Math Minds: Holly Krieger

This week, Center of Math intern Tori sat down with Dr. Holly Krieger, a post-doctorate fellow and instructor at MIT. Dr. Krieger discusses her research, her favorite math class, and her time with web series Numberphile below.

Dr. Holly Krieger

I’ve done a little bit of background research on your research topics-  is there any reason you chose algebraic geometry and dynamical systems as your focus?

I was always interested in Number Theory, just because it’s an incredibly appealing subject. You can state some of these problems very simply, but they’re very intricate and deep mathematics. I got into dynamical systems through coursework I took in graduate school. It’s sort of a new field, it’s not something that [undergraduates] would necessarily take a class in, but once I became aware of the field it was of course interesting, and that’s how I ended up going that direction.

Can you explain Diophantine geometry for us? 

It’s essentially the study of Diophantine equations, which is the question of looking for integer solutions (rational solutions) to integer equations. So just to give you an example: x^2 + y^2 = 0. This is an integer equation because to write it all down, all the coefficients are integers. If you were interested in “What kinds of rational solutions does this thing have?” Well there’s only one, because x^2 and y^2 are positive or zero. What about x^2 + y^2 = -1? Well this has no rational solutions whereas x^2 + y^2 = 1 has lots and lots of rational solutions. So, Diophantine geometry is the study of what kind of behavior can happen when you’re trying to solve equations (maybe more than one, maybe multiple equations simultaneously). You’re trying to find rational solutions to equations like this, and you’re answering questions like: what kind of behavior can happen? How many [solutions] can you have? What do they look like? That kind of thing.

When did you first become interested in math? Was there a specific moment when you knew you wanted to pursue this field?

Funny answer: I did not know that I was going to be a mathematician for most of my life. I always thought I’d do something else. I went to my 10 year high school reunion not too long ago, and when people found out I was a mathematician, I thought that everyone would be sort of shocked, and instead everyone was like “Oh yeah , that was really predictable, we all knew you were going to do that.” So everyone except me knew that I was going to be a mathematician.

Here’s the real answer to your question: I had gone through a bunch of majors in college and nothing clicked, and I was a computer science major and I ended up having to take an honors version of a math course that was an introduction to proofs. And there was actually a moment where I was studying for the final, and it was the first time that I really understood a proof, and it just clicked. There’s a quote, I don’t know it exactly, about how that moment of understanding is “like the click of a well made box” and it’s exactly like that. I was looking at this proof and suddenly it was like, “click.” It was incredibly satisfying.

Do you credit anyone in particular for guiding you to this career?

There was a professor- I did my undergraduate degree at Univerity of Illinois Urbana-Champaign,  and there was a professor there, Rick Laugesen, [who] encouraged me to go into the honors program there, and to apply to graduate school. In particular he was a counterbalance to some of the discouragement that I got, and it was due to him that I ended up going to graduate school and continuing on.

What was your favorite math class that you’ve ever taken, and why?

Probably my favorite class that I’ve ever taken was that one where I first realized I wanted to get into math. The introduction to proofs course. It was all new to me at the time, I had taken some logic and computer science classes, and I had also taken some liberal arts classes that I didn’t like as much, and it was my first time feeling like [math] just fit. And the instructor was great.

If you could attend a class taught by any mathematician or professor, living or deceased, whose class would it be?

I will say David Hilbert. He wasn’t necessarily the most brilliant mathematician that ever lived, but he was very well known for being able to bring together a bunch of different mathematical ideas. He was sort of a connector for a bunch of different mathematical ideas and I think that’s a powerful thing to have done.

So I actually learned about you first when I watched your appearances on Numberphile. The first one I watched was “63 and -7/4 are special.” Then I realized you were located at MIT and reached out to you! Tell me a little about your experience with Numberphile. Did you choose your topic?

Numberphile was great! I did mostly choose my own topics. Of course since I had never done it before, I ran it by Brady [Haran] because I didn’t have a good feel for the audience. But yes, it was a great experience. He will let you sort of go on as you want to as you’re talking about math, but then he’s very talented at detecting when you’ve crossed the threshold of what a listener might no longer follow. He gives a good perspective on what the non-mathematician might think.

Do you have any general advice for students thinking about pursuing a degree in mathematics?

Yeah, unfortunately it’s the same advice that I got and ignored, but it’s really true: the way to be successful is to involve yourself in the math department. Spend extra time going to office hours, ask professors to recommend extra things to you if you feel like you’re already understanding things well, interact with grad students and post-docs, go to seminars even if you have no idea what’s happening. So really that’s the strongest advice I can give. To start becoming a member of the math community.
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Thank you so much for meeting with us, Dr. Krieger! Learn more about her work by visiting her MIT page. If you missed our first Interview with a Mathematician, click here to read about Dr. David Massey.