Interview with Dr. Nathan Ilten

Department of Mathematics (Algebraic Geometry)

Joined SFU in July 2014

Dr. Ilten’s research focuses on understanding algebraic varieties (e.g., toric and Fano varieties), deformation theory and Hilbert schemes, and linear subspaces of varieties. His group tackles geometric problems using combinatorial techniques related to polyhedral geometry, representation theory, and Gröbner basis theory. Dr. Ilten is also engaged in developing computational tools in algebraic geometry; for example, he has authored several packages in the Macaulay2 software system.

What early experiences influenced you to pursue a career in science?
I was one of the fortunate people who had really good math teachers through grade school and high school. Two teachers really stood out: my ninth grade algebra teacher and my calculus teacher in twelfth grade; they conveyed to me the excitement and passion that can be found in pursuing mathematics, and this got me very interested in math.

What is the best part of the work you do? What is the downside?
Mathematics, for me, is really about understanding complicated things. What gives me the most satisfaction is the moment of realization of what's going on in a particular problem. Often it's a problem I've thought about for a long time and have not necessarily been making progress on, but then suddenly after all those hours something clicks and I start to figure out what's happening.

On the downside, it's certainly frustrating to be stuck on a problem and not make progress. It happens quite regularly. You just plug on; it takes a certain amount of diligence and dedication and optimism to believe that it will eventually work out.

Which algebraic varieties occupy your thoughts the most?
There are many different algebraic varieties. The ones that are most interesting to me are those that are somehow very concrete. An algebraic variety is a geometric object that is defined by a set of equations of a certain form. Many people study varieties where they don't really know what the equations are—they are nebulous abstract things—and if you ask them to write down the equations they probably couldn't do it. I am interested in more concrete varieties where the defining equations are given explicitly, or are at least accessible in some form.

Do you find algebraic varieties intuitive?
For me, a large component of studying mathematics consists of developing some sort of intuition for the objects with which one is dealing. This intuition is not present at the start, but the more I do, particularly in my field of algebraic geometry, the more I have developed intuition for dealing with these objects. It’s something you always wish you had more of, and it continuously grows stronger.

What algebraic geometry research tools have you developed?
One tool I've been involved in developing is the software system called Macaulay2. I've been involved in writing a number of packages for the software system. This is a great system for people who are interested in getting into the field of algebraic geometry because there is a large community of mathematicians involved with it who provide support for users; you can just post a question and you will get a response within hours.

How do you see your program developing over the next decade?
At the moment, I am not directly attacking some of the problems I'm interested in because I have to develop the required machinery first. So hopefully in a couple of years this theoretical machinery will be developed to a sufficient degree and I'll be able to attack the problems that motivated me to develop the machinery.

Could you give a few examples of the problems you can't wait to get to?
There is a field of mathematics and computer science called complexity theory, which explores how difficult certain tasks are to do. The most important problem in this field—and one of the most important problems in all of mathematics—is ‘P versus NP’, in which for two different classes of tasks, the aim is to decide whether there is an inherent difference in complexity between the tasks in theses two classes.  I am interested in applying some of the tools I’m developing to attacking problems related to this notion of complexity, which might hint at what can be said about this P versus NP problem.

What attributes of the Math Department made you choose SFU?
I was very impressed with how friendly everyone is in the Math Department. There's a sense of community, a very collegial atmosphere. Certainly there are a number of faculty whose research interests overlap to a certain degree with mine as well. The collegial nature was probably the most attractive feature of the Math Department.

What specific strengths/backgrounds do you look for in prospective graduate students?
A strong background in Mathematics. Specific traits that are desirable include a strong sense of inquisitiveness, persistence, and some capacity for abstract thought – that is, the ability to take a step back from the concrete and think about things in more abstract terms.

If someone came in with a Physics or Computer Science degree their expertise could be useful. We are interested in problems related to these fields; nonetheless they would still have to have taken a fair amount of mathematics to prepare themselves for further studies.

What contemporary scientific issue worries you the most and desperately needs more attention?
One issue that certainly worries me is climate change. It is getting a fair amount of scientific attention, but I view it as more of a political problem than a scientific one – this is most worrisome to me.

I am also concerned about disease control. Bacteria and viruses are constantly evolving and getting better at beating whatever means we have to control them.

What other area of science completely boggles your mind?
My brother is a particle physicist and I cannot for the life of me understand what he is doing. That is something I've always struggled with. I always wanted to understand more physics.

What instruments do you play and what type of music do you listen to?
At the moment, I admit, I'm most enjoying playing the tuba, a recent acquisition. I also play trumpet, ukulele, baritone and bass guitar. I listen to a lot of jazz, classical, and other music, too.

As an avid sailor, are you constantly shaping your experience and surroundings to algebraic varieties, or do you just get to enjoy sailing?
Definitely the latter – I just enjoying sailing. In fact sailing, especially smaller boats like those I'm interested in, requires you to do a lot of things at once, very quickly. The only way this is possible is by internalizing the tasks so that it all happens as if by instinct. If you're constantly thinking about what you're doing, you're going to get into trouble. So really it's all about developing instinct, letting your unconscious mind take over, and enjoying yourself.


Dr. Ilten adds strength to SFU’s expertise in toric geometry and deformation theory and bridges the gaps between combinatorialists, algebraists, and geometers. His versatility is reflected through an impressive range of collaborations, addressing problems ranging from algorithmic implementations of geometric calculations to coding theory. In addition to providing fundamental insights in pure mathematics, his research has applications in applied mathematics, physics, and computer science.

Read more: Dr. Ilten's personal website, profile on the Mathematics website and the New Science Faculty page

Interview by Jacqueline Watson with Theresa Kitos