Climate modeling – a non-expert’s thoughts

Dear readers,

I recently attended a panel on the role of science in society. A highlight of the evening was hearing one of the panelists, a biologist, talk about her experience of being asked for an interview about climate change by a TV channel because she was “not biased”. She, of course, responded that the channel would do best by asking an expert in climate modeling instead of her, since they would have the needed expertise. Her point was that just because you happen to be an expert in something doesn’t mean you’re necessarily biased (and that asking a non-expert showed a lack of understanding on the channel’s part).

Well, in today’s post I’m going to try to do what she refused to do, in other words, provide my thoughts on climate modeling without having any expertise in it. The reason why I feel it is reasonable for me to do so is because of my expertise with mathematical modeling in general. However, in order to ground myself in the subject matter at hand I’ll frame this post as a discussion of this article in The Economist.

The first point the article makes is that there is a mismatch between the predictions of climate models and the temperature trends observed over the past decade, namely, that “surface temperatures since 2005 are already at the low end of the range of projections derived from 20 climate models”. This is used as a way of bringing the models’ validity into question. I disagree with this statement. Although agreement with observational data is frequently used as a litmus test for the validity of a mathematical model, such agreement is neither necessary nor sufficient for a model to be useful. A model’s main task is to provide new insights into the behavior of a system, not to mimic it exactly. In fact, when a model fits the (historic) data too closely, this may be a warning sign that it is being “overfitted”, which means that the model’s parameters are selected to produce trends too similar to the input data. This may be a problem if the model is going to be used beyond its original context (see, for example, the Titus-Bode law which appears to be simply a nice coincidence). At the same time, a model that can predict critical features such as the periodic nature of climate trends without having them “built in” may be providing valuable insights without necessarily producing perfect agreement with data. Finally, agreement with data needs to be evaluated on a longer timescale; being off for one decade is not a model’s death knell…

The second, related, point the article makes is that there are two different kinds of models used to represent climate, namely, bottom-up models known as “general circulation models”, and top-down models known as “energy balance models”. The former are essentially mechanistic models, detailing the processes by which different factors influence climate, including various feedback loops. According to the article, “Their disadvantage is that they do not respond to new temperature readings.” The latter are less complex, and “do not try to describe the complexities of the climate”, but they do “explicitly use temperature data to estimate the sensitivity of the climate system”. Thankfully, the article does not make any conclusions about which type of model is “better”. Indeed, it shouldn’t. Different types of models have different ranges of applicability. While I don’t claim to understand the complexities of the distinctions between the two types of climate models I can draw a parallel with biochemical system models, with which I have extensive experience from my PhD days. There are also two main classes of models there: the more complex kinetic models, and the simpler flux-balance models. Kinetic models are detailed and describe exactly the rates at which each reaction happens and how the concentrations of the various molecules change over time. Their drawback is that they require a lot of prior knowledge which is often difficult to obtain experimentally, such as kinetic constants. Flux-balance models, on the other hand, require a lot less detail, and simply assume that the concentrations of all the molecules have reached a steady state and derive conclusions from there. Neither class of models is “perfect”. The main point, however, is that both types of models provide valuable information and insights about the biochemical system being analyzed, even though they do answer somewhat different questions about it.

The third point is that there is disagreement about the actual value of “equilibrium climate sensitivity”, the long-term amount of temperature increase due to a doubling of atmospheric CO2 levels. Curiously, the article cites estimates by Julia Hargreaves published in 2012, that a cursory search of her website’s publications page has failed to turn up (let me know if you have better luck), as well as “Nic Lewis, an independent climate scientist”, who turns out to be a semiretired financier with a background in math and physics with only one published paper on the subject. It is disappointing that The Economist uses a less-than-credible source like Nic Lewis on par with actual climate scientists. Granted, there may well be disagreement about the actual value, as there should be, but evidence needs to be properly weighted. Call me elitist, but I don’t see how giving equal weight to an amateur scientist with one publication and an expert in the field makes sense. Another source that is mentioned as providing evidence for a lower climate sensitivity is an “unpublished report by the Research Council of Norway, a government-funded body”. If there is one thing that I don’t think should belong to an article about science, it is unpublished reports (that being said there is a recent article by the team behind this research that’s freely accessible). My point here is that not all sources are made equal, and even if peer review does not always guarantee sensible publications, The Economist appears to exhibit a surprising bias in favor of dubious sources.

Another point that I want to make based on my own recent experience with epidemiological models of infectious diseases in Sub-Saharan Africa is that reconciling the predictions of multiple models (or even understanding the sources of the discrepancies between them) is extremely challenging albeit necessary to inform policy decisions (see this article examining the impact of antiretroviral therapy on HIV in South Africa predicted by 12 different models). Nevertheless, if a large fraction of the models predict a particular range of results, the laws of probability suggest that the correct answer is within that range even when several new models appear to give predictions outside that range. Of course, the really hard decision that needs to be made is on the policy level, which in the case of climate change largely falls into the buckets of “mitigation”, meaning a significant reduction in CO2 emissions, and “adaptation”, meaning adjustment to the change in climate, which would make sense if that change were less severe. If I know little about climate science, I know even less about policy, but in this case I tend to agree with William Nordhaus of Yale University, quoted in this article as supporting drastic interventions as a sort of disaster insurance protecting humanity against fairly unlikely events with catastrophic consequences.

In conclusion, The Economist does an excellent job of describing the scientific challenges of modeling climate in simple terms, but appears to overestimate or overstate the disagreement between different models, partly based on its use of less-than-credible sources. It also highlights the difficulty of taking a scientific conclusion and translating it into policy. But it seems to be missing some fundamental points about models by evaluating them purely based on their agreement with observations. Furthermore, it is misleading in suggesting that newer models may be more accurate than older ones, when they actually seem to be based on a less detailed, rather than more detailed, representation of the system in question. I hope that, by educating the general public about the development and use of mathematical models, as well as the conclusions that can and cannot be drawn from their results, we mathematicians and scientists may one day help it reach a level of understanding that will make such articles superfluous.

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.

Other blogs I read

Dear friends,

In today’s post, I want to share with you some of the other blogs I read regularly, both mathematical and non-mathematical. I have six in each category, and they will eventually form part of my blogroll once I figure out how to create it. I hope you will enjoy them as much as I do. And although it currently seems to be inactive, I want to mention Steven Strogatz’ blog with the New York Times, which was a source of enormous inspiration for my own.

Mathematical blogs

1) Terence Tao’s blog contains lots of technical material from Terence’s work in various areas of math, but also advice to mathematicians and fairly accessible expositions of new mathematical developments.

2) Timothy Gowers’ blog discusses interesting mathematical topics, but also projects such as Polymath, a collaborative platform for mathematicians, and issues such as open-access mathematics journals.

3) Peter Cameron’s blog discusses interesting connections between combinatorics, algebra and discrete mathematics, and also touches on issues such as mathematics education and the running of universities.

4) Scott Aaronson’s blog focuses on complexity theory, the area of mathematics concerned with the amount of resources needed to solve various problems on a computer, as well as quantum computing.

5) Maria Monks’ blog collects various mathematical “gemstones”, which are either problems with nice solutions or useful techniques, and classifies them by difficulty level; it offers something to everyone.

6) Sam Bankman-Fried’s blog contains mainly game-theoretic analysis and simulations of sports events and elections, but also his reflections on rational philosophy and other not strictly mathematical topics.

Non-mathematical blogs

1) Hillary Rettig’s blog contains a wealth of valuable advice for dealing with productivity barriers and other work (and life) issues directed at activists, writers, academics and all people with ambitious goals.

2) Chris Dippel’s blog, Retronyma, analyzes the latest developments in the field of global health, more specifically the biotechnology and medical innovations being made available in low-income countries.

3) Pragya Bhagat’s blog, The Road Not Taken, documents her return to her native India and her work on understanding and addressing the sources of various socioeconomic problems in rural parts of India.

4) Assaf Urieli’s blog, Moyshele, describes his various projects, such as translations of songs between English, French and Yiddish, publication of a book of riddles, and the adoption of a child from Russia.

5) Brooke Shields’ blog, Veggie 365, lists her delicious vegan recipes for breakfast, lunch, dinner, and, of course, dessert. Every recipe is accompanied by a photograph and some are quite easy to make.

6) Jonathan Sharman’s blog, which he just started, describes his experience as an experimental and theoretical physicist (some of the projects will also appear on video) and his thoughts on science today.

What other blogs have you found particularly interesting? Please let me know in the comments!

Acting, academia and industry: an interview with Melodie Mouffe

Dear readers,

It is my pleasure to introduce to you today Mélodie Mouffe, a young mathematician from Belgium whose career path so far has started in academia and ended in industry, and is somewhat the opposite of my own. I hope to present to you the different challenges faced by mathematicians in the two environments by discussing with Mélodie her experience in both, to give you a better idea of the tradeoffs between the two.

Mélodie’s interests extend beyond mathematics research into teaching, as well as music and acting. We were able to touch on a number of interesting topics in our conversation, which I hope you will enjoy! Here it is.

Mathematics of Relationships

Dear readers,

Many people around the world today will be celebrating Valentine’s Day, a day that celebrates romantic relationships. It may be surprising to you that mathematics can say something helpful on that topic, a never-ending source of complexity that resists clean definitions. Nevertheless, mathematics can in fact provide useful insights in some specific contexts, which I’m going to talk about in today’s post.

Perhaps some of you are considering making a marriage proposal to your significant other, or will have to respond to one (for those of you who don’t like the idea of marriage, feel free to substitute civil union or whatever other form of commitment works for you, as long as it entails long-term exclusivity). In case you have to respond to such a proposal, what is the best way to decide whether to say yes or no? This is a difficult problem, involving a host of variables such as compatibility, emotional connection, similarity of values, and chemistry. One of the challenges of this situation is the fact that, assuming that you’ve been in several other relationships before, your current partner is unlikely to be superior to all your previous romantic partners on all these criteria. In fact, it’s easy to see that the more partners you’ve had, the less likely the current one is to simultaneously maximize all these criteria. But suppose for the moment that you’re able to compare any two of your partners. What’s the optimal decision then?

There is a simple mathematical answer to this question, originally called the fiancée problem, solved in 1958. It depends on a number of critical assumptions. First, you have to be able to give each partner a “suitability score”. Second, you have to assume that the partners present themselves in an order that’s random with respect to this score. Third, you have to respond immediately and can’t change your mind later. Lastly, you need to know how many partners you might have to choose from over your lifetime, which we will call n. Under those conditions, the optimal strategy for choosing the best possible partner turns out to be to say no to roughly the first n/e partners, then to say yes to the first one who happens to be better than every other one you’ve considered so far. Here, e = 2.71828… is Euler’s number, the base of the natural logarithm. This strategy gives you a roughly 1/e (or about 37%) chance of finding the best partner for large n. For small values of n, there is an explicit formula. For example, if you expect to have 8 to 10 partners, you should say no to the first 3 and then yes to the first one who is better than all the previous ones, which will give you an approximately 40% chance of saying yes to the best one.

Of course, finding the best possible partner is not a guarantee of the stability of the marriage, because it may happen that other options will present themselves to either yourself or your partner later on, and an incentive to deviate from exclusivity (or towards commitment with a different person) will appear. Can we guarantee that this doesn’t happen? Not an easy task, when both infidelity and divorce rates are so high. But part of the problem lies in the fact that we as a society have more options now than we ever had, and we can no longer exhaustively evaluate all of them ahead of time when making a decision. Suppose, however, that all of us do know all of our options ahead of time, and can rank them without any ties. Is there a way to guarantee stable marriages in this case? It turns out that there is, provided, of course, that there are as many men as women (I do realize that this is a heteronormative restriction, but it will make the discussion easier – in general, the important thing is that there are two distinct groups of people who can only form marriages with someone from the other group, not from the same group).

This problem, known as the stable marriage problem, was solved in the early 1960s by David Gale and Lloyd Shapley, the latter being a co-laureate of the latest Nobel Prize in Economics for his varied work. The solution is both natural and very elegant. Once again, we use n to denote the number of men (and women) who need to be married. Each woman has a ranking of the men, and each man, of the women. Initially, none of them is engaged. The solution proceeds by rounds. In each round, an unengaged man proposes to the highest-ranked woman on his list to whom he has not proposed yet. The woman always provisionally says “maybe” to him if she is not engaged or is engaged to someone she prefers less (in which case the man she is currently engaged to gets the initial “maybe” changed to a “no”), and “no” otherwise. At the end of this process, every man will be engaged to some woman, and marriages occur. These marriages will be stable in the sense that no man and woman will want to run away with each other from their spouses. Indeed, if a man, say Bob, likes a woman, say Alice, more than his wife, this means that he proposed to Alice before he proposed to her. But then, Alice must have said “no” to him at some point (otherwise they would be engaged at the end), so she prefers her assigned spouse to Bob.

An interesting side note is that the marriages that result from this process are optimal for the men (in the sense that no man can do better in a stable solution than the spouse he ends up with). This is due to the fact that a man is in principle able to propose to every woman on his list, while a woman may only get a limited number of proposals. From that point of view, it pays to be proactive in making proposals (though I’m not necessarily advocating for a reversal of the “traditional” model based on this analysis).

Many of you will be quick to object that the two problems I discussed so far – the fiancée problem and the stable marriage problem – are fairly unrealistic and do not reflect the way our society works. That’s a fair criticism, to which I can only respond that all mathematical models, especially those that have an elegant analysis or solution, involve simplifications of reality. But there is also a more serious approach to the mathematical modeling of relationship dynamics that appeared a few years ago. It is based on a system of differential equations, and, like any good model, it has parameters that can be estimated from experiments and makes falsifiable predictions. The book describing this approach has made it onto my reading list, right behind Nate Silver’s book that I mentioned in an earlier post, and I’ll report on it as soon as I get to read it. Meanwhile, I wish you all a happy Valentine’s Day, no matter your relationship status! And remember that mathematics is like love; a simple idea, but it can get complicated.

Chocolate, chance and choosing a problem

Dear readers,

So far in my blog I’ve described various areas of mathematics, discussed common stereotypes and misconceptions about mathematicians, and even interviewed several of them. However, I’ve never quite managed to give you a sense of what it is that professional mathematicians actually do. Today, I’m going to try to do just that, using a problem I came across recently as an example. I’ll describe my own process from beginning to end; keep in mind that this process may be quite different for other mathematicians.

Let’s start with the problem choice. I was at a workshop on entrepreneurship where the workshop leader made us play a simple game. We were given a bag with 10 light and 10 dark chocolates, and had to call the type of chocolate we were going to draw next. We always got to keep the chocolate that we drew, and if our guess was correct, we also got to continue playing; otherwise, the game ended there. This game was used as an example of a fully predictable situation. The best strategy also seemed pretty clear: guess at random if there are as many light chocolates as dark ones, and guess the more abundant kind if there is an unequal number to stay in the game longer. When I saw this game, though, a new question immediately came to my mind: what was the expected value of this game? In other words, if I value a chocolate at a dollar, how much should I be willing to pay to play the game?

I took out a pencil and a sheet of paper and within a few minutes, I was able to calculate the value of the game, which was slightly above 2 dollars. This calculation was pretty simple; it used a recursion relating the value of the game to the values with either one less dark or one less light chocolate (which were equal since all the chocolates had equal value), and relating that value to the value of the game with one less dark chocolate and one less light one. While I was glad to be able to solve this particular question, another one immediately came to mind. What if I valued dark chocolates more highly than light chocolates? In that case, my recursion showed something slightly unexpected: the optimal strategy is still to guess the more abundant type of chocolate, but if there are equally many of each kind, it’s better to guess light even though I prefer dark. This becomes obvious when there is only one of each kind, but less obvious when larger numbers are involved.

At this point, the game had piqued my interest enough for me to try to find a general formula for the value of the game for different relative preferences over light and dark chocolates. At first I computed some values by hand, but couldn’t see a pattern. I sent the problem with my preliminary results to a few of my fellow mathematicians, but none of them saw a pattern either. Then I decided to ask MAPLE, the symbolic algebra software, for help. It didn’t find a simple pattern, but by working with it, I eventually saw a pattern myself, which it helped me confirm. I then checked by hand that the formula I had found satisfied the recursion. At that point I was ready to write down a proof of my formula, involving recursion and some case analysis.

Interestingly, the formula involved a famous sequence of integers known as the Catalan numbers; they count various mathematical objects, such as tied two-candidate elections where one candidate is never behind the other in the partial counts, the ways a polygon can be divided into triangles by joining vertices, and the ways an expression involving only one operation such as addition or multiplication can be bracketed. I was intrigued by that and had a hunch that something interesting would happen in a more general scenario, say with three kinds of chocolates involved. This turned out to be the case.

There were actually two ways of generalizing it, of which I picked the more natural one (where you had to guess the right kind among the three to stay in the game). Strategy-wise, the same result turned out to be true; namely, you should always guess the most abundant chocolate, breaking ties in favor of the one you like the least. The new formula, discovered with the help of MAPLE and a lot of trial and error, involved a new sequence of numbers that did not seem to be known, which I tentatively called the poly-Catalan numbers. Proving the formula turned out to be rather challenging, not only because of a much larger number of cases to analyze, but also because of the complexity of the formula itself. However, I finally managed.

I believe my work on this problem illustrates several common approaches to mathematics. First, I found an interesting situation which I could turn into an equally interesting mathematical problem. Second, I spent time figuring out a solution to this problem, unsuccessfully at first, and successfully later when I got a computer involved. Third, I found an interesting feature of the problem’s solution that made me interested in making it more general. Fourth, I developed the tools needed to solve the more general version, and discovered a previously undiscovered sequence of integers. Finally, I wrote up my solutions and I’m currently putting the final touches on it so I can try to get it published in a journal. In the process, I also raised a number of new questions that I hope will be answered by other mathematicians someday.

As for whether this kind of work is actually useful, I don’t know for sure. It was certainly useful for me on many levels: it taught me to be more patient, made me learn some new tools, and involved the help of a computer. The new sequence of numbers I discovered in the process may well turn out to be important in a completely different area of mathematics or the sciences; however, even if it doesn’t, I really enjoyed this work, and ultimately, that’s the most important justification for doing it.

The Missing Equation

Dear readers,

For many people, the word “mathematics” immediately evokes equations. Equations certainly stand for something mathematics does very well – take several quantities in symbolic form and join them together into a single, concise yet powerful statement. Equations can provide tremendous new insights into the way the quantities relate to each other, and revolutionize subsequent developments in science.

This is precisely the argument made by the fantastic mathematician and writer Ian Stewart in a recenty published book. The book describes the meaning 17 equations that literally changed the world. Six of them are simply definitions – logarithms, derivatives, imaginary numbers, the normal distribution, the Fourier transform, and Shannon’s information entropy, all fall into this category. Three others uncover relationships between mathematical quantities – the Pythagorean theorem, Euler’s formula, and the logistic growth model. Seven others are fundamental physical equations – Newton’s law of gravitation, the wave equation, the Navier-Stokes equation, Maxwell’s equations, Einstein’s famous E = mc2, and the Schrödinger equation, as well as the second law of thermodynamics, which is actually an inequality. The last equation is the Black-Scholes equation from economics, whose incorrect application arguably contributed to the financial crisis of the recent years; I’m looking forward to reading more on this soon.

While this selection of influential equations is necessarily limited, I was somewhat disappointed not to see any equations from my field, mathematical biology. It is true that there aren’t as many equations in mathematical biology as there are in physics, and their role is somewhat more limited. But there is one equation which I believe deserves a mention, perhaps in the next edition of Ian Stewart’s book: that is the defining equation of the Hardy-Weinberg principle: p2 + 2pq + q2 = 1. Incidentally, the first part of the principle’s name comes from the same Hardy who I quoted in an earlier post stating his pride in the fact that none of his work in mathematics would ever have any applications. After explaining what the equation means, I’ll show how it applies to blood types, forensics, and the search for genetic markers of diseases, in spite of Hardy’s claim. Weinberg was a German scientist who independently discovered this principle at the same time as Hardy, as seems to so frequently happen with good ideas.

Suppose that we have a single human genetic locus (position), which can have either one of two alleles (forms). These two alleles are commonly denoted by A and a; for convenience, A is called dominant and a, recessive. Since every person inherits one copy of the gene from their father and one from their mother, their genotype (underlying genetic makeup) can be either AA (homozygous dominant), Aa (heterozygous), or aa (homozygous recessive). The Hardy-Weinberg principle tells us what happens if the allele frequencies do not differ between males and females, and the mating happens randomly (to be contrasted with assortative mating, where individuals prefer to mate with those whose genes are similar to their own). In this case, if p is the frequency of allele A and q = 1 – p, the frequency of allele a, the frequency of AA is p2, that of Aa is 2pq (the factor of 2 is needed because we ignore the order, ie. Aa and aA are the same), and that of aa is q2. The equation tells us that these frequencies add up to 1; the principle itself tells us that these frequencies do not vary from one generation to the next under the conditions of random mating, provided there is no mutation, migration, or natural selection happening.

This idea can be generalized to more than two alleles. For instance, the blood type can be described as a trait with three alleles, A, B and O, with O being recessive. This means that there are four possible blood phenotypes: A (which could come from genotypes AA or AO), B (which could come from BB or BO), AB or O (which could only come from OO because O is recessive). Knowing the frequencies of the A, B and O alleles (say, p, q and r) would allow us to calculate the frequencies of each blood type: p2 + 2pr, q2 + 2qr, 2pq and r2, respectively (you can easily check that they add up to 1 if p + q + r = 1). In the same way, knowing the frequencies of each blood type can give us the frequencies of each allele. We can also test for significant deviations from the equilibrium frequencies in a population to see if the mating between people of different blood types happens randomly. For example, if we look at Canada, where the fractions are 46%, 42%, 9% and 3% for each of the 4 blood types, we get estimates of p, q and r that are not consistent with the Hardy-Weinberg law, and conclude that the mating is not random.

When forensics has to answer the frequently asked question of how likely a person is to have the same genetic variants as the person who committed a certain crime, the Hardy-Weinberg principle plays an important role. It is typically used to justify the assumption that the probability of a person having a given set of genetic variants is the product of the individual probabilities (frequencies) of the variants. However, in some specific cases, this assumption may not be justified because of deviations from the equation we have been discussing, and this may (at least in principle) lead to some false convictions.

One final application of the Hardy-Weinberg equation comes from the search for the genetic causes of complex diseases. In this situation, we typically look at a large number of people who have the disease (called cases) and a large number of people who don’t have it (called controls). A genetic variant that deviates from the Hardy-Weinberg equation in the cases, but not in the controls, is very likely to be associated with the disease. Of course, after some spurious discoveries in the early days after the Human Genome Project was completed, the association is now considered to be confirmed only when it has been found in a group of cases distinct from the initial group (called a replication cohort). As for the variants that deviate from the Hardy-Weinberg equation in the controls, they are usually discarded from the analysis as unreliable. In this way the equation is an important tool in the analysis of GWAS, or genome-wide association studies, the tool of choice for the discovery of genetic disease markers.

I hope I convinced you that the Hardy-Weinberg equation deserves to be included among the most influential equations of our times. What other equations would you suggest adding to Ian Stewart’s list?