**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.