I’ve had a number of fine surprises this year, but Piper Harron’s PhD thesis in number theory is at or near the top. Why takes a bit of explaining.
Math is core to almost everything we do, but sadly it has become abstracted for too many people. Part of the blame falls on the mathematicians. The failure rate of math undergrads in many schools is very high. You come into realizing how little you know, but often bootstrapping is too difficult. Some of its fields have become exceptionally technical. No one in their right mind would do a PhD thesis in number theory unless they were extremely brilliant as the “good” problem space has become the territory of genius. These departments have become formidable ivory towers.
So along comes Piper - a young woman with an interest in number theory who understands math for what it is - a deeply cultural and human undertaking. After all, you have to convince others that your proofs work - something that is seriously broken these days (the abc conjecture in number theory comes to mind as do any number of computational proofs).
Her thesis is *very* readable and pedagogical. A sophomore math or physics major should be able to grab enough of a hold to bootstrap themselves through with a bit of work. A senior level student should find it a delight. There aren’t many math thesis that grab you. I was sent a copy on Christmas Eve day and its style drew me in. It isn’t a terribly deep result, but it is different from conventional math papers in the sense that Minute Physics or Physics Girl are different from physics for high school students and college non-majors. I’ll hazard a guess that it will be widely read and will have a greater impact than any other math thesis this year - perhaps this decade.
Some of you have serious math skills, others have studied other areas with not much exposure. If you know math jump it, otherwise just read the part I quote and perhaps skim the paper for cartoons and structure … the point is not so much to read the paper as to understand what she is doing culturally.
Maybe, just maybe, math and math education is changing and becoming less elite. That is exciting.
download it (1.8 MB pdf)
The prologue and early parts of the introduction say it all:
Respected research math is dominated by men of a certain attitude. Even allowing for individual variation, there is still a tendency towards an oppressive atmosphere, which is carefully maintained and even championed by those who find it conducive to success. As any good grad student would do, I tried to fit in, mathematically. I absorbed the atmosphere and took attitudes to heart. I was miserable, and on the verge of failure. The problem was not individuals, but a system of self-preservation that, from the outside, feels like a long string of betrayals, some big, some small, perpetrated by your only support system. When I physically removed myself from the situation, I did not know where I was or what to do. First thought: FREEDOM!!!! Second thought: but what about the others like me, who don’t do math the “right way” but could still greatly contribute to the community? I combined those two thoughts and started from zero on my thesis. What resulted was a thesis written for those who do not feel that they are encouraged to be themselves. People who, for instance, try to read a math paper and think, “Oh my goodness what on earth does any of this mean why can’t they just say what they mean????” rather than, “Ah, what lovely results!” (I can’t even pretend to know how “normal” mathematicians feel when they read math, but I know it’s not how I feel.) My thesis is, in many ways, not very serious, sometimes sarcastic, brutally honest, and very me. It is my art. It is myself. It is also as mathematically complete as I could honestly make it.
I’m unwilling to pretend that all manner of ways of thinking are equally encouraged, or that there aren’t very real issues of lack of diversity. It is not my place to make the system comfortable with itself. This may be challenging for happy mathematicians to read through; my only hope is that the challenge is accepted.
1.1 Notes to My Dear Reader(s) 1.1.1 The Layperson: Math 101
I will always be honest with you.
The hardest part about math is the level of abstraction required. We have innate logical abilities, but they are based in context. If you give people a scenario of university students drinking beverages at a bar and give them information either about the person’s age or about the person’s beverage, most people know instinctively which students’ drinks or IDs need to be checked to avoid underaged drinking (i.e., if the person’s 22 you don’t care what they’re drinking, but if the person has a vodka tonic, you need to know their age). Take the logically equivalent situation of cards with a color on one side and a number on the other. Suddenly it takes some work to figure out which cards have to be turned over to satisfy a given condition (say, all even numbers have red on the back). Just one level of abstraction and the untrained, but educated, person will have a good amount of difficulty even understanding the situation. Now try doing Number Theory.
