The Millennium Problems:

The Seven Greatest Unsolved Mathematical Puzzles of Our Time

(A brief review)

Popular math and science books have to walk a very fine line: on the one hand, if the material gets too complicated or abstract, you’ll lose readers. The corollary to this is fairly straighforward: editors and publishers tend to be wary of the complicated and the abstract, because they’re afraid of losing readers. (Stephen Hawking was apparently informed that each equation he included in A Brief History of Time would halve the sales. He therefore limited the number of equations to one, E = mc2.)

On the other hand, in the quest to make everything as manageable and benign as possible, writers can oversimplify to the point that they misrepresent the actual material. This is both irritating (and/or confusing!) for a reader who does have some background in the subject, and – in my opinion – does a disservice to the then (misinformed) public and (misrepresented) scientific community.

I have immense respect, therefore, for those authors who manage to produce engaging, accurate, comprehensible, and accessible accounts of their research – especially considering that most of the scientists and mathematicians who undertake to write these popular books are not trained writers.*

The latest entry on my reading list does a superb job of carrying off this balancing act, and quite honestly, it’s a book that I’d love to go back in time and give to my fourteen-year-old self, or that I’d recommend as a gift for any budding young student with a good grasp of high school math.

The book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, by Keith Devlin, who is currently a visiting professor at Princeton, and who normally teaches at Stanford, where he is the Executive Director of the Human Sciences and Technologies Advanced Research Institute.

The premise of the book is quite simple: the Millennium Problems are, as the subtitle suggests, seven of the greatest mathematical problems for the 21st century, which have baffled all of the world’s leading mathematicians for up to a couple of hundred years. In 2000, the Clay Mathematics Institute, after conferring with leading experts around the world, announced a competition: they selected a list of seven problems, that are not only extraordinarily difficult, but whose solutions will have extraordinary implications in advancing the frontiers of mathematics, physics, computer science, or engineering. Anyone who can solve one of these problems will be awarded a prize of $1 million. (Now almost fourteen years after the contest was announced, exactly one of those problems – the Poincaré Conjecture – has been solved.)

It should be clear right now that none of these problems are remotely easy. To give you an idea of the challenge Devlin took on in agreeing to write a book that would explain all these problems to a general audience, let me quote the technical formulation of the Hodge Conjecture, one of the seven problems:

Every harmonic differential form (of a certain type) on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles.2

I suspect even those with undergraduate or graduate degrees with mathematics might have difficulty understanding that one! Some of the other problems – the Riemann hypothesis or P vs. NP, for instance – are a little less complicated at first glance, but none of them could ever be termed “easy”. Nonetheless, Devlin does an admirable job of introducing the history of each problem, sketching in some detail the implications that a solution would have for modern science and technology, and translating what is often highly technical mathematics into intelligible English prose.

The book is also an accessible (!) tour of some of the most fascinating branches of modern mathematics that students are rarely exposed to unless they are studying to become mathematicians themselves – like algebraic topology (third-year class at most universities), complex analysis (also a third-year class), number theory (ditto), and partial differential equations (fourth year/graduate level). Do you have to be a mathematician to read this book? Absolutely not – Devlin assumes only a reasonable knowledge of high school level mathematics; some background in calculus would be helpful, but is not – in my best judgment – essential. That being said, if the idea of any equation harder than E = mc2 has you running for the hills, this is not your book: the level of mathematics presented is a good deal higher than in most popular books I’ve encountered. (So do be prepared to do some thinking – these problems have been giving the professionals headaches for years!)

But if you’d like an introduction to the frontiers of mathematical research today, or you happen to know a student who likes math and is wondering what one might, well, do with that interest, I’d say that this would be a great place to start. It’s also sufficiently accurate and well-written that (as a reader who does have a somewhat more advanced background) I found it a really worthwhile and engaging read.

(Now I need to go find a textbook on cohomology classes …)

1Some other names that come to mind, off the top of my head, of scientists who are also excellent writers for a general audience: Leonard Susskind, Stephen Hawking, and Richard Feynman (physics); Jeffrey Rosenthal (statistics); Ed Frenkel (mathematics – Langlands program).

2 Devlin, p. 214


What Happened to Mathematics?


When I was four years old, this was my favourite book:


Apparently I carried it everywhere, including up trees, to playgrounds, and whenever I went over to play at my next-door neighbour (and best friend) J’s house.  One of my earliest memories, actually, concerns a certain science project that I wanted to do, involving a simple circuit, a light bulb, a switch, and a buzzer. J had to ask her mother’s permission, because it used electricity, and I distinctly remember flipping to the relevant page in the book to explain just what it was we wanted to do. J, as it turned out, was then forbidden from getting involved because her mother was worried about us getting electrocuted.

