Math Minds: Pablo Portilla

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.

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Thank you very much for meeting with us, Pablo! We look forward to hearing you talk at the Center! 

Math Minds: Farid Tari

Last week we had Professor Farid Tari of Instituto de Ciências Matemáticas e de Computação visit the Center to give a talk about umbilic points. Professor Tari was born in El-Kseur, a small town 20km south of Bejaia. He has many years of math experience as both a researcher and professor, working in Denmark, the United Kingdom, and Brazil during his career. Co-ops Ben and Kelsey interviewed him shortly after the talk which can be found on YouTube here.

What would you say is your mathematical specialty?

I work in singularity theory and its applications to differential geometry and differential equations.

What research are you doing and why did you choose it?

If you look at the surface of full cup of coffee or tea, you will see that the reflection of light on the surface of the liquid form a curve. The curve has some sharp turning points called singular points. Some singular points of curves split in several other singular points or disappear completely as we deform the curve. Right now I am trying to understand how these singular curves deform but making sure that the deformations  keep some properties of the the shape of the curve. The problem is part of an ongoing research activity by various people on the differential geometry of  singular submanifolds of Euclidean spaces and of other spaces such as Minkowski spaces.

Do you have a favorite mathematician, living or deceased?

Not one but several. I cannot name only one as they are my friends. Mathematicians are human beings and my favorite are humble geniuses.

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

When I was at school, about 13 years old in a small village in North Africa and I managed 

to solve a hard problem that my teacher set for the class. After that I tried to solve all the problems in the textbooks.

Is there anyone in particular that you would credit with guiding you to mathematics?

My school teacher Mrs. Thomas. I hope she is still alive and well. She went back to her country 

after she taught me for one year. She was the teacher who gave us the hard problem that I managed to solve. She was full of praises and I didn't want to let her down after that, so I started doing all the exercises in the textbook. Mrs Thomas always found time to check my solutions and made encouraging remarks. Now a teacher myself, I learned from her how important positive feedback is as a tool for student learning.

What was your favorite upper-level math course that you’ve ever taken, and why was it your favorite?

I did my undergraduate studies at the University of Science and Technology Houary Boumedian, Algies, Algeria. The syllabus then was very much influenced by the Bourbaki school (too abstract!). Nevertheless, I enjoyed most of the courses. In particular, the courses that are still relevant to me are those on Differentiable Manifolds, Topology and Group Theory. I also took some heavy courses on Functional Analysis that I enjoyed then but do not use much now, except that they help me follow talks in seminars and general conferences on the subject.

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

There is this professor at Durham University (United Kingdom) named John Bolton.  We were colleagues when I used to work there (now I work at the University of Sao Paulo in  Brasil). John was the star professor and got the best feedback (questionnaires) from the students. I observed some of his lectures. John has the ability to make the students comfortable  with learning new material. He magically conveys the key points of the lecture without being bogged down by difficult mathematical language and notation. When I started my job at Durham University I was given a course to teach in the second term.  As it happened, the course was an annual one and was taught by John Bolton in the first term. When I got the end of the year questionnaires from the students, one of them wrote "Bring back John Bolton"!

Do you have any general advice for students looking to pursue a degree in mathematics or a career in the field?

My general advice to students is to do what you like and give your best to the task. That way you will get more out of your experience at university. If you like mathematics,  then do it! There is a wide choice of career for you after you graduate as you will be equipped with transferable skills.

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Thank you for the wonderful talk, Farid! We hope to see you again in the future! 

Interviewing a Math Student

Do you want to major in mathematics?

As freshman students spill through the doors of their respective colleges each fall, they are faced with a whole new world of opportunity, excitement, and exploration. However, before even taking a step on campus, some students are asked to select their major or program of study. This can be a daunting task, as choosing a college major begins a path of education specialized towards a particular career. This decision revolves around a personal inventory of skills, interests, and goals. A heavy demand for employees in the STEM fields has caused many students to select a major rooted in the math or science department. Sometimes, students are unaware of the career options they may have, preventing them from launching themselves into a math or science program. Mathematics can fall into this category, as many students don’t fully understand the major or opportunities it may lead to. Even students who have an affinity towards math may shy away from it at the college level wondering, “What does a mathematician do, anyway?”. At the Center of Math, we want students to embrace mathematics. We thought, what better what to see what a college math major is like than to speak to someone in the program! We interviewed Northeastern student Chloe Weiers, an intern at the Center of Math, to field our questions about college mathematics. 

WCoM: So Chloe, tell me about yourself briefly.

Chloe: I'm a rising third year student at Northeastern, and I'm a Music and Math double major. I play the flute. Also, I'm from Minnesota.

WCoM: Music and Math? That’s a very interesting combination. Did you know before you started college that combination was possible?

Chloe: No, going into college I thought I was just going to be a music major. I had always liked math but never considered it as a major before college. I've always thought math was beautiful, but not in the same way as music. Then I got the chance to look at math and music under the same light, I was better able to appreciate the symmetries and beauty in both and how they complement one another.

WCoM: What exactly made you change your mind and add math to your major.

Chloe: Once I got to school I had a great class called The Algebra and Geometry of Music, and after the course finished was invited to do research with the professor of the class on a topic of my choice. I chose dynamical systems and mathematical modeling, and then ran with it. After than, I got really interested in math and added it as a second major.

WCoM: Okay so I’m going to gear in a little more on the math side of your major now. Do you feel that you were aware of the career possibilities that were possible for math majors when you entered college?

Chloe: No, I had no idea what the options were. Many people seem to know going in, "Oh, I want to do specifically this or that", or like, "Oh, I want to be an actuary or an accountant or a professor", but I had no idea.

WCoM: Did that make you nervous at all?

Chloe: No, because I never really thought about it in that way. It seemed natural that I should add it as a major, because it’s one of those things where the more you learn, the more really interesting stuff you realize you don’t know, and its kind of addicting. I couldn't imagine never taking another math class. I have to keep growing as a mathematician.

WCoM: You mentioned 3 careers so far– becoming a professor, actuary, or accountant. What would you say to someone like yourself, who isn’t necessarily interested in any of those specific fields?

Chloe: Well, it's important to consider all the options, and sometimes that involves taking things into your own hands. There are tons of jobs that use math, but they aren't necessarily going to come to you- sometimes you have to seek them out. But when you do, the magnitude of really cool options is very encouraging. Also, when I was a first-year undergrad, my math professors kept encouraging me to learn some mathematical computing skills, like MATLAB and Python, and even LaTeX for typesetting mathematics. I was so reluctant, because I never considered myself even the slightest bit competent with computers. But it's so necessary. Even just learning the basics of something will help you get your foot in the door and help bolster your confidence. You don't have to be some coding expert, but having the ability to tackle new challenges with confidence instead of shutting down is an excellent thing to have going for you. The modes of thinking involved with all of these programming languages is completely logical and can even help you streamline your thought processes in other ways. So it's a win all around.

WCoM: Have you had a moment yet that you realized, yes I definitely made the right decision?

Chloe: I would say it’s less a moment than a series of realizations, usually coming when I’m working on or solving problems. That feeling you get of accomplishment is a really powerful, albeit selfish, indicator for me that I’m doing the right thing. Also, when you get into courses like Real Analysis and Number Theory and start learning the derivations behind the mathematics you have known and accepted your entire life, and then turning it inside out with new axioms or rules just kind of blows your mind. It’s also always very challenging and very interesting, so you’re never idling. Mathematics even helps me think rationally and logically in other areas of my life.

WCoM: Wow, so it seems like you have no regrets about your decisions so far, academically speaking. Is that correct?

Chloe: Definitely.

WCoM: So there is heavy discussion about women in STEM, or lack thereof. Do you notice this in your classes?

Chloe: Yeah, definitely. I mean, most of the time less than 30 or 40% or the class is women, often even less. It’s not necessarily a lack of opportunity for women in college, but rather a lack of opportunity and encouragement leading into the university setting that I think prevents many women from even considering math as a potential career field.

WCoM: Obviously there isn’t necessarily an easy fix to this issue, but what do you personally think can be done to encourage women in the field?

Chloe: I think that it needs to start much earlier than college. If you are looking at trying to get college-aged women into STEM fields you’re already too late. It needs to start as early as elementary school. Girls should be encouraged and celebrated as active participants in their math and science classes, so they can build confidence with their own abilities and learn about their options early on.

WCoM: Building confidence from an early age definitely makes sense. Do you have any advice for students thinking about majoring in mathematics, male and female alike?

Chloe: Try to get as involved as you can with extracurriculars and such. When I say extracurriculars, I mean pretty much anything- robotics, math Olympiads, just recreational math at home or with friends, coding, whatever. It doesn’t have to be these high level competitions you see kids doing. Math competitions aren’t for everyone, and they’re certainly not for me, so don’t feel pressured to participate in math like a competitive sport. There are often cool summer programs that I wish I would have known about as a high schooler, and they’re offered all over the summer, so definitely look into those if you’re interested in really finding out what interests 

WCoM: It’s nice to hear you so encouraging to young mathematicians. So what have you been up to lately at the Center?

 

Chloe: I've been working with Ruairi, who works with digital media here at the Center, to create a video series on mathematical music theory. We're calling it Musimathics, and it will be available soon on our Youtube channel. The videos give an overview of some of the especially interesting areas of mathematical music theory, explained in a casual setting by myself. You don't have to be a super advanced mathematician or skilled musician to understand what's going on the the videos. They're meant for everyone! So stay tuned for more information about that.