I like to imagine abstraction (abstractly ha ha ha) as pulling the strings on a marionette. The marionette, being “real life,” is easily accessible. Everyone understands the marionette whether it’s walking or dancing or fighting. We can see it and it makes sense. But watch instead the hands of the puppeteers. Can you look at the hand movements of the puppeteers and know what the marionette is doing? A puppeteer walks up to you and says “I’m really excited about figuring out Fermat’s Last Thumb Bend!” You say, “huh?” The puppeteer responds, “Oh, well, it’s simply a matter of realizing that the main thumb joint has several properties that distinguish it from...” You’re already starting to fantasize about the Zombie Apocalypse. Imagine it gets worse. Much, much worse. Imagine that the marionettes we see are controlled by marionettoids we don’t see which are in turn controlled by pre-puppeteers which are finally controlled by actual puppeteers. NEVER HAVE A CONVERSATION WITH THESE FICTIONAL ACTUAL PUPPETEERS ABOUT THEIR WORK!
I spent years trying to fake puppeteer lingo, but I have officially given up. My goal here is to write something that I can understand and remember and talk about with my non-puppeteer friends and family, which will allow me to speak my own language to the puppeteers. To you, the lay reader, I recommend reading this introduction and then starting each subsection of laysplanations (the .1s) and reading until you hit your mathiness threshold (stopping to think or write something down is encouraged; even math you know won’t necessarily make sense at the speed at which you can read and understand non-math), then skim/skip to the next lay portion. Depending on how you feel with that, you should look at the math parts (the .2s) which will look familiar if you were able to finish the lay sections. I can’t promise they’ll make sense, but things should be vaguely readable. Maybe. The weeds (the .3s) contain extra information (some lay, some math) and calculations, more for answering questions than for reading. Enter at your own peril.
1.1.2 The Initiated
Welcome mathy friend! Depending on the extent of your initiation (and your sense of humor), this thesis may be exactly what you’ve always wanted to read! Skim the laysplanations (.1s), but if they are too math-less for you, it’s okay to only read the math sections (.2s) and just go back to the lay stuff if necessary (several things are introduced/motivated in the laysplanations, including explanations of my Formula in the .1.1s). You may also be interested in the weeds (.3s) which are appendices with things that weren’t strictly necessary to get through the proof of the Main Theorem, but were necessary personally for me to get a hold of things. The weeds aren’t to be read straight through, but you might find an explicit calculation or extra explanation there.
1.1.3 The Mathematician
Dear Professor, thank you for showing interest in my thesis! Your introduction awaits at §1.3. For results, however, you may find the fluffless arxived original [BH13] easier to read (certainly quicker!) than this thesis.
1.2 This Thesis (Problem) 101: A Mostly Layscape
Every thesis is a question and (very long) answer. My question in layspeak is: “How many” “shapes” of certain degree n “number fields” are there?
The naive short answer is: Infinitely many! But of course, though true, that is not nearly enough information. What we will show is that the infinitely many shapes we find are actually “equidistributed” with respect to the “space of shapes.” In other words, if you think of the collection of possible shapes as being a blob (a “space”), then wherever you look in this blob, you will find shapes of number fields in equal quantity.
Equivalently, though somewhat less to my liking, a thesis is a claim and a (very long) proof. My equivalent claim in layspeak is: “Shapes” of certain degree n “number fields” become “equidistributed” when ordered by “absolute discriminant.”
In what follows I hope to do enough “laysplanations” to make the whole argument approximately readable by approximately anyone. Approximately. In addition to laysplaining and “mathsplaining,” I will also, where appropriate and not too horrifying, have some “weedsplanations” where I wade into the weeds with examples and explicit calculations, sometimes with extra laysplanations that were not strictly necessary to the main argument.
People often say kids are natural scientists. It isn't true, but they have bushels of curiosity and that is a key component of science and math. Somehow this goes away in somewhere around the 5th grade. It shouldn't be - perhaps it is useful to look at where STEM education came from.
With the cold war came a recognition that the country needed scientists, engineers and mathematicians in much greater numbers than ever before. The curriculum was changed to ensure that a pipeline was created to identify and nurture those who would ultimately end up with Ph.D.s from the best universities in the country and programs within the universities were greatly expanded. In two decades the number of physics Ph.Ds. increased from under 200 to about 800 a year. There was this one little problem. What do you do with the millions of people outside the pipeline?