I’m not quite sure why that incident was so memorable – possibly because even at four, I was conscious of the injustice of the situation: there was absolutely no way we were going to electrocute ourselves with two 1.5 volt batteries, a pile of short wires, and a light bulb. On the other hand, my ability to explain this cogently and convincingly to J’s mother when I was only a couple of feet tall … was undoubtedly more than a little questionable.

The reason I’m bringing this up is that despite the fact that I’m an English student, I’ve discovered enough of a lecture scene here in Toronto that math and physics and astronomy are going to crop up quite frequently. And, well, I know from experience that when I mention math in front of literature students, I generally get one of two reactions:

“… That’s weird, and you’re crazy.”


“… That’s kind of cool, and you’re still crazy.”

(or, more recently, from a fellow MA student: “You could have said you did anything else in your spare time, and I would have been more able to relate.”)

So before I confuse everyone who knows me almost exclusively as a writer and an actor, I’m going to talk a little bit about my own background in mathematics, because it’s eccentric at best, and really downright odd compared to the usual system.


Elementary School

I took exactly one mathematics course, pre-university, in the conventional school system, and exactly two mathematics courses as part of my undergrad degree – the rest of my mathematics background is entirely attributable to a combination of homeschooling, self-study, and online coursework.

My parents chose to homeschool me for a number of reasons – the foremost of which was that the school system where we lived at the time was terrible, to the extent that the Grade 1 teacher at the local elementary school, a friend of my mother’s, advised her to “put her [aka me] anywhere but here.” Living in a small town where the high school was well known to be the worst in the province (the elementary school and middle school weren’t much better), the options for other schools were extremely limited, and my parents decided to homeschool on the premise that they would take things one year at a time, and if they couldn’t do better than the traditional school system, then I should be put back in school immediately. To that end, I wrote yearly standardized tests that covered all aspects of the normal public school curriculum, and my parents would use my results on these tests both to gauge how well I was doing compared to my cohort, and to plan the next school year. The premise was that if I was doing really well in reading comprehension and writing skills, but my social sciences scores were lower, we’d do more history, geography, and economics the next year.

It quickly became apparent, however, that the tests were useless for distinguishing which subjects I was good at and which subjects I needed to work on, because my scores were so uniformly high that the test at my grade level didn’t give any useful information. So my parents simply got me to write tests above my grade level – eventually, five years above my grade level became the new norm.

In Grade 6, I scored above the 99th percentile on all math-related sections of the Grade 11 test. (My reading/editing/verbal scores were all downright mediocre by comparison!) This was possible because for the first several years of elementary school, I did two grade levels of math per year. When we visited my cousins in Ontario when I was nine, my oldest cousin was in the ninth grade – and for years, the way I remembered how much older she was … well, was based on the fact that we discovered partway through the visit that we were using the same math textbook. After Grade 6, I worked through a textbook called Saxon’s Advanced Math – which, in retrospect, is roughly equivalent to the contents of the Grade 12 advanced functions and data management courses, plus the vectors part of the calculus and vectors course. In Grade 8, I then took a short detour by doing a full year of geometry, which gave me a rigorous background in proofs, before starting calculus in the fall of Grade 9.

Up until this point my mother had a pretty good system going: she would hand me a textbook, and assign me a lesson to do each day. I would read the textbook, learn the material, and do the problems; only if I had any questions would I ask my mother, who would then sit down and explain the concept to me. For the most part, however, I worked completely independently. After every four lessons there was a test; these were supervised and timed, and any score lower than eighty percent was unacceptable. These lessons were further supplemented by daily speed computation drills. By the time I got to Advanced Math, I was bored by the endless repetition in the textbooks we used – in a daily problem set of twenty questions, there might be five or six on that day’s topic and the rest would be review of previous topics – so my mother let me use my own judgment to decide which problems I needed to do, and which ones I had already mastered. As long as my test scores remained high, this continued; if my test scores had been lower than 80%, I would have had to go back and do the review problems.

This system began to fail when I started calculus at the beginning of Grade 9. In general, I was pretty good at teaching myself from a textbook. But when I did get stuck on something, I’d always been able to ask my mother, who studied science in university and was a bit of a math whiz kid (her profs thought she should have been an accountant). When I got to calculus, my mother realized that she didn’t have all the answers anymore – although she’d done quite a bit of math in university, she would have had to relearn it in order to continue teaching me.

So she did a bit of research, and, for the latter half of Grade 9, signed me up to do first-year calculus with Stanford University, since it was offered as an online course through their Educational Program for Gifted Youth.

