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! 

Math Minds: David Craft

Dr. David Craft is a modern-day renaissance man. He graduated from Brown University with a degree in Mechanical Engineering, then went on to a mathematical Ph.D. in Operations Research from MIT, and currently works with Harvard Medical School in oncology research. Among his many interests are Gallery 263, foraging, and creating music. He is also working on a new book- it will be titled Something About Infinity- and it presents interesting mathematical topics in an enjoyable, easy read.

He sat down with Tori and Zach this past week to discuss the math side of his life, and to get his perspective on the challenges of math modeling in medicine.

Math Minds: Scott Kelley

Last weekend, the Center of Math team went to the NCTM conference held in Boston. At the conference, we met hundreds of math teachers from the United States and abroad (the first person to stop by our booth was visiting from Perth, Australia!), and one of our interns, Zach, jumped at the opportunity to interview a few of the visitors. 

Here's his interview with Scott Kelley, a math instructor at the Texas School for the Blind and Visually Impaired. Read on to find out about how the blind "see" math problems, how his students use technology, and what simple advice Mr. Kelley offers to all students...

Zach talking math with Scott Kelley at NCTM

Math Minds: Marcus Fries

Marcus Fries is an Assistant Professor at Eastern Nazarene College in Quincy, Massachusetts. He received his undergraduate degree in mathematics from North Dakota State University, his Ph.D. from Northeastern University, and is now a friend and visitor of the Center of Math. He sat down with our interns Tori and Zach last week to discuss his research, his favorite class, and a lecture he would have liked to attend.

Are you currently involved in any research?

Yes, I work with cotangent bundles of the grassmannian.

So when did you first become interested in math?

At a very young age. But I actually started off at college as a chemistry major, and I did a year of chem research, learned that under no circumstances do I want to be a chemist. In Calculus II I had a great instructor, and power series just interested me to no end. So that’s when I made the switch.

That was kind of “the moment?”

Yeah, power series were just so much fun.

And do you credit anyone in particular for guiding you to this career?

Jim Coykendall and Jim Olsen, great mentors.

What was your favorite math class that you’ve ever taken, and what made it that way?

Probably representation theory with Andre Zelavanski. I took that in grad school. It wasn’t my favorite at the time, but it’s one of my favorites now. I’ve really come to enjoy the field. It’s a part of what I do now- I use representation. The spaces that I deal with, the grassmanian and other spaces are all related to representation or Lie groups. I find it really interesting.

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

I would have loved to attend Emmy Noether’s lectures on algebra. That would be my top pick. Apparently she was just very stream-of-consciousness, talking about whatever she wanted to talk about that day. And it was such an exciting time, it was very foundational to commutative algebra especially.

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

Realize that math is much broader than what you’re taught in high school. There are lots and lots of different areas to study, and there are a huge number of careers that you can go into with a math degree.

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Thank you so much for meeting with us, Marcus! 

Math Minds: Lê Dũng Tráng

Over the past week, we've had retired Professor Lê Dũng Tráng visiting the Worldwide Center of Mathematics. Born in Saigon, Vietnam and raised in Paris, France, his work in mathematics extends over almost 50 years and across 4 continents. Lê has been a friend of the Center of Math since its founding, and has contributed two research lectures to our YouTube channel (view them here and here). He sat down with interns Tori and Zach for an interview.

What is your current position?

I’m currently retired, both from UNESCO and from France, and I am visiting a university in Brazil. I belong to the Federal University of Ceará in Fortaleza, in the  Northeast of Brazil.

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.

Math Minds: David Massey

David Massey, a full professor of mathematics at Northeastern University and the founder of the Worldwide Center of Mathematics, sat down with our math intern Tori for an interview this week. He is the first in hopefully a string of mathematicians who will be interviewed for this new blog series.

David B. Massey, Ph.D

Are you currently involved in any research?

Yes - I study abstract notions of space in any number of dimensions. I prove things about what those spaces look like at places where they’re not smooth [singularities].

How do you picture those?

You don’t really. You picture what happens in low dimensions and hope that it gives you intuition for the higher dimensions. You can’t picture the higher dimensions, so you prove theorems that describe them. I’m interested in singularities because they’re a great blend of topology, commutative algebra, algebraic geometry… a lot of stuff.