Jordan Ellenberg is the John D. MacArthur and Vilas Distinguished Achievement Professor of Mathematics at the University of Wisconsin, Madison. He works in arithmetic algebraic geometry, and is well-known for his research connecting geometry, number theory, and topology. He is highly awarded, and is a Fellow of the American Mathematical Society, as well as a Guggenheim and Sloan Fellow.
Besides his excellent research credentials, Jordan is a New York Times Bestselling author of the popular mathematics book How Not To Be Wrong and the novel The Grasshopper King. He is a much sought after speaker with dozens of invited addresses. We are excited to have him give the Public Lecture at this year’s Canadian Discrete and Algorithmic Mathematics Conference held at Ryerson University this June.
AB: When did you first realize you had a talent for mathematics?
JE: I was one of those kids who was interested in mathematics from a very early age—as long as I can remember. I think, however, there is a stereotype that all mathematicians are like that. But it’s really not so.
AB: Was there a person or teacher who influenced your early mathematical career before college?
JE: My parents influenced me and they are both statisticians, so it was a “mathy” household already. In addition, there was Eric Walstein who was a teacher in Montgomery County, Maryland where I grew up. He made a habit of travelling around the county finding young kids who were mathematically advanced and working with them one-on-one. I think I might have been the very first kid who worked with him. Several mathematicians worked with him such as Jacob Lurie at Harvard.
AB: And you were also involved in the Putnam competitions?
JE: Yes, that was at college. Again, it is a stereotype that doing contests is identical with doing mathematics. Those are different things. There are great mathematicians who are not into contests, and there are people into contests who are not interested in making new mathematics. I was involved in the International Mathematical Olympiad in high school and was on the Montgomery County mathematics team.
AB: How did you end up working with Barry Mazur at Harvard for your doctorate?
JE: When you start graduate school, you don’t really know anything about mathematics or the kinds of mathematics you like. I think that anyone going into graduate school could go into any kind of mathematics. I don’t believe people have a taste for a certain kind of mathematics as they would have a taste for a certain kind of food. I think you acquire those tastes for your field later on.
When I began graduate school, it was the mid 1990’s and it was the age of Andrew Wiles. Everyone was going into number theory. Many of us wanted to learn algebraic number theory and the mathematics around Fermat’s Last Theorem such as elliptic curves.
AB: Your research is on arithmetic algebraic geometry and number theory. Would you provide a summary of your research program in layperson’s language?
JE: My work has spread out a bit over the years, but at the very center is the same question that has been the same for thousands of years: the Diophantine question, which asks whether equations have integer solutions. The answer to that simple question is mostly a complete mystery. Over the years, we’ve mildly diminished the amount of mystery around that question.
AB: What research topics are you working on now?
JE: I am working on several things. Today I am finishing a paper with a student of mine about rational points on curves, which is a classical subject that has many open questions. We are getting some new results using ideas of Minhyong Kim who is at Oxford and is a wonderful, extremely original mathematician. We are applying his techniques in areas it didn’t apply to before.
AB: I noticed you a part of the Wisconsin Institute for Discovery, where you are a part of the Machine Learning group. How did you get involved in this group and what does it do?
JE: We all have the drive to be interdisciplinary and connect with scientists and even non-scientists. It’s hard to make that work. We are lucky that we have a group of engineers, computer scientists, statisticians and mathematicians that talk about these problems. On the one hand, they are engineering problems about making things happen, but the problems are fundamentally mathematical in nature. It is incredibly fun for me hanging out with these guys and I think of myself there as an honorary graduate student. I am constantly learning from them.
There’s a completely new field of statistics being developed right now. It’s very exciting, and it’s my desire that mathematicians be in the room while the subject is being created. The subject will be better if there are people from classical mathematical fields such as geometry and algebra involved. I want to be part of the process.
AB: Ingrid Daubechies said something similar when I interviewed her, how it is important for mathematicians to be part of the development of machine learning.
JE: Ingrid was there from the beginning! I’ve only been learning about this subject over the last five years or so.
