Mathematician

Dr. Samit Dasgupta’s passion for number theory blossomed when he was just 12 years old. He has pursued mathematics ever since, going on to place fourth in the 1995 Westinghouse Science Talent Search as a high school senior with a project on Schinzel’s Hypothesis. Today, he studies algebraic number theory at Duke University. Check out his interview to learn about what it is like to be a math professor, sharing the beauty of mathematics with the next generation of bright minds.

STEM to the Sky

Dec 23, 2020

- Can you introduce yourself and briefly describe what you do?
- What exactly is algebraic number theory?
- At what point did you decide to pursue STEM, in particular, math?
- What do you enjoy the most about your job?
- What is one change you have noticed in the math field overtime?
- What is one thing you wish you were told before you started your career?
- Can you describe a “day in the life” if we weren’t in a pandemic?
- Besides the technical math skills, what other skills would you say are important for a role like yours?
- What would you say to a student who is interested in pursuing math as a career?
- What are some advancements that you think will happen in your field in the future?

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My name is Samit Dasgupta, and I’m a mathematician at Duke University studying algebraic number theory

Being a professor has three different aspects:

**I. Research:** There are a lot of classical questions that have motivated branches of research in number theory, and I’m working in one of these directions and trying to push the frontier.

**II. Teaching:** There are many levels of teaching. I teach undergraduate and graduate students in classes that are usually upper-division. I also work with graduate students and postdocs.

**III. Service:** University service consists of working on committees on campus, in ways that we can make the university function better. There’s also service within the profession. I’m an editor at a few journals. I handle submissions of journal papers for people’s research papers, helping decide whether or not they should be accepted to the journal and making sure they’re well-written and up to the editorial standards. Then, there’s broader service to the community; I serve as a math judge for the Regeneron Science Talent Search, a well-known science competition. Additionally, I give lectures sometimes to the community and more often within the university to different clubs that are interested in my research.

Algebraic number theory encompasses a lot, but one of the central motivating questions is:

*How do we solve equations with solutions that are rational numbers or integers?*

The first example of this that we usually see in school is the Pythagorean equation: x^2 + y^2 = z^2, where x, y and z are integers. This is an interesting equation because it has a geometric interpretation. If x, y, and z satisfy the relationship, x^2 + y^2 = z^2, then that means that there’s a right triangle where the legs have sides x and y, and the hypotenuse has side z.

We’re interested in knowing the answer to the geometric question, and then we turn it into an algebraic question. We might be interested in what kind of right triangles have integer sides. There are some famous solutions to this equation like 3^2 + 4^2 = 5^2, and 5^2 + 12^2 = 13^2, but you might ask: can we generate all the solutions? Can I write down a formula that tells me all the triples X, Y, and Z, such that x, y, and z are positive integers, and x^2 + y^2 = z^2? The answer is yes.

This is one of the first examples of a problem in the field that’s called arithmetic geometry. The solution to this question is fascinating, and you use geometry to solve it. First, you take x^2 + y^2 = z^2, and if you divide by z^2, you get (x/z)^2+(y/z)^2 = 1. If you graph this in the plane, then that’s just a circle. So what you’re really asking is: what are all the points on that circle that have rational coordinates? And then use some nice geometry to solve that question. It’s a nice blend of geometry, and algebra, to give a solution to this equation. And if you work it all out, you actually get a total solution to that original Pythagorean equation.

For more complicated equations such as cubic equations, which are still big, open questions within number theory today. We have some proposed solutions, but it hasn’t been proven that these algorithms that exist to try to study these equations always work. Those are the type of questions that number theorists like myself think about.

I was actually really interested in humanities/literature as a kid, reading books and writing stories. In third grade, I was exposed to some computer programming, studying some aspects of fractals and bifurcation, and I wrote some computer programs related to the Mandelbrot set. I became interested in STEM then, but I don’t think I was set on being a mathematician or even necessarily in STEM at that point.

When I was 12, I went to a 8-week summer number theory program at Ohio State University called the Ross Young Scholars Program. It was my first exposure to number theory, and I totally fell in love with mathematics and the whole concept of doing proofs and the overall material.

I started getting more into math contests, getting better and better every year. By my senior year in high school, I was one of the winners of the U.S. Math Olympiad. These contests really got me into the community of kids doing math. I went to a program after my junior year called the Research Science Institute at MIT. I kept doing math and science programs every summer, and later majored in math at Harvard. I did expand my other interests, spending a year doing some economics for a year after I graduated from college. But, even then I knew that I was going to go back to math.

This is my story, but it’s not the only story. There are lots of people that have very different paths to becoming a mathematician, some that go through art, music, or even magic. I know a mathematician who was a taxicab driver for a while!

There’s a lot of aspects in the profession that I really love, the first being just thinking about math. My favorite thing in the world is to sit on my couch, stare at the ceiling, and think about these beautiful, deep structures that govern classical number theory questions. But I’ll be honest with you, if that was all I did, it would be a lonely and I think difficult profession to handle because that sounds like a pretty isolating thing, right?

The other aspects of the profession are really important. Almost all of my papers are collaborative. I love getting together with mathematicians and just talking about math. There’s a social aspect; I’ve made lots of great friendships on every level with my collaborators, my peers, my advisors, and my students. I really do love the interaction with students and the teaching. I love transferring my passion onto the next generation, which is something I’ve learned from my teachers.

Something else that I really value is that, personally, it’s fun. It’s fun interacting with students. It’s fun teaching things. It’s fun presenting some mathematics that you find is beautiful, and it’s fun seeing someone learn something for the first time and appreciate its beauty. It really means a lot to me when students write to me saying that my course inspired them to go on to a career in STEM.

One positive change over the 20 years that it’s been since I graduated with my undergrad degree has been the increased participation of women and minorities within the role of STEM fields, but in particular math. I can say that it is a very positive and welcome change. Accompanying that, there’s been a culture change within the field to be more open and to be more welcoming and accommodating to typically underrepresented groups. I’ll say that the change doesn’t happen fast enough, and I think we’re all doing our part to try to make the field as open as possible. I’d like to encourage other people to take part in that. That’s a change and something that needs to continue to change.

Even though math seems pretty cut and dry like there’s a right answer or a wrong answer, it’s certainly not the case. What makes math interesting is not just getting the answer, but how you get the answer. That is the proof, and the elegance of the proof is very important.

Another thing is that you have to love what it is that you’re studying and really find beauty in it. Otherwise, you’re not going to continue to have the passion to work in the field. The second aspect to that is to realize that it’s subjective and that there’s a culture at play. It’s good to get exposed to the current culture of the field to understand what types of things people think are interesting and why. You want to be able to think outside of the box and do something different that other people aren’t doing, but at the same time, you don’t want to be on an island where no one cares about what you’re doing. Art is better if it’s more widely appreciated, and I think that mathematics is an art in many ways. Trying to maintain that balance between those two factors, and realizing when you’re swinging too far in one direction, is something that I would have wanted to realize earlier than I did in my career.

I would typically drop my daughter off at school around 8:00 AM and come to the department. Because it’s the morning, I try to spend the first few hours focusing on research because that’s when you can get it done before the knocks on your door start happening. Once everyone shows up at the department, the day is usually a whirlwind.

Sample day:

8:00-10:00 AM ––––– Research

10:00-11:00 AM ––––– Teach a class

11:00-12:00 AM ––––– Meeting with Postdoc

12:00-1:00 PM –––––– Lunch with number theory group

1:00-2:00 PM –––––– Talk about research with visiting speaker

2:00-3:00 PM –––––– Editorial work

3:00-4:00 PM –––––– Listen to seminar

4:00-5:00 PM –––––– Meeting with grad student

5:00 PM –––––––––– Go home

The next day, you might do completely separate aspects of the job that I didn’t even mention. I’ll have office hours for my class and there might be a student who wants to meet with me because they’re struggling or interested in something beyond the course.

During the middle chunk of that day, there are always people knocking at your door, and you’re getting emails about letters that you have to write. One thing that people don’t appreciate, for example, is that the field works a lot with letters of recommendation. You’re writing letters of recommendation for people all across the board, whether it’s students, undergraduate students, graduate students, or tenure track candidates. So, especially during letter writing season in the fall, you spend several hours a week just writing recommendation letters. It’s really a lot of little things that come together that take up all your time.

Being an effective communicator is important. You have to teach, present, and speak about not just your research, but about general mathematical topics. Beyond teaching, you want to explain to other mathematicians why your research is important and what’s interesting about it through writing papers that are clear, understandable, and convey the beauty of the mathematical research you did. I think being an effective communicator, both orally, and in written form is really important.

Another skill is being able to work well with others. If you want to do collaborative research, it’s important to know how to not step on the toes of a collaborator and at the same time, bring out the best in both of you. You also have to know how to work with students. Most of my interactions, by the nature of the job as a professor, are with people that you’re in some level of authority over. It’s very important to understand how to interact with students in a healthy, professional way. To be honest, that’s something that we don’t get a lot of training in (we probably should)!

Math has a lot of applications beyond the academic world. Math is used all over aside from at research universities. Nowadays, there are so many tech or finance companies that are all using math in a huge way. I think when you’re in high school, you should just learn mathematics and decide later what aspect of what avenue you want to take to turn into a career.

The simplest recommendation is just to follow what you love. Make sure you’re studying and learning about things that you could see yourself being passionate about, or hopefully are already passionate about, whether it’s math or anything else. Unless you really have the commitment and the passion for something and realize that it is what you want to do, it’s hard to push through that. The people that are the most successful are the people that love what they do.

In terms of specific programs, there’s a program called Promise at Boston University, and there’s still the Ross Young Scholars Program that I went to at Ohio State University. There are tons of other programs out there too. I think doing extracurricular mathematics is really valuable because there are always different ways of thinking about Math beyond the traditional high school approach. You can’t go wrong if you’re spending your summers doing something that stimulates you academically within math, STEM, or really anything else. You’re only progressing, learning, and expanding your horizons.

It’s hard to know where the future is. The biggest conjecture in mathematics I’ll say is the Riemann Hypothesis. It is different from my field, but I would love to see that proven in my lifetime. It has been an open conjecture for hundreds of years and is so important in number theory.

Closer to what I think about is the Birch and Swinnerton-Dyer Conjecture, which has to do with the solution to cubic equations (elliptic curves). There are some disagreements on if the Riemann Hypothesis or the Birch and Swinnerton-Dyer Conjecture will be proven first, but I would love to see either. It would be the biggest breakthrough during my career!