From biology to mathematics via online learning – an interview with Artem Gulyaev

Dear readers,

It is my pleasure today to introduce you to Artem Gulyaev, who has been a friend of mine since primary school in Russia. Artem and I lost touch when I left for Canada, but reconnected when he also moved to Canada after high school. His original interests lay in biology, but he ended up developing an interest in mathematics and changed fields.

Artem has gone on to obtain an advanced degree in mathematics in Canada, but then returned to Russia where he is currently working. However, he manages to combine his job with online mathematics education, and he is in fact the only person I know who has experienced that learning medium.

This interview was conducted on Google chat, so I’m presenting it in transcript form. It touches on topics ranging from the challenges of switching from biology into mathematics to the differences in the North American and the Russian attitudes towards mathematics, and, of course, goes into detail about online learning.

LC: Hi Artem, and thanks for taking the time to speak with me today! I’d like to start by asking you what motivated you to study mathematics.

AG: To start with it was envy. Some of my classmates went into mathematics and physics, and one friend in particular got into the famous MIPT. Famous in Russia that is, although I’m pretty sure it’s got a name in certain circles abroad as well.

‪LC: What does that stand for, if you don’t mind?

AG: It stands for Moscow Institute of Physics and Technology. And although I was sure at the time that I would go into biology and that was the right thing for me to do, I still felt sort of inferior for not being able to do proper math. I had a feeling that mathematics is a thing (maybe a language) which if you possess you can talk about a whole lot of things and if you don’t you’re sort of mute.

‪LC: So math felt like a more challenging subject to you?

AG: Yes. I wasn’t good at it in school and it seemed to me that people who could do it well were able to come up with unmotivated tricks to solve problems. I could never get how they do that. I also had this romantic notion of a proper biologist who should know math and physics and bring that knowledge and methods into bio. So, I bought the textbooks they use in Moscow and brought them to Canada and started reading them. And I tried to take honors classes in both math and physics in my first two years of college. By the end of the second year three things became apparent. First, I was doing far better in math and physics classes than in bio. Second, nobody seemed to care about any kind of theory-building in biology – they taught us various experiments and expected a lot of memorization. Third, I was getting nowhere with the Moscow books. So, I put down the first analysis textbook and took up another one, which was “Mathematical analysis” by Zorich. And that did it – it’s my favorite book until this day. It is so well written and insightful that I felt like it was showing magic to me. Not magic tricks, but actual magic. And so I went and switched majors.

‪LC: Very interesting! Other than your friend who went to MIPT, did you have any other role models at that point in time?

AG: Role models: at the time no, I didn’t have any. I just kind of wanted to be on the same level with my classmates.

‪LC: Wow! Switching fields like this is quite a courageous thing to do! What resources did you find helpful along your mathematical path?

AG: The first push was a series of private lessons I took over the summer vacation with the professor you actually know. He taught me the basics of analysis.

‪LC: Right, so a private tutor…

AG: Then mostly books. There were a couple of very good profs in my university, from whom I was able to learn, and I’m sure there would be more If I could make better friends with classroom learning. But, since I don’t do it very well, most of the things I know came from the books I read. Nowadays, of course, there are tons of resources like mathematics stackexchange, where a student motivated enough to properly think about what questions they want to ask can get a lot of insight.

‪LC: That sounds like a good resource!

AG: I find it helpful, but I only recently started using it.

‪LC: Sure! Can you comment on the differences in mathematics education between Russia and Canada? Or, if not the education, then at least the attitudes people have towards mathematics.

AG: My knowledge is very limited here. From what I can tell, they try to be very gentle with students in Canada and many courses are sort of skeletal – they only teach what they absolutely have to. Also very little in the way of heavy calculations or calculation tricks. It starts very slowly and picks up the pace by the third of fourth year and grad school. In Russia, on the other hand (where there is still decent math faculty that is), it goes in reverse.

‪LC: How so?

AG: The first two – three years seem to be the ones where a lot of mathematics is unloaded onto the student, very often without telling them why they should care. They go deeper and learn to solve tougher problems. All of it very classical though. A lot of it simply irrelevant by now. Exams are hard and there are a lot of them. Trouble is though, most students adapt by developing the ability to load a lot of stuff into short-term memory, pass the exam with flying colors and totally forget most of it by next month. Very little actually sticks.

LC: Sounds like biology to me :)

AG: Yes, I guess it’s a bit like that. Although, again, this is a personal impression looking at it from the outside.

LC: So would you say there is an earlier filtering that happens in the Russian system?

AG: Absolutely, yes. You have to pass difficult examination in order to get into a mathematics faculty in the first place.

‪‪LC: And what about the attitudes that people in general have towards mathematics? Educated people, but not necessarily mathematicians. Do you have any impressions on those? Any differences there between Russia and Canada?

AG: I think it’s pretty similar. Most students see it as a subject you encounter in college and have to pass and never ever see again. Students of mathematics either love it unconditionally (very few) or see it as something they have a knack for and can make a profession out of (like become a programmer, or something applied). In fact, many employers will hire you for an unrelated job (to mathematics, that is) if they know you’ve got a degree in math, simply because they know you can do a lot of work. This also influences the attitudes of students. The general public is simply unaware of mathematics. I think all of it goes for both Russia and Canada, except the hiring policy.

‪LC: Fair enough, thanks for sharing this! Switching gears a little bit, I was hoping you could tell me about your online learning experience.

AG: Sure.

‪LC: What motivated you to try it, and how has it been working out for you so far?

AG: Well, I’ve been in awe of this school ever since I’ve heard about it, and when I came here I met some students online by chance, who told me that distance education is possible, and told me how it’s done (the guy I talked to actually lives in Ukraine). I thought to myself that between reading on my own and having my learning directed by people who know what they are doing, the second option is far better.

‪LC: Makes sense!

AG: The “online” component is by necessity – I live in a different city from where the school is. But the program is not set up as an online one specifically. They just film their lectures and make other materials available. So, it’s kind of a “pretend online”. As for the experience: I am loving it. Mostly because of the content – they really do push you. Well, they push me. There are perhaps people who find it a cake-walk.

‪LC: What specific challenges have you encountered with this type of learning? And perhaps you can mention any advantages as well?

AG: So, the main challenge for me is that my professors are on the screen and my classmates I don’t see at all. It feels a bit empty and communication with your more experienced colleagues is very important because it’s very inspirational (I never found communication with classmates very rewarding). We’ve all had the experience of coming home very excited after a short talk with a good professor and this charges you emotionally, it’s a resource you can spend. And you have to spend it, because mathematics isn’t just a bit hard intellectually, it’s emotionally demanding as well.

‪LC: In terms of the struggle it requires?

AG: Yes. For most of us it means coming face to face with the fact that we’re idiots on a daily basis, which is unpleasant. There are some that don’t, but very few. As for advantages: you set the pace to some extent, although I think this is school specific. I imagine that in a proper Western online program there are just as many deadlines and they are just as strict. But at least you don’t have to be able to make it to class, which I am not. You can choose when to watch a lecture, and if 2 AM is your favorite time, then you can do that. There are no repercussions. In a proper university setting people like that usually just fall through.

‪LC: Based on your experience with online learning so far, what changes would you like to see in the way courses are taught to make it better for you?

AG: Oh, that’s interesting. I haven’t really thought about that, so let me try and think how I feel… In my case (and therefore in the case of my particular school) I’d like to have more written material available. The courses cover a great deal of ground and they are almost entirely theoretic. Obtaining the necessary experience dealing with this theory is entirely the responsibility of the student – we teach ourselves about solving problems. I have to sift through a lot of books in order to do that, and it would be nice to get some pointers on where to look, as well as more clarification on subtler points. So, I guess, I miss the kind of handholding one often gets in Canada.

‪LC: I see. And now, I want to ask you this. Having experienced traditional classroom learning, private tutoring, self-directed learning and online learning, which of these four would you say you liked the most?

AG: Private tutoring, naturally! It’s as good as it gets – one teacher, one student. If they both enjoy each other’s company the lessons are great, very rewarding for both and the student gets to learn a lot quicker because everything is set up around their specific needs.

‪LC: But don’t you miss the ability to discuss the material with your peers?

AG: Honestly, I never did. This is very atypical, I think, but I never really knew a lot of people in my classes.

‪LC: Fair enough… and what about collaborative problem-solving?

AG: Again, this very rarely happened for me. I either knew how things are done, and told people, or I didn’t and then I tried to go talk to the prof during office hours. Or, in some cases, I was hopeless at the subject and didn’t bother at all.