AB: You are well-known for your popular mathematics writing. Would you tell us about your New York Times Bestselling book How Not To Be Wrong? What inspired you to write it and what were your goals?
JE: I thought about writing the book for a long time. Since I was in graduate school, I’ve been doing journalism, and writing shorter pieces. Going back further, I did many writing courses in college and have a Master’s degree in creative writing from before I began my PhD. I’ve written for Slate and other magazines.
Then I started to get into writing longer pieces. My editor Nick Thompson from Wired took a chance on me and let me write a longer feature where I wrote five or six thousand words. In mathematics, everything connects to everything else. There is a natural, discursive way the subject works which you cannot capture in 1,200 words. In that many words, you have one idea which you express and then close the door. Mathematics doesn’t come in these discrete, quantum pellets.
I was interested in the idea of what it would look like if I had room to stretch out in a longer form. Then I had a sabbatical. The great thing about a sabbatical in our universities is that allows you tackle a more ambitious project, which is not the same thing that you’re presently doing.
I wrote a long proposal and I sold the book in advance with the help of my literary agent. When I signed the deal, the publisher said you have a deadline of eighteen months. I thought I could do it in six months but it does take eighteen months as I learned. I didn’t have an accurate concept of how much work it would be.
AB: You’ve also written a novel The Grasshopper King that came out in 2003. How did that come to be and what was the experience like writing fiction?
JE: It was written even earlier than 2003 when I was in graduate school in creative writing in the 1990s. The book sat in the filing cabinet for about ten years. I wanted to explore writing a novel and I’m happy with what I wrote. While I was writing the book, I missed mathematics, so it was a great experience as it helped me realize I want to be a mathematician.
AB: Do you think you have another popular mathematics book in you?
JE: Yes, I can see that, but I don’t know what it will be. I have another sabbatical coming up, so I may write it then. Now that I know how to do it, I can do it faster. Famous last words.
AB: Mathematics can inspire strong and even negative feelings from the public. As a researcher and author, what are the main messages for people that they may not already know?
JE: Even at a very basic level, I would like people to know that there is a field of mathematics. When you consider how mathematics is viewed, it isn’t people not grasping some fine points of algebra. Most people literally don’t know that there is a mathematician who’s alive and not a dead person in a robe. Even the fact that mathematics still exists and it is thing you can do, and that there are thousands of people working today in the field, is a message to take away.
Another message, as I describe in the book, is that there is no magic, mathematical way of thinking. People are already thinking mathematically about things. What I am doing in the book is not saying how to think mathematically, but rather I am showing you how you are already seeing the world mathematically.
AB: There seems to be a trend in films to showcase mathematicians such as in Hidden Figures and The Man Who Knew Infinity.
JE: The movie Gifted coming out in a month has me in the trailer.
AB: What is the movie about and how did you get involved?
JE: It’s about a child math prodigy. I am writing on the board for a few seconds at the end of the movie.
I wrote an article in the Wall Street Journal about how we should think about children who are very advanced in mathematics. The producers wanted to talk to people who know something about the topics in the movie, and they spoke to several people including me. I spoke to the director Marc Webb and they needed someone to be a math professor in the movie. I didn’t have to develop many acting skills to portray someone giving a math lecture!
[AB: Here is the trailer. Jordan is at 1:31, writing on the whiteboard.]
AB: I’d like to close with looking forward. What would you say are some of the major directions for mathematics in the future?
JE: I think one direction is the development of machine learning and data science. It will be a mathematical subject but it won’t be part of mathematics, and it will be called something else. I think it is a very exciting development and every mathematics department will have people working on these kinds of topics.
The other thing is that we really don’t know what kind of mathematical tasks computers will be able to do for us. I am not one of those people who think that computers will be replacing mathematicians. I do think that there will be certain mathematical tasks that will be considered “just computation” in the future. It will be similar to the situation in the 19th century, where you could publish certain results in journals that now you can just ask a computer to do. We will change the definition of what is mathematics and what is computation. Humans are good at racing ahead of what machines can do. There will be new things we can do that we couldn’t do before without computer aid.