To close out the semester, here is a conversation I had with fellow math PhD student Michael Catli. Michael and I both started the PhD in Fall 2023, although I knew him slightly before that, via a mutual friend at Seton Hall University, where Michael completed his undergraduate studies. Michael now works in the group of Professor Alexei Miasnikov, and is a Teaching Assistant for the first-year calculus sequence. I found this conversation, edited for conciseness and clarity, to be particularly fascinating, so a more complete account of it can be found on the Stute website. Happy last day of classes, and good luck on finals!
Charlie: What was your experience with mathematics growing up? What about the subject excited you, or made you want to pursue it at the university level?
Michael: I really liked math along with science, and as a kid I actually wanted to be an astronaut because I just loved space and was so excited to think about things that existed outside of the planet. In elementary and middle school, I used my parents’ laptop to look up math stuff beyond what I was learning in school, and got a lot more into mathematics after courses in algebra and geometry in high school. I got to teaching myself calculus from the internet in junior year, and senior year I bought Cornel Ioan Valean’s “Almost Impossible Integrals, Sums, and Series,” and would work on problems from it all day. This prompted study into other areas of mathematics used to evaluate integrals, and I came to see mathematics as the number one way to improve the ability to think. That’s a loaded statement, but I feel like it’s really true, and improving my ability to think made math the most stimulating subject for me, and that can be attractive for anyone as well.
C: That perspective on math you give at the end is really interesting! I know a lot of people lump mathematics into the sciences thanks to STEM being such a popular acronym, while others (typically mathematicians) believe that the subject is more of an art, or at least has more artistic elements, due to its abstractness, compared to the other sciences. It seems like you’re saying mathematics to you is a kind of philosophy – could you comment a bit more on that?
M: You could view math as a big circle, and then the sciences and engineering fields can make up a part of that circle, in the sense that they depend a lot of times on being able to make conclusions utilizing mathematics. I’m not saying that math is any better than these fields, because you can be very stimulated by these other fields too, but I think all of them require the ability to mathematically think. And this mathematical ability really strengthens overall cognitive ability and reasoning skills. Ultimately it’s hard to describe this mathematical ability in words – like somebody once said “Mathematics is what mathematicians do” – and that’s all you can really say about it. But this ability to think or desire to strengthen your ability to think is what drives your ability to study the subject.
C: That’s a really great explanation on the connection between mathematics and logic and reasoning skills! Now I want to ask about the work you’re currently doing for your PhD. What is your current research focus?
M: It’s still in the very beginning stages, but I’m currently trying to prove results about what we call the decidability of certain problems, specifically problems within group theory. I’m trying to show that in certain groups, there’s no easy algorithm for finding a solution to any given system of equations. For those not familiar with group theory, the idea of a group is that you have a set of elements, and then you can perform what’s called a binary operation on any two elements to get a different element in the set. You can also “divide” by any element, and there’s some element that acts like a “1,” both with respect to this operation. An example of this is multiplication of two nonzero numbers: each number has a reciprocal, and you can multiply by the reciprocal to obtain the “division” I just mentioned; 1 acts as the identity, since any number times 1 is again that number.
C: What about this problem interests or excites you?
M: I think it’s exciting to work towards making these algebraic structures less abstract and more something you have a grip on. The more results we can show about groups, the better we understand them, which is what I’m trying to do. I’m trying to get a sense of, “Okay, how hard is it to do this task in groups? And based on that, would the group be more similar to more familiar algebraic structures we know?” Also, I’ve always had a particular interest in group theory, going back to my undergrad, so it’s cool to push the limits of understanding in that subject.
C: This is already quite interesting, and you already touched on your self-study of integrals going back to high school. Are there any other mathematical results or areas of math that you find particularly exciting?
M: I was thinking of sharing for this question that all finite simple groups have been classified, via thousands of pages of mathematical proof due to around 100 authors. Simple groups are really important since they’re like building blocks for more general finite groups, and thanks to these mathematicians, we have basically a periodic table but for group theory. This is very interesting, but it’s also quite beautiful, because I think it shows how fruitful these types of collaborative efforts can be between hardworking people who have the same goals in mind.
C: The level of collaboration really was impressive for this classification project – you mentioned the periodic table which is a great analogy, and I’d also like to analogize it to the sequencing of the human genome, which was deemed “complete” in a certain sense in the early 2000s, coincidentally around the same time the simple groups were fully classified. To switch gears a bit, what are your long-term goals in mathematics?
M: If we’re just speaking practically, it would be great to make a living out of it, that would be really nice. What I had in mind coming into the PhD program, and what I think are some natural ideas, that I could try to become a professor, or I could work in industry. I don’t think it’s clear at the moment what direction I’d go in. I think if my research lends itself to certain industry-related positions, then maybe that would be more natural. Or if it’s not as industry-focused, then perhaps it would make sense to try to become involved in academia, although getting a professorship is very hard, so I don’t want to just casually throw it out there. I think it will be clearer to me what I’ll do when I’m in the later stages of my research.
C: Absolutely – it’s the same with me, I hope to continue in academia but I will hopefully have a clearer idea of things by the final stages of the PhD! To finish up, what advice or recommendations would you give to others studying mathematics, either as their primary field or as a tool for understanding their primary field?
M: I actually have a few. First, as a practical recommendation, I would say to time your study in mathematics – set a timer for 30 minutes or an hour and during that time do only the math. For me it’s a totally different studying environment when I do that, since it really helps with my attitude and focus going into study. Some other things that are more long-term: getting used to math, or any new topic, is always going to be foreign and hard at first, and it takes time to familiarize and learn the subject. Talking with other people about mathematics or going to online forums, especially when you’re stuck on a problem, is also important in this learning process. I would recommend learning to understand the subject, which takes more effort but really pays off and makes the learning fun. If you’re struggling, recognize that others are struggling too, and that you’re not alone and in fact doing a very noble thing to admit you’re struggling. Finally, I’d say all of this stuff applies not just to studying mathematics but life in general. You could time yourself for anything and set a schedule to do certain things you’re passionate about. You won’t know the answer to everything, but you can and should ask others for help when you don’t know. To learn anything you really need to understand it, and there will be struggles in that learning process, but everyone goes through those struggles at some point. This goes back to what I was saying at the beginning, where getting better at math makes you a better thinker, and I think it makes you better equipped to handle life.
C: That’s all great advice and a great way to end! Thank you Michael for this great conversation!
M: My pleasure!
