This interview is a start in what I hope to be a series of posts illustrating the human side of mathematics. Mathematicians as a group have distinctive cultural features. We have our own specialized humour and shared experiences that bring us together and make us laugh. A week ago, I read an interesting article by John Swallow entitled “Mathematical Community” in the Notices of the American Mathematical Society, which reminded me of the unfortunate truth—very few people outside of the mathematical community really knows what being a mathematician is about.
I hope that this will change, and I feel that it is very important to inform the general public about the work of mathematicians. And by this I don’t mean the technical details, but the culture and the general ideas that we work with every day. As part of this initiative, I thought I would interview a few of my fellow graduate students at McGill. My first interview is with Benjamin Smith, who is a PhD student in geometry.
1. Ben, can you tell me a little bit about the field you are working on now?
I work in the area of mathematics widely known as geometry and topology. More precisely, I am interested in problems about describing the topology of the moduli space of holomorphic vector bundles over Riemann surfaces and real algebraic curves. My thesis topic is to provide a Kobayashi-Hitchin type correspondence theorem for monopole moduli spaces of principal bundles. A lot of people would classify this subject as string theory, but those who work on it most likely wouldn’t be able to give a clear indication as to why that is.
Much of the ground breaking historical landmarks that I am building my research out of was done began in the 1980’s by Michael Atiyah and Raoul Bott in there paper entitled “The Yang-Mills equations over a Riemann surface”. [Link]
Some frequently occurring core mathematical concepts that I use to solve these types of problems are: advanced linear algebra, group theory, vector calculus, algebraic geometry and topology.
2. How did you choose your field and what attracts you to geometry?
I was initially attracted (in the gravitational sense) to the subject when learning about gravitational physics from an extraordinary lecturer Richard Epp. This was a purely observational course with a required level of mathematics at a second year undergraduate level. The connections between geometry and reality were truly astonishing and I just had to know more. I was concurrently taking a course on curves and surfaces with Ruxandra Moraru (who was eventually my masters supervisor) and the use of a combination of calculus, linear algebra and topology made the subject of geometry seem like an all-encompassing field. Being the curious fellow that I am, geometry seemed like the place to be.
3. So, what caused you to apply to graduate school? Was it seeing geometry or did you already know earlier that you would apply?
I have always known that I was destined to become a teacher. Having prior interest in mathematics and physics, it wasn’t difficult to determine what to teach. The choice about graduate studies occurred during my first year of undergraduate. It seemed that teaching at a high school level would result in forgetting all of the interesting stuff I was learning. Overall, I wouldn’t be able to teach people about factoring quadratic polynomials as a career.
4. Do you have a favourite theorem?
A favourite theorem? Now everyone knows that choosing favourites is a difficult task and perhaps not even worth while. If I were to mention a theorem I am thankful for that comes in handy every day, it would have to be Harder-Narisimhan filtration for holomorphic vector bundles. This is essentially like a Jordan Hölder theorem for vector bundles which allows one to see the relationship between sections of the bundle and maps into flag varieties. It brings together several great concepts in mathematics and allows one to move forward in the classification problem for holomorphic vector bundles.
5. How did you get into mathematics?
There are several reasons I have chosen the wonderful world of mathematics as a career. One vivid memory was looking forward to those annual math contests put on by the University of Waterloo and being recognized for good performance as well.
I also had the advice of my mother ringing in my head while making decisions about the future. She would always say things like “work with you brain not your back or you will be too tired to enjoy yourself in your old age” or “make sure you enjoy what you do for a career, because then it’s not work”.
The final determining factor in my choices fall under the category of being a person who talks too much. Furthermore, I am incapable of sitting still or being quiet. It was absolutely mandatory that I find a career where talking and communicating with people is in the job description. This being necessary, teaching has always been an ideal goal and teaching mathematics just tops the cake.
I should also mention a few remarkable teachers I had during high school who were entertaining and supportive to say the least. Here’s to Mrs. Greaves, Mr. Gregorio, Mr. Deluca (R.I.P), Mrs. Shenton and Mr. Zelinka (in that order).
6. Speaking of math contests and competition and regardless of how much competition actually occurs in mathematics, do you feel mathematics to be very competitive in your life as a graduate student?
Relatively speaking, with respect to the business-oriented sector, mathematics would certainly not be considered competitive. Surely there is some competition involved in obtaining funding to perform mathematical research, but there seems to be enough funding available for those who truly desire to work in this industry. In terms of other fields of research, I would like to argue that mathematics is among one of the least competitive. One may notice that publications in mathematics often have several authors which suggests collaborative as opposed to competitive research. I like to think of performing mathematics as a recreational sport played with a group of friends who are just trying to work together as a team and play a good game, rather than a sport which is nationally televised and people get fired for loosing.
7. Being a graduate student is mathematics sometimes pretty intense, especially with the many courses we have to take an the reading we have to do to become familiar with a problem. How do you find enough time for research, and what advice would you give new graduate students to avoid being overwhelmed?
Math is hard there’s no doubt about it. Being a graduate student has been a significant change of atmosphere from undergraduate work. Undergraduate mathematics, no matter how difficult, was easy in comparison to this. Doing a PhD in this subject requires dedication, and innovation (in that order). The courses and qualifying exams in the beginning should be praised a nice smooth introduction and confidence builder.
Finding enough time for research boils down to making enough time for research. It requires being interested enough in the subject to essentially be ‘hooked’. Imagine your favourite television show, one of those shows that is packed with twists and turns then leaves you hanging at the end and doesn’t air again for another week. That yearning to get back to it because you just gotta know is an absolute must in this field.
That being said, I would advise any graduate student to practise a well-rounded and sociable life style. I find that a healthy diet, frequent exercise and good sleeping habits can be quite beneficial regardless of the pressure you may feel to be working constantly. It also helps to communicate with people outside your realm of work and keep in tune with the world around you. I have also noticed some people choosing political, musical or spiritual outlets for entertainment outside of their work.
Thank you to Ben for doing this interview. I immensely enjoyed thinking of these questions and reading these answers!