, a number theorist.

“Barry was a great adviser and a very learned guy,” Ellenberg said. “One of the things he showed me is that it’s OK to be interested in things other than math. Through him I saw that being in a university isn’t just about being in the math department, but rather being part of a whole world of scholarship.”

Ellenberg has taken that view to heart, finding mathematics to explore in everything from internet fads to voting rights. He has interacted and even collaborated with colleagues from many different fields and departments, while keeping up his writing — academic papers for math journals, and popular articles for newspapers and magazines. In 2001, he started writing a column for Slate called “Do the Math.” Many entries are not typical mathematician fare, such as “Algebra for Adulterers,” “Cooking the Books on Virginity,” and “What Broadway Musicals Tell Us About Creativity.”

His latest book, Shape, is all about geometry — though, as you might expect, it departs significantly from the traditional geometry of your high school days. Proving the congruence of triangles and the like, he said, bears little resemblance to the work of modern geometry. In the book’s introduction, Ellenberg confesses that it was a curious subject for him to have taken up: “Reader, let me be straight with you about geometry: at first I didn’t care for it.”

Quanta spoke with Ellenberg earlier this month about geometry, electoral math and creativity. The interview has been condensed and edited for clarity.

When did you first realize there was something special about math?

When I was 6 years old, I was in the living room, gazing at the rectangular pattern of holes on a speaker where the sound comes through. I noticed there were 6 rows of 8 holes and, equivalently, 8 columns of 6 holes. I knew that 6 × 8 equals 8 × 6. But at that moment, I grasped that this was a fact about the world, not just a fact from the multiplication tables. Mathematical knowledge, I realized, was something that existed on its own — something you could directly apprehend — and not just something you were taught.

That, for me, offered an early glimmer of the power of mathematical thinking — and the emotional force that comes with it. As teachers, we aspire for every kid to have that kind of experience of mathematical knowledge.

Mathematics is a diverse field. How did you decide to focus on number theory?

I went to graduate school not really knowing what I would work on. It was just after Andrew Wiles proved Fermat’s Last Theorem. There was so much energy and enthusiasm about number theory at that time. It seemed to be the most exciting thing going on.

Students often ask me: “How do I figure out what area of math is right for me?” I tell them that it’s all interesting. Every field of research has deep wonderful ideas. You just have to see what you fall into. And wherever you fall, there is excitement to be found.

Of all the possible subjects in math, why did you write a book on geometry, especially when you admit to having a mental block when it comes to visualizing things?

It’s true, I didn’t really take to high school geometry. There was a certain style — the Euclidean “theorem, statement, proof” approach — that did not vibe with me. That approach is certainly a part of geometry, but it happens to be a tiny part.

It’s also true that I have difficulty with some geometric things. For example, when you have to put a credit card into a machine, I can’t follow the diagram and instead end up trying all four possibilities. If I’m on the first floor of my house and am asked about the layout of things in the room above me, I can’t really picture that. But it turns out that those skills aren’t so important when it comes to doing geometry.

Even though I steered clear of geometry when I was young, I later learned that you can’t maintain a dislike for any part of mathematics because all of its branches touch each other.

You also like to find mathematical connections even among ideas that don’t seem too mathematical, like pondering how many holes a straw has. Why bother answering that?

Well, it’s kind of an internet craze [with more than 60 million hits on Google]. It goes viral all the time, and you may wonder why people are so captivated by such a weird question. I’d say it’s actually a deep mathematical question, not a triviality nor a cut-and-dried matter. You could say one hole or two holes — or zero holes if you think about taking a rectangular piece of paper (with no holes in it) and rolling it up. It’s a way of getting people to understand topology and homology groups, which involves classifying objects based on the number of holes they have.

It turns out there is a mathematical answer to this: Topologists would say the straw has just one hole. But the point is not just to give people an answer, but rather to show them why it’s an interesting problem. Although this question is settled, many of the things that are now settled in mathematics may not have been settled 100 or so years ago. People have fought hard over almost every single conceptual advance.

In 2019, you and 10 other mathematicians signed a brief about gerrymandering that was submitted to the Supreme Court. What does math have to do with that? . . .

Continue reading. There’s more.

Written by Leisureguy

29 May 2021 at 9:30 am

Posted in Books, Daily life, Math

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