# Samit Dasgupta '95: Algebraic Number Theory

*As a senior at Blair in 1995, **Samit Dasgupta** won fourth place in the **1995 Westinghouse Science Talent Search** with a project on the Schinzel Hypothesis. He has been studying Number Theory ever since, as an undergraduate at Harvard, a **graduate student at UC-Berkeley**, and as a **professor at UC-Santa Cruz**. He spoke with Ted Jou '99 about his research and his memories of Blair:*

*How long have you been working in Algebraic Number Theory?*

My focus in Algebraic Number Theory began as an undergraduate. I have been interested in Number Theory for a long time. In high school I went to math summer camps, and there was one at Ohio State where I learned the basics of Number Theory. When I was in high school, I was more interested in Analytic Number Theory, and I did my Senior Research Project in Analytic Number Theory. When I got to college I focused more on the Algebraic side, which I guess was a fine-tuning of my interests. So I have known for a while not only that I wanted to be a mathematician but that I wanted to be a number theorist.

*How would you describe Analytic Number Theory vs. Algebraic Number Theory?*

Analytic Number Theory uses methods of Calculus, more specifically Complex Analysis, to answer number theoretic questions such the distribution of the prime numbers. When you’re actually trying to prove a theorem, you’re doing a lot of Analysis, which is higher level Calculus. In contrast, in Algebraic Number Theory you’re not usually doing Calculus, where you’re taking derivatives and integrals, but you’re using more algebraic-type structures. The algebra involved is just a higher level of the Algebra that we have all learned in high school. So the difference between analytic and algebraic number theory has to do with what type of foundational mathematics is applied to solve number theoretic questions.

*What does Number Theory look like on a day-to-day or week-to-week basis for you?*

It depends on what stage of the process you are in on a project. On any given day, you might be working on several different projects at different stages. Usually at the beginning of a project, what happens is you are reading a paper or you go to a talk or somehow you get exposed to a problem, and you realize that some expertise that you have might shed light on a problem, either with a solution or just some partial progress – you might not know until you carry out the research.

The research part of it then becomes just thinking. Once you realize that there is a problem that you might be able to say something about, you apply the methods that you’ve learned to see if they can be used to attack that problem, and some of that might involve doing literature searches and reading papers that are either about that topic or about some method that you think might apply to that topic. Eventually you just work on the problem, and you sit with a pad and a pen and go to work. You get stuck sometimes and you go talk to people for ideas, but that is basically what it comes down to.

I don’t want to glamorize it, but **usually there is an “aha” moment when you get the key insight that will enable you to prove the result**. It happens in jumps and steps. In math research, you don’t necessarily make clear progress in each day of work – it's very difficult to have the patience for it. On the one hand you have to continually work and work and keep at it with a lot of determination, but you can go months and months with basically no progress or no visible signs of progress. Obviously you’re making progress – you’re thinking and you’re increasing your understanding of the difficulties of a problem – but you might not actually come up with a solution after months and months of thinking about it. And then finally, in one moment it comes, and it kind of feels like if I just had that moment months earlier then I would have saved all that time. But really it’s all the months and months of putting the time into it that enables you to have that “aha” moment. For most papers there’s probably one or two moments you can look back at and say that’s the breakthrough that I had that enabled me to write this paper.

*I noticed on your **website** you had a list of some conferences that you helped to organize. Can you tell me about that?*

The main thing that I organize (twice yearly) is a conference called the Bay Area Algebraic Number Theory and Arithmetic Geometry Day, which is a conference that takes place between UC-Santa Cruz, Berkeley, and Stanford. The location rotates between the three universities, and we invite all the Number Theory and Algebraic Geometry faculty and postdocs and grad students from the three universities to come to these talks. We really try to gear the talks towards the graduate students. The idea is to bring the community together from the three universities, and also to give students – who aren’t even sure if they want to do number theory in graduate school – a sense for the type of research people are doing. We have one coming up in December at UC-Santa Cruz.

*What advice would you have for undergraduates or even high school students with an interest in math and are thinking about pursuing academic math?*

There are a lot of things to say ... My first advice these days is to get a very solid foundation in the basics of Mathematics. Students these days seem to know at least the words for really advanced topics at a young age that before we only thought of as graduate or even postgraduate material. Now, even undergrads are talking about them. Obviously, there’s only so much anyone can process and understand at a young age, and if they’re learning these advanced topics at least superficially, that means there is something more fundamental and basic lower down that they are skipping or glossing over or not giving adequate time to.

So my advice for younger people is actually to make sure you understand the basics well before you move on to more advanced topics. **It’s not a race. It’s more important to understand something really well**, the basics, so that when you get to more advanced topics you’re very comfortable, than it is to think about advanced topics from a really young age and not be able to make progress because you don’t understand the background or the history behind why people are thinking about these advanced topics. If you want to be successful, it’s important to have a solid foundation.

The other thing, very generally, is the kind of mindset you need to do academic research. You have to be very determined and you have to be willing to stick to something for a really long time. As I alluded to earlier, you have to think about research problems for a really long time.

One reason I mention this is that people get into mathematics often because of contests, and that’s true for me too – I took part in a lot of high school and college math competitions. These are the types of competitions where you’re given a problem that you have to solve within 20 or 30 minutes at most, if not 5 or 10 minutes. If you don’t solve it in that time, you just move on to the next problem, and whoever can solve the most of those problems 10 to 15 minutes at a time wins. Math research is not like that at all. You’re usually thinking about things for months and months at a time.** It’s not necessarily the quickest mathematicians who are most successful by any means.** The most successful mathematicians are those who are able to have keen insights into deep problems, and those come about by ruminating on things for really long periods of time and becoming an expert on something.

*I wanted to ask you a few questions about your memories of Blair and the Magnet. Is there anything you learned or did in the Magnet that you still carry with you today?*

I can say one thing, which is that the Magnet Program was an incredible training ground for me. I can tell you that I went to Harvard for undergraduate after Blair, and **I thought Blair was harder than Harvard**. It really prepared me for college and afterwards, so that’s one thing.

The other thing is that I really took for granted the diversity that you had at a school like Blair, both racially and culturally and equally importantly, intellectually – just the different ideas that people have and different interests that they have. The person next to you might be interested in computer science and the person on the other side might be interested in poetry. And there were so many ideas just from your friends, not to mention all the amazing teachers that we had. I was kind of used to that, and I took it for granted. Unfortunately, once you’re older, you’re often in situations where you’re not surrounded by such diversity.

*Are there any particular teachers that stand out in your memory?*

Since I was really focused in math, of course Eric Walstein had a big influence on me, even more through working with him on the County Math Team than through the courses that I took. However, I remember that the biggest influence on me was really my Computer Science courses. Even though I didn’t follow in it in college, I really loved computer science courses in high school. I learned the most from them, and they influenced the way that I thought about problems in a way that has been very valuable to me.

The two teachers I really remember are Ms. [Mary Ellen] Verona and Ms. [Susan] Ragan. They were really great teachers, and I remember I learned things in high school that computer science majors in Harvard didn’t learn until their second or third year. So that was the level of advanced teaching we had in the Magnet program.

Sometimes you don’t appreciate these kinds of things, but for me I’ve been very cognizant of the extent to which that training has helped me in my career and in my life in general, even though I didn’t go into Computer Science ... It helped me understand how to think about problems from a problem-solving point of view and how to create algorithms and structures to attack the problem. That’s something that can help you not only in mathematics but in your life in general: **How to break down a problem into pieces, and compartmentalize and how to do things efficiently –those are all important things that serve one well both in one’s career and in life.**

*Can you talk generally about your research?*I study Number Theory, which is a branch of pure mathematics. Within number theory I study Algebraic Number Theory. Broadly speaking, Number Theory has to do with two types of problems: One is finding the solution to equations in the rational numbers, and the other is studying the distribution of prime numbers.