Being a mathematician is a lot like being a movie star. (Here I must admit that while I have never actually been a movie star, I have no doubts that they are completely similar. I must also confess that I can't think of any analogies between the two either, but my certainty remains unshaken.) Whenever I tell someone I'm a mathematician---or a movie star---they become immediately fascinated and ask me what I do all day. Here's a example:
Q: So, what exactly do you do all day?See what I mean? Unfortunately, it is difficult to accurately portray the grandeur and romance of the world of mathematics in a sentence or two. In the elder days, neither mortal nor god would even attempt to describe it outside of poem, song or interpretive dance. (Personally, I have devoted several odes to Mathematics, but the only surviving work is to be found here.) However, since we have moved into the Age of Reason, I will try to answer this question by recounting a day of my life.
As a quick background, I'm currently in my first year as a postdoc at Columbia. A postdoc is essentially a short-term assistant professorship after your Ph.D. which will not turn into a permanent position. Postdocs are expected to do research and most postdocs in math also require teaching. My postdoc is three years with no teaching this year and half-time teaching the next two years. (I'm slated to teach Calculus I next fall and Calculus II in the spring).
I'll throw around some math terms which I won't explain and I don't expect you to understand, so don't worry about it. That's part of the mystique. I will try to give you some idea of some things I'm thinking about, but I will try to do it with a poetic vagueness that inspires you to imagine what I really mean. That is, after all, what Mathematics is about.
March 22, 2005, Tuesday.
Reveille, 07:00. Lights out, 07:04. Re-reveille, 07:30. I dreamt it was time to give my first calculus lecture, only the time I thought class started was an hour and fifteen minutes after all my students thought it started. But for some reason, when I get there 6 or 7 students are still there. I enter through a hatch in the ceiling, and other students file in and out.
Gradually, I realize I'm awake but still tired. I think about some things I have to do: taxes, shopping, math. I think about this Galois representation problem I was talking to Edray about at the Kentucky conference last week. It's not a problem I'm really working on now, but it's a potential project, though I don't have any ideas how to approach it. It's not quite in my area of familiarity, though it is number theory. Someone at Princeton originally suggested it to me, and when I go to Boston next month I'll talk to a professor there about it. I spoke to Urban, one of the Columbia professors, about it once before. He had related results; though they won't apply in this situation, I decide that I should go ask him for his paper today.
After breakfast, showering and taking care of some things at home, I head off to Columbia around 10am. It's especially warm and sunny out, so I take off my jacket and walk the long way to the Math Department. I happen to see a couple of my old students from Caltech who are here on spring break and I stop and talk to them for awhile. Then I head to the Math Library to look at one of Jacquet's papers. Jacquet is my mentor/host at Columbia, and we're working on a project together. He gave me a paper by Raggy to look at, which is related to what we're doing, and told me to report back to him. I happened to meet Raggy at Kentucky last week and he asked me if I'd read this old paper of Jacquet. I hadn't, and this is the paper I was looking for. I skimmed through it and discovered many results similar to Raggy's, but I didn't have time to look at it too carefully.
At 11am, I go to Urban's class. It's not quite related to my research---he's lecturing about counting elliptic curves satisfying certain properties---but it's good material for me to learn anyway. After class I go to a nearby Chinese place for lunch, where I mull over some problems. On my way back, I stop off at the bookstore to look for a math book which I don't find, but I do pick up a couple books in topology and geometry on some material I want to review, partly for my own research, but mostly to understand what other people are doing.
After I get back to the office, I begin studying these books a little, partly out of fear that if I don't start now when I'm motivated, I'll never get around to it. Then I start thinking about my work with Jacquet. He wants me to write up our results thus far, but I still need to work out some details of the proof. The part I'm thinking about now involves distributions (which you may have encountered in probability or statistics). Let me try to explain the idea in a simpler context.
Let's suppose it snows (with no wind) on a perfectly round dome or hill in a flat open field. You'll notice that there's more snow piled at the top of the dome than lower on the dome because the further down the side of the dome you go, the more the snow will roll down. Also, the snow that rolls off the dome will pile up around the edge of the dome, making it higher than snow pile-up in the open field. This pattern of snow accumulation is a distribution. In this case, the distribution tells you how much snow there is above any point in the field or any point on the dome. (Put another way, the distribution tells you the probability that a snowflake will land in given spot.) Now if I forget about the boundary of the field, or if I assume the field is round with the dome in the center, then the distribution is rotation invariant, meaning that if you were to rotate the field (including the dome) around a vertical pole through the center of the dome (before the snowfall), the distribution would look exactly the same. However, if, instead of rotating everything, we shifted (or translated) everything over two feet then we would not get the same distribution of snow (even forgetting about the edges of the field)---the distribution would essentially be shifted over two feet---because the dome would be in a different place. Thus we say the distribution is not translation invariant. (Now if our field were a grid and the snow accumulation in every block of the grid looked the same as that in every other block of the grid, then the distribution would be translation invariant. Of course this doesn't mean that any translation would leave the distribution invariant, only translations in the x and y directions by the length of a grid block.) If you think about it, you can imagine that the only way a distribution could be both rotation invariant and translation invariant is for it to be constant (flat) everywhere. One can rigorously prove this if things are made precise.
I am trying to prove something similar in a more esoteric setting. I'm trying to prove that a certain distribution is zero because I know it is invariant under certain symmetries and I already know it's zero in most of the region. To figure out how to do this, I've been reading a certain paper Jacquet gave me where someone else does something similar. So I spend awhile at the blackboard trying to work through his argument, but I don't understand the end of it. At this point I get a little down and wