*Dan Dugger is a professor in the Department of Mathematics at the University of Oregon. He received his Ph.D. in mathematics from the Massachusetts Institute of Technology and his B.A. in mathematics from the University of Michigan. Professor Dugger is a graduate of Project MEGSSS of Broward County, Florida, which used the mathematics textbooks that underlie the EMF Math online program. As a high school senior, he submitted an original research paper entitled "On the Homomorphisms between Finite Abelian Groups" to the 41st International Science and Engineering Fair, winning–among several other honors–the first place award from the American Mathematical Society.*

**When did you know that you wanted to be a mathematician? Who or what experiences influenced you in that direction?**

When I was in elementary school, there was a class for gifted students run by a remarkable teacher. Her idea was that the students should be in charge of their own education, so she had them each pick a subject they wanted to study. At first I studied space exploration, and then I studied whales. The next year I studied computers for a long while (this was in the days before anyone had a computer in their home!), and the year after that I switched to lasers. One day I was reading an explanation of lasers in a physics book when an older kid came along and in a tone of complete awe asked, "Danny, you're studying *physics*?" I sheepishly explained otherwise, and he went about his business, but I came away with the idea that physics was a difficult and impressive subject! So the next year, in fifth grade, I was determined to study physics. After that, I was convinced I wanted to go to Caltech and become a physicist.

At that point, I honestly had no idea that there were such things as mathematicians. I liked math, and I liked it when my dad showed me mathematical things he knew that were over my head, but I had no idea a person could make a career of it. Then in middle school I went into Project MEGSSS (the precursor to IMACS) and started learning from the *Elements of Mathematics* curriculum (the basis for the EMF curriculum). At that point I felt the wonderful beauty and poetry in mathematics. I had great teachers like Burt Kaufman and Ed Martin who were encouraging and really made this material accessible, and thanks to them I was hooked. So pretty much from then on I wanted to be a mathematician.

The important point that I want to get across is this: I remember fairly little of elementary school, but I remember every subject I studied in the four years of that gifted class. That strikes me as remarkable. And in a very direct way I see that class as having put me on a path that led me to my eventual career. It's really hard to overestimate the effect that good, caring teachers have on children.

**You attended the University of Michigan at Ann Arbor for your undergraduate degree. At what point did you start looking into grad schools? What were you looking for, and how did you decide on MIT?**

As an undergraduate, there were times when I was uncertain that I still wanted to go to graduate school, but I also didn't know *what* I wanted to do. A kind math professor grabbed me one day after I had taken his class and said "Are you planning to go to graduate school? Because you really should!" At his suggestion I participated in a summer research program at Michigan with another great professor. The next fall this professor sat me down and said, "Okay, here are the places you should apply to graduate school: Harvard, MIT, Princeton…" I was pretty shocked! I had no idea I would be able to get into schools like that.

So I applied for graduate school, although I was still not exactly sure that this was what I wanted to do. For a while I was convinced that I wanted to go to University of Chicago, but when I visited there, it didn't quite click for me. I remember quickly calling up MIT afterwards and asking to arrange a visit! Luckily, during that visit something *did* click. What I mostly remember was the very pleasant atmosphere among the graduate students at MIT. Seven or eight of them took me out to dinner, and I remember they spent the whole evening gossiping about the professors in the department. That doesn't sound like a good reason to choose a graduate school, but somehow it worked.

I think the real lesson here is that visiting schools is very important. It is quite difficult to know, or to predict, when things are going to click or not click. Every school is different, and different things work for different students.

**Tell us about your years at MIT. How did you choose your advisor, or is it the advisor who chooses the student? How do you decide on the topic for your thesis?**

I came to MIT determined to study a subject called commutative algebra. But once I was there, I found myself more and more in the company of topologists. There weren't many people studying commutative algebra at MIT, but there were numerous topologists–it was a very active and exciting community. My best friend was a topology student, and I always seemed to be hanging around the topologists and their interesting seminars, so at some point it seemed that I should just become one of them!

Unbeknownst to me at the time, other forces were also at work. During my first semester at MIT I took a seminar in topology run by a retired professor, and he decided that I should become a topologist. (He didn't tell me this at the time.) He somehow set things up so that I started talking regularly with a brilliant topologist named Mike Hopkins, and after a while Mike became my advisor. I never formally asked him to be my advisor, and he never formally invited me to be his student; it somehow just came to be. However, this is somewhat atypical. Most students just choose someone who is in the area in which they want to work, possibly having already taken a class from such a person, and simply ask. It's typical for the student to do some independent reading under the professor's guidance to see if things are a good fit before anything is set in stone.