Every math course I’ve taken since has had the burden of trying to measure up to Stanford. It was simultaneously the most rigorous, most exhausting, and most mentally exhilarating course I’ve ever taken. I did have all the necessary background, but Stanford stressed independent and creative thinking in a way that my previous textbooks hadn’t. As I said, when I got stuck on something at home, I asked my mother and she explained the concept; I then went and did the rest of the problems. When I got stuck on a twenty-step complicated derivation in Stanford’s course, the tutor provided a hint for how to tackle step one, and then would expect me to work the rest out on my own. Roadmaps were not, in short, provided – instead, you were expected to think.

My background in geometric proofs stood me in good stead; so did all those speed math computation drills my mother had insisted upon (Stanford entirely banned the use of calculators); and so did a very strong work ethic. All the same, for those three months of the year, I spent four hours a day doing calculus. At the end of it, I had both my mark from Stanford, and my second perfect score on an Advanced Placement exam (my first was microeconomics, written back in Grade 7).


High School

That summer, the summer I was fourteen, I was reading a high school physics textbook in my spare time (from Apologia Publishing, to be precise), and partway through the section on the structure of the atom, we got to orbitals and electron clouds. The textbook pointed out that these were described by an equation called Schrödinger’s equation, but that the solution of said equation was far beyond the scope of the text, since students would have to do a couple of years of mathematics beyond calculus in order to have any hope of understanding it.

Those who know me well will know exactly what’s coming next. Telling me I’m not capable of learning something is a pretty effective way of getting me determined to learn it: I have, after all, spent the last fifteen years of my life operating under the very firm conviction that there is no (academic) subject that I cannot master with enough hard work, and I haven’t been proven wrong yet.

Ergo, I armed myself with the Internet, my Stanford textbook, and a score of my parents’ old math and physics textbooks, and set out to master Schrödinger’s equation for the hydrogen atom.

In the process, of course, I had to teach myself a fair bit about differential equations, which were only briefly treated in the Stanford course, and I became quite fascinated by everything that had anything to do with quantum mechanics (and quantum field theory, string theory, cosmology, high energy particle physics more generally…)

That fascination would be sufficient for another post entirely, but for the moment I’ll stick to the mathematical side of things. My further studies in mathematics were seemingly derailed when I went to high school that fall – I spent two years at a school noted for its strong Advanced Placement program – and I was exempted from taking any math classes based on my calculus marks.

The one math course I did take in high school was AP Statistics/Grade 12 Data Management, which was an exercise in futility from beginning to end: the class only covered about two-thirds of the AP curriculum, and it was the two-thirds that I already knew from the Advanced Math textbook. I ended up teaching myself the rest of the AP material, and spent class time working through the rest of Stanford’s calculus textbook – the first-year course had only used the first eight chapters of a sixteen-chapter textbook that the university also used for higher-level courses, so I worked my way through multivariable calculus while my classmates were learning about standard deviations and how to work a graphing calculator.

My course load during those two years left little time for casual study outside of class, but I didn’t abandon math entirely – multivariable calculus and differential equations became my recourse in any class that was exasperatingly easy.


Gap Year

I wanted to go to Princeton, or Waterloo, or Toronto … or virtually anywhere, I must confess, other than Ottawa, and since my parents weren’t particularly comfortable with the idea of me taking off on my own at age sixteen, they encouraged me to take a gap year before applying to university. In fact, they actively encouraged me not to study – to take a break and relax, as it were.

This lasted from the end of June until the middle of August, whereupon I became thoroughly bored of doing nothing, and decided I would write a few more AP exams, using free lectures posted by the Massachusetts Institute of Technology online to prepare.

While visiting the University of Toronto’s bookstore that summer, I also picked up Quantum Mechanics: An Accessible Introduction, by Robert Scherrer, which was, at the time, the textbook that Toronto’s physics department used to teach their third-year quantum mechanics course. I was working at a fast-food place for most of the year, and I took Scherrer with me and spent my breaks reading through the text and working out the problems in a little notebook. (My managers probably thought I was more than a little eccentric, but I worked hard when I wasn’t on break, so they were happy in the end!) In the process, I had to acquire a basic knowledge of linear algebra; fortunately for me, Scherrer’s text assumed no previous knowledge of it, and spent a couple of chapters explaining, in accelerated format, all the linear algebra that was needed in order to understand the rest of the book.



So, at this point, the question might (justifiably) be posed … how exactly did I end up in English?!

Well, I applied to universities as a double major in physics and English – chiefly because I couldn’t decide whether I wanted to do a doctorate in medieval literature, or a doctorate in theoretical physics. (I distinctly remember a conversation in Montreal in which a fellow actor had to convince me that doing two doctorates would be patently absurd.) My interest in medieval literature has a similarly lengthy story, but in brief: blame Tolkien, the national spelling bee, and a very persistent fascination with other languages.

The reasons I switched out of the physics major, though, were more complicated. Analyzing my own decision-making process is never an easy task, but there were basically three reasons:

1) I really wanted to act. And so if I wanted to work in theatre, why wasn’t I studying theatre?