There is a deeply rooted assumption that science literacy is good for society. That somehow the pre and pre-pre professional courses designed by Berkeley and MIT would create a citizenry that could deal with science and society issues. Unfortunately there isn't any empirical evidence to support the assumption.
Just how do people interact with science? About ten years ago stem cell research was a hot topic in the news. There were diverse group. People with parents who have Alzheimer's, certain fundamentalist Christian sect members and biotech investors were some of the groups with differing motivations driving their interests. Global warming is another example - depending how you cut it there are at least a half dozen distinct groups (I have some nasty scars on this one). And now we need to strap in for CRISPR. Science and the public issues are important, but the reality is we have many publics each with their own issues.
Another huge issue is science is not a single monolithic subject. Science education teaches about the scientific method. Science doesn't work like that - reality turns out to be a different beast. Scientists agree on hypothesis testing and many methods, varying across branches and subpecalities. Trying to sort out the methodological differences between black-footed ferrets studies, climate modeling, and theoretical particle physics is a non-trivial task to anyone outside of science.
Most people don't worry about science on a regular basis but rather when something comes up and they would like to use it. Your two year old is showing signs of autism, your tap water has so much natural gas in it that you can set it on fire, the license on the nuclear power station twenty miles away is up for renewal, you are worried about losing weight and eating better... Text book learning isn't terribly useful and pronouncements by experts and pundits are often at odds. What is taught involves a regurgitation of memorized "facts" and the ability to plug in some numbers into equations to answer an artificial problem set. You have learned this thing science so you can score well on standardized tests. Generally people have little ability, five years out of school, to form a meaningful question and come up with a reasoned path that might be useful. We need to teach a useful science literacy. I was going to add an aesthetic science (and math) literacy, but I think that comes with study.
I was very lucky in high school. It was a time and place where some of the teachers were allowed to create their own courses on their own. Not every class, but a few were enough to keep me excited. My two best teachers taught history courses. Neither taught a bag of places or dates that we had to recite and as such that probably didn't help us on standardized tests. One had classroom discussions, both lectured and both required papers every few weeks on a topic or two of your own choice. They were trying to get us to understand history in context and to relate it to the present - which happened to be a very spooky time. I still can't remember dates and places well, but I think I have a better appreciation for the flow of history than the average high school graduate - enough that it helps me think about societal issues and begin to work out how to think about politics and policy. Again - I wouldn't claim to be great at it, but the education I had was much deeper. Had it not been for some native curiosity and some out-of-school mentoring I probably would have thought about majoring in history. (and that may have been good)
The good news is people are experimenting with programs that allow students to find and interpret science in the context of real world problems as well as judge the credibility of scientific claims. Courses like these may need to find interests of the student that can be used to apply science and this may lead to paths far from what is currently measured in standardized tests. There are barriers to be broken, but it is exciting to contemplate a much larger population that has a bit of real science literacy rather than the current artifact of pipelining. Some exciting work is going on: Problem-Based Learning, Place-Based Education, Science-Technology-Society approaches and probably a dozen I'm not aware of.
In college we have examples like Piper's thesis. I think physics and astronomy are in better shape, but more needs to be done. And outside of formal education the seeds being planted with laysplaining from YouTubers like Physics Girl and Minute Physics are just.plain.wonderful!
and with that a Happy New Year to you!
Fabulous this time of year
Lemon and Oil Spaghetti
° 2 large wax-free lemons to zest
° handful of flat-leaf parsley, finely chopped
° a pound of spaghetti
° 1–2 garlic cloves (depending on your love of garlic - I go for two) finely chopped
° a small dried chilli, finely chopped
° ~6 tbsp olive oil
° grate the zest from the lemon and then mix with the parsley and set aside
° bring a large pan of salted water to a boil, add the spaghetti and cook until al dente.
° in a large pan, gently warm the olive oil, garlic and chilli over a low flame until fragrant – do not let it burn.
° once the spaghetti is cooked, use a sieve or tongs to lift the spaghetti and a just a little residual water into the frying pan. Stir, add the lemon and parsley, a pinch of salt and if you like a squeeze of lemon, stir again,
° plates and eat immediately!