Concerning choosing a thesis problem, most advisors will assign the student a problem based on the student's interests. Usually it is a problem the advisor has some ideas about, so that he can suggest a plan of attack. Some advisors, however, have the students find their own thesis problem. In some ways the latter is a better, albeit more challenging, technique because eventually in his or her career a student will have to learn to do this anyway. In my own case it was somewhere in between these extremes, but closer to the latter.

**Were there challenging times when you just hit a wall with the research? How did you keep yourself motivated?**

It is not just that *sometimes* there is a wall to get past; it would be fair to say that almost *all* of the graduate school experience is a wall. Let me try to explain. Everyone goes into graduate school assuming that when they write their thesis it's going to be fun and beautiful like the mathematics they fell in love with, but that is an idealized view. When you take mathematics courses as an undergraduate, you are presented with a very beautiful, developed theory. But it's important to realize that what you are seeing in these courses has been refined over hundreds of years–you are almost never seeing "raw mathematics" as it comes out of the ground, so to speak. Instead of polishing a beautiful diamond, a working mathematician spends most of his or her time digging in a mine, carving away at the rock. In addition, when you learn mathematics in a course you are carefully protected from all the false trails that mathematicians took to get there. Actual mathematics research has many false trails, or trails that lead a little ways but then peter out. Someone once described their life as a mathematician this way: "I spend my days writing lots of stuff on the blackboard, most of which turns out to be wrong." This description, although exaggerated for a humorous effect, is not that far off.

So in some sense graduate school ends up being largely about making a transition in how you see mathematics. It is not easy to do. And a challenging factor that accompanies this is that the gulf between working mathematicians and students becomes particularly large here. Working mathematicians made this transition long ago in their careers, and many are not quite aware of it anymore. This can make communication between students and their professors somewhat frustrating.

The question of how to keep motivated in the face of all this is the crucial one. It is very difficult, and that difficulty never goes away. Not many people get PhDs in mathematics, and far fewer are still engaged in research ten years afterwards–and I think this is largely because of the motivation issue.

I don't want to give the impression that this is an impossibly difficult career path; obviously, there are new math PhDs awarded every year. I do think it is important to understand that life as a mathematician is very different from life as a mathematics *student*. When you are a student doing exercises from a book, the distance between what you know and what you are trying to find out is relatively small. In mathematics research, that gap is multiplied by a hundred. It can seem like you are walking from San Francisco to New York, but with the added twist that there are no roads or maps and that you don't really know that New York is even there! Everyone has to find their own answers for how to keep motivated on a journey like that. You can look at it as an insurmountable feat, or you can look at it as an challenging adventure, or somewhere in between. For what it's worth, I will tell you the answers that have worked for me. For myself, it always comes back to the fact that I like teaching; I like understanding neat things and sharing them with people. This is where my motivation comes from, even for my research. I'm always out to find that next "neat thing" to explain.

**How did you know when you were ready for your thesis defense, and what was that like?**

In most U.S. mathematics departments the thesis defense is, to some extent, a formality. Basically, you are ready when you have proven a new theorem. As long as you can convince three or four professors that it really *is* a new theorem, and that your proof is correct, you will get your PhD. The defense consists of you giving a lecture about the topic and about what you've done, and as long as you can do this intelligently and answer a few of the professors' questions, you get your PhD.

In my case, I was ready for my defense much earlier than I realized at the time. I had a new result that would have made a perfectly fine thesis, but it wasn't the result I was after—it wasn't the kind of result I wanted my thesis to be about. But looking back, having seen much more of the mathematical world since then, I realize this was pretty silly. I ended up spending about eight more months working on it until I got a result that I was happier with in the end (although still not really the one I wanted!). It wasn't really wasted effort, but it also wasn't necessary to put myself through the extra struggle.

**So you officially become Dan Dugger, PhD. What's next in the process?**

This is an important question. The first thing you need to realize is that getting a PhD is just the beginning. Someone once described it to me as a "license to practice mathematics." Getting your license doesn't mean you're ready for the Indy 500, or even rush hour on I-95. In mathematics, recent PhDs do not immediately go on to become full colleagues in a math department somewhere. Usually there is an apprenticeship process called a post-doctoral position that lasts two to three years, and, depending on circumstances, a person might end up doing more than one of these. (I did two of them for a total of five years.) During these years a person needs to establish his or her own research program and prove (by writing several papers) that he or she can do research without having an advisor. Assuming you do this, you can get a tenure-track position in a research department. These are highly competitive jobs, so you go where you are lucky enough to get an offer.

**And what interesting research are you working on now?**

I'm afraid this would take a very long time to explain. It is the curse of being a mathematician: it is very hard to explain your work to people outside the field, in a short amount of time and space.

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