‪LC: It sounds like doing mathematics is a solitary pleasure for you. I actually think it is for most people. Just that sometimes it can also be helpful to bounce ideas off of others (at least for me).

AG: Oh, absolutely! I find that banging my head against the wall is often far less fruitful than just posing the question to someone else. Even if they just listen. But it’s hard to find a conversation partner even among the mathematically educated people. It’s mostly just too much work for them to turn on their math brain, get to the part where relevant information is stored and unpack it. The knowledge might even be there, but it’s archived, not active. You, by the way, are a happy exception to this rule.

‪LC: Thanks, I appreciate it even though I feel it’s not fully deserved… Well, in addition to mathematics you do have other serious interests. In particular, the classical guitar (which you also inspired me to take up back when we were both 8 or 9). So, I’d like to ask you this. Have you found any parallels between learning complex mathematical concepts and a complex piece on the classical guitar?

AG: I seemed to notice that my memory got better from both. Other than that – not much. I’m very basic in my approach to guitar, I don’t deconstruct the piece, but just learn it as a sequence of notes. So, there’s very little thinking involved. I can certainly imagine doing the same things to a piece of music as you do to a mathematical theory, finding connections between different parts and all. But in guitar I don’t have that kind of training.

‪LC: So no patterns or structures that are inspired by your knowledge of mathematics?

AG: When I play – no. Listening reminds me of mathematics, but not playing.

‪LC: That actually makes me feel better, because I don’t see those either (though I’m much less experienced with the guitar than you are). Can you elaborate on how listening to music reminds you of mathematics?

AG: A mature mathematical theory – classical Galois theory is a good example here – is usually presented as a series of ideas, where each individual one is very simple and very clear. And then you start seeing their interplay and a beautiful and complex structure arises. Now, if each idea is a voice, you get a fugue.

‪LC: Wow! That’s a really nice description!

AG: So, music by Bach, say, always reminds me of mathematics. Not because the music is dry, as a lot of people seem to think, but because a mathematical theory appears as polyphonic music. This only applies to subjects which are well established. The newer ones probably look like contemporary music, which I find it almost impossible to listen to.

‪LC: Haha, I can relate to that! Though I also sometimes feel like a certain harmony emerges after I’ve spent some time trying to wrap my head around some new concepts. An almost musical harmony. Even with less-established fields.

AG: There certainly must be harmony, since “there is no room in the world for ugly mathematics” (either Hardy or Polya said something to this effect allegedly). Every mathematical theory moves naturally towards a state where its definitions pretty much imply most of the theorems. I am sure you can see glimpses of that even where no theory yet exists.

‪LC: Definitely. Well, now that we’re talking about different fields of math, since you ultimately want to become a mathematics researcher, what field or fields appeal to you the most, and why?

AG: This is tough. Until recently I thought that analysis is king and everything else is just ugly. Then I saw that algebra, like some very resistant type of weed, takes over everything soon after it gets just a little bit of ground. So I very reluctantly started looking at it and found it to be enjoyable (in a different way from analysis). Now I am very interested in what they do together. I still think that if I have any ability at all, it’s more suited to questions of continuous nature. One thing I can say for sure is that I will do all I can in order to not do any applied mathematics.

‪LC: Interesting! Well, I promise I won’t take it personally, but would you mind explaining why?

AG: I guess the simple answer would be that I hate to calculate. And feel uncomfortable with the “real world” in general. That mystic place where ideas live just seems so much cleaner and happier.

‪LC: I see… well, I can’t argue with you on the second point, because I’ve also at times felt quite uncomfortable with the real world. But I will disagree that applied mathematics requires calculation. I’d say that most of the calculation is done by computers nowadays. The main task of an applied mathematician is to build models.

AG: Oh, I don’t mean by hand! I meant that I don’t like calculation as an essential tool. As something you need to very seriously consider, even if to tell your computer how to do it. To give an oversimplifying example – I like existence theorems and don’t like numerical methods which actually find approximations to those points which are guaranteed to exist. And to clarify: I am not saying it’s bad mathematics. Just that I enjoy the other kind more.

‪LC: Fair enough, I now remember having an argument with you about this sometime ago… but since we agreed to disagree back then, let’s go on to my final question. The question is the following. If you were to give advice to someone who is considering switching from their field of study to mathematics, with limited prior experience in it, what would you recommend?

AG: Think whether or not mathematics makes them happy. It’ll definitely make you very unhappy very often, and there must be something to justify that. It can’t be success because chances are that you will not succeed, so it must be something in the process of doing it, which rewards you. If it makes you happy, it’s worth trying.

‪LC: Wonderful, on that note I’d like to thank you once again for taking the time to speak with me today. I’ve really learned a lot from talking to you, and I hope my readers will, too!

AG: My pleasure and thank you for having me.

Poetry and Mathematics

Dear readers,

Today is World Poetry Day, and I would like to draw some parallels between poetry and mathematics. It’s certainly true that they are quite different, but the connections are much more significant than one might expect. I will describe some of them, drawing on my experience as a reader and (former) writer of poetry.

The most superficial connection is in the fact that both poetry and mathematics are abstract arts. While the kind of abstraction is different between the two, it is nevertheless impossible to appreciate either one without a proper understanding of the abstraction (though arguably, poetry is more accessible to those without a special training than mathematics). Both poetry and mathematics illuminate a certain part of our experience – poetry, of the real world, and mathematics, of the conceptual world it describes. Just like a good poem can provide insights into particular situations, a good work of mathematics (this can be a model, a theorem, or an analysis) can provide insights into particular areas it is examining. Of course, I’m using “good” in a subjective way, which illustrates another similarity between poetry and mathematics – their beauty is subjective, and repeated exposure leads one to develop a particular taste for the kind of poems, and the kind of mathematics, that one enjoys the most. For instance, I tend to enjoy form poetry with a regular rhythmic structure and containing unusual imagery; similarly, I tend to enjoy mathematical models and methods with a small number of tunable parameters, explicitly stated assumptions and that can be examined analytically rather than exclusively with numerical simulations.

This leads me to the next point – both poetry and mathematics deal with patterns. While a lot of contemporary poetry is written in free verse, historically, most poetry has been written using pre-defined patterns, or forms, such as the sonnet, the villanelle, the rondeau, or, in the East, the haiku, the ghazal, the rubai (plural: rubaiyat), and so on. Each of these forms imposes particular restrictions on one of the following poetic “parameters”: length (number of lines per stanza and in total); rhyming pattern (which lines line with which other ones); meter (which syllables bear stress and which do not); and number of syllables in each line (such as the 5-7-5 pattern of the haiku). One of my poems included in an embarrassing treatise on writing poetry addressed to my friends that I put together 10 years ago (incidentally, while taking breaks from studying abstract algebra), made use of 3 of these restrictions. The entire poem had 6 stanzas, each stanza had 6 lines, and each line had 6 syllables. Mathematics can not only suggest other interesting forms, but can also be used to count the number of possibilities. For instance, the number of possible rhyming patterns of a certain length is counted by the Bell numbers, while the number of possible non-crossing rhyming patterns (eg. excluding ABAB) is counted by the Catalan numbers, mentioned in one of my earlier posts.

But, just as poetry can inspire a mathematical analysis of structure, mathematical concepts have also inspired numerous poems. There is an entire blog dedicated to poems inspired by mathematics. I’m a big fan of using mathematical ideas in poetry, and still recall bits and pieces of a love poem I wrote in undergrad starting with the words “You are not isomorphic to any other”. But, of course, the unrivaled champion of mathematical poetry is, in my opinion, “The finite simple group of order 2” from this post.

Despite these various connections, there has only been one person to my knowledge who has made a significant contribution to both mathematics and poetry – the great Persian philosopher of the 11th century Omar Khayyam. He did a lot of fundamental work in algebra, discovering what we now call Pascal’s triangle, and proposing a systematic method for solving cubic equations by finding intersections of conic sections: parabolas, hyperbolas, ellipses and circles. In addition to this and other mathematical accomplishments, he is most well-known as a poet for his rubaiyat (quatrains), which are very concise yet profound. Here is one of my favorites (original here), translated by Edward Whinfield:

Of mosque and prayer and fast preach not to me,
Rather go drink, were it on charity!
Yea, drink, Khayyam, your dust will soon be made
A jug, or pitcher, or a cup, may be!

On that note, I hope you all enjoy the official beginning of spring as well as World Poetry Day!

Genetics, genomics and languages: an interview with Mathieu Blanchette

Dear readers,

I’m thrilled today to interview someone who introduced me to the fields of computational biology and bioinformatics, which I am still working in today, Mathieu Blanchette. I met Mathieu in 2003, when he taught the Introduction to Computer Science course at McGill. I was then still under the illusion that I could end up double-majoring in mathematics and physics, and computer science was only a backup plan. However I quickly realized that I enjoyed computer science much more than physics experiments.

Mathieu not only made the basic concepts (algorithms and data structures) in his class fun and exciting; he also convinced me that I could actually enjoy programming (I had struggled a lot in a previous class on programming). So when he mentioned his own research in the last lecture of the semester, it was an easy choice for me to ask him to take me on as a research assistant during the summer. That project led to my first rejected paper submission, but eventually it resulted in a paper published with 3 co-authors.

I stayed in close touch with Mathieu after my undergraduate years, and we regularly bounced around ideas – and ping-pong balls – during my visits to Montreal. I also had the chance to interact with a few of Mathieu’s postdocs, an experience that has always been enriching. Mathieu has always inspired me by, on the one hand, his outstanding intuition for and a sharp focus on his field of bioinformatics, and on the other hand, by his breadth of interests within that field. Here is the conversation I had with him.

To find out more about Mathieu’s work you can visit his homepage. Hope you enjoyed this interview, and happy Pi Day!

Addendum: The transcript of the interview is now available here!

Some inspiration from George Polya

Dear readers,

I’m currently at a conference near Stanford, and have been spending some time at the Stanford mathematics library. In today’s post I’d like to share with you my impressions of an exhibit I saw there.

This exhibit is dedicated to George Polya, one of the 20th century’s most distinguished, prolific and versatile mathematicians. While it focuses primarily on Polya’s career as a professor at Stanford, it also documents his earlier life, as well as some of his correspondence with other mathematicians of his time. There are three things that particularly impressed me about Polya’s life work.

The first one is his versatility. He started out his academic career primarily interested in philosophy, but his philosophy professor encouraged him to look into mathematics, in which he ended up doing his doctoral work. He also had an interest in physics that he maintained throughout his life. He is well-known for his work in analysis, with books such as this one, co-authored with Gabor Szego that are still used today in advanced analysis classes. He also made his mark in probability theory, being an early proponent of the idea of a random walk, a key ingredient in many applications, including Google’s PageRank algorithm that many of you use every day when searching for information online. He is also well-known for his contribution to combinatorics (the branch of mathematics that deals with counting families of objects), the Polya enumeration theorem. Other notable areas of his work include algebra, geometry, number theory and numerical analysis. In today’s highly specialized world, few mathematicians are as versatile as George Polya, with the only exception I can think of being Terence Tao, whose blog I mentioned last time.

The second one is his wide-ranging collaborations. He maintained an active working relationship with a number of other outstanding mathematicians throughout his life, working in many disciplines across mathematics as well as many different countries. This meant that he had an active role to play in the development of new mathemtical ideas all around Europe and, later, the United States (his adoptive home after the start of World War II). In addition, Polya published mathematics papers in six different languages: Danish, English, French, German, Hungarian and Italian! Kind of puts all these complaints by doctoral students in mathematics about the second-language requirements in perspective, doesn’t it?

The last thing that struck me was his desire to engage with the broader community, by taking a deep interest in mathematics education. He adapted the Eotvos competition, a contest written by high school students in his native Hungary, to Stanford, which became a popular idea that spread across the country. He also investigated the way mathematicians think, detailing his findings in his famous book Mathematical Discovery. His most famous book, however, remains How to Solve It, well worth reading for those interested not just in mathematics, but also in problem-solving in general. The ideas in this book inspired several artificial intelligence projects, and it remains in print to this day!

In conclusion, I believe that Polya’s long and prolific career is an inspiring example, albeit one difficult to emulate. To end this post on a humorous note, try to find the flaw in Polya’s famous inductive proof of the statement that all horses are the same color; a small prize for the first person to find it correctly!

  • Basis: If there is only one horse, there is only one color.
  • Induction step: Assume as induction hypothesis that within any set of n horses, there is only one color. Now look at any set of n + 1 horses. Number them: 1, 2, 3, …, n, n + 1. Consider the sets {1, 2, 3, …, n} and {2, 3, 4, …, n + 1}. Each is a set of only n horses, therefore within each there is only one color. But the two sets overlap, so there must be only one color among all n + 1 horses.