Pablo Portilla is a PhD student at Instituto de Ciencias Matemáticas and is currently visiting Northeastern University. This Friday, he will give a talk on Tête-à-tête graphs and Seifert manifolds which you can find more information on here. Ben and Kelsey interviewed Pablo about his research and interests.
What research are you doing and why did you choose it?
When I was an undergraduate student I didn't have that much "margin of action". When I finished my bachelor's degree I had a very narrow perspective of all the different kinds of mathematics being researched so as to say that I could actually "choose". I knew I liked topology and geometry, and I really enjoyed a course on differential topology that my current PhD advisor (Javier Fernández de Bobadilla) gave during my master's degree. It turned out that he works on singularity theory but he usually works more from the algebraic point of view. The good thing about singularity theory is that it intersects many different areas of mathematics. So he could propose to me a problem that, within singularity theory, is completely topological, and I was happy with that.
What would you say is your mathematical specialty?
As I said, I work on singularity theory from a differential topology viewpoint. I study things that happen in the space near the places where "abrupt changes" occur, that is singularities. More concretely I am trying to understand some mathematical objects associated to singularities from a combinatorial viewpoint. Hopefully this will be useful for computing invariants of singularities that are difficult to compute now or it will provide tools to attack other problems in mathematics.
Do you have a favorite mathematician, living or deceased?
If I had to choose I would choose René Thom (1923 - 2002). He made the bulk of his career in differentiable topology with very valuable contributions in the theory of characteristic classes and cobordism theory which are now essential to many branches of geometry. Then he settled the foundations of what is called "catastrophe theory" which could be understood as a part of singularity theory. He dedicated the last 20 years of his career to write philosophy and epistemology by "revisiting" much of the work of Aristotle (of who he considers himself a descendant). This last part is very often not valued by mathematicians which tend to see work outside mathematics as a waste of time. Nevertheless, I find inspiration in a man that had not only a great mathematical mind but also a huge interest for other areas of knowledge.
When did you first become interested in math? Was there a specific moment when you knew you wanted to pursue this field?
I remember that being a kid I used to think of studying computer engineering. Then, at some point in my very early teens, I decided I was going to pursue a career in mathematics. In the beginning I was moved by a platonic idea about mathematics thinking they provide a "path to the truth" (or something like that) and I also liked how the "arguments from authority" just didn't work in mathematics. Then I realized that the kind of truth mathematics talks about is not the kind of truth I was thinking of in the first place. I also realized that since mathematics is done by people, "arguments from authority" are done sometimes at seminars or talks and personal relations influence the publishing (or not-publishing) of a work. I guess in the end, people enjoy doing things they are good at doing, and you don't have to find deep motivations to do anything. I just feel that making a living from doing research will make me happy.
Is there anyone in particular that you would credit with guiding you to mathematics?
I have to give particular credit to a couple of teachers that I had in high-school: Mercedes was my mathematics teacher when I was 14 or 15. She was also involved in a program that "trained" young students to participate in the mathematical olympiads. I think she was very good at taking care of the individual needs as well as to boost the potential she found in some students. It was definitely very motivating to have her around. I also remember very kindly Luismi, a physics and chemistry teacher I had in high-school. He always put reasoning over knowledge and you could tell he really enjoyed his job. I think he transmitted to me his passion for science and discovery.
What was your favorite upper-level math course that you’ve ever taken, and why was it your favorite?
My favorite upper-level math course was the one I said before that my current advisor gave during my master's degree. It was a course on differential topology, and the goal was to prove a central theorem in this area which is known as "The h-cobobordism theorem" (originally proved by Smale). I really liked it because we started basically from scratch, defining the notion of differentiable manifold and we ended up proving highly non-trivial facts about manifolds in higher dimensions. The core of the proof relies on a result called Whitney's trick that tells you that you can "untangle" spheres of complementary dimension when the ambient space in which they lie has dimension big enough (for instance greater than 4). The implications of this theorem are deep, in particular it tell us that spaces of dimensions 3 and 4 are in some sense much more complicated than spaces in higher dimensions.
If you could attend a class taught by any math professor living or deceased, whose class would it be and why?
I think I would pick any class by John Milnor. I watch some famous recorded lectures he gave on differentiable manifolds at Cornell University [https://www.youtube.com/watch?v=1LwkljjLBns]. He starts from the definition of smooth function and ends up stating precise deep results in differentiable topology. It is just delightful to hear him speak. He goes at the right pace making the right remarks and emphasizing the important parts.
Do you have any general advice for students looking to pursue a degree in mathematics or a career in the field?
Mathematics can be very arid sometimes, but it is really rewarding to understand things, and much more rewarding to prove new results so... keep up the good work! Because it is worth it.
Thank you very much for meeting with us, Pablo! We look forward to hearing you talk at the Center!