2) I became progressively more convinced that, as an alternate to theatre, English and writing were the way I wanted to head. A major in physics became less of ‘career preparation’ and more of ‘something I really like studying as a hobby’.

3) This was, to a large extent, because the math and physics classes that I took in my first year at UOttawa were a bit of a disaster.

Those who know me well might not be surprised to hear me quote Sherlock Holmes on this one:

 “My mind rebels at stagnation. Give me problems, give me work, give me the most abstruse cryptogram, or the most intricate analysis, and I am in my own proper atmosphere. But I abhor the dull routine of existence.”

I was able to skip first-semester calculus and first-semester physics, chemistry, and statistics, based on my Advanced Placement marks, but the University of Ottawa will only grant a half-year exemption for AP courses that cover a full year’s worth of material.*

The physics class, therefore, covered exactly what I’d done previously via MIT. The calculus class (supposedly for scientists and engineers) was taught by a professor who had to remark the entire first midterm after realizing that she’d taught a major unit on real-world applications of the integral … and had completely forgotten to account for the existence of gravity. She made similarly egregious errors on a regular basis when ‘explaining’ proofs, so sitting in class, at least for the first third of the term, was an exercise in sitting on my hands and biting my tongue. I survived with my sanity intact by skipping the last two-thirds of the classes and showing up to write the final, on which I apparently got an 98.

Linear algebra, in contrast, was well-organized, and well-taught – the professor was excellent, and so was the textbook he’d chosen.

Unfortunately, compared to what I was used to, the class also moved at the pace of the proverbial snail, and I have never felt so much like Hermione in my life – she has a tendency to get insanely high marks, well over 100%, on tests and exams. In the case of linear algebra, the professor included bonus questions on all assessment items: it was entirely possible to get 123.5% on all three assignments and 107% on both midterms.

(I have no clue what I got on that final because I never saw it afterwards, but I’d bet quite a bit that my average in that class was well over a hundred.)

Obviously I can’t fault the professor on this one – about half of the class either failed or dropped the course, so clearly it was at least moderately challenging for the average incoming student! But when I am not challenged, I get bored. When I really, really am not challenged, I turn off.


So … I’m impatient, now what?

My lack of patience with the trivial is undoubtedly by some accounts a serious character flaw. Granted. For better or for worse, my need to be challenged and my continual desire to learn something new, are part of who I am, and appear to be here to stay.

To put things in perspective, though, by the end of my first year at university, the last time I’d taken a math course that actually challenged me was the Stanford class when I was fourteen. I wanted to learn; I desperately wanted to be challenged; and my classes weren’t giving me that. Everything interesting that I’d learned about mathematics since Stanford had come from my own study: from textbooks and online lectures freely available to anyone in the world with a working Internet connection. Is it all that surprising that I came to the conclusion that if I wanted to study math and science, I would do better to buy the textbooks and work at my own pace, rather than sit through another round of boring second- and possibly third-year classes until we finally got to something I hadn’t learned?

Or so went my logic at the time. If I’d accepted Waterloo or McGill, where I would have been placed straight into second-year courses, would I have made the same decision? I don’t know.

Perhaps it’s simply that I like too many things: abandoning theatre for too long makes me miserable, and the same goes for English, writing, and literature … and exactly the same is true for math and physics. Never yet have I encountered a university that would actually let a student study everything in depth! (Though Toronto is coming close – more on this in future posts.)

But yes, I am a writer who likes quantum mechanics, and an actress who can derive Laplace’s equation in the polar coordinate system on a blackboard if you hand her a piece of chalk, and an essayist who spent her childhood building radios and telephones and airplanes and running wires up four flights of stairs to construct a telegraph because her parents wouldn’t let her drill a hole in the ceiling.

Crazy? No. Eccentric? Possibly. Way too much fun? Yes.

I spent most of today writing an article for The Varsity about the importance of fostering youth engagement in science, and the importance of promoting math and science literacy, so there will be more on this topic eventually, but this is also something I believe quite strongly: obviously not everyone takes the route I did. Most people, in fact, don’t take the route I did. But everyone should learn basic math and science skills. Being interested and engaged in current developments in physics, chemistry, biology, aerospace – that shouldn’t be something just for “nerds”, or just for STEM students. It’s not “irrelevant” or “unnecessary” or “weird”, and I shouldn’t have to be doing a PhD in physics to be interested in Arkani-Hamed’s recent discovery of the amplituhedron  – this knowledge is essential, in more fields and occupations than I think people realize. And the innovators in these fields will and have and continue to change the world.




*While this was obviously problematic, in UOttawa’s defense I must say that they are remarkably generous, compared to other schools, about the total amount of AP credit they will allow a student to count towards their degree – I was granted a full year’s worth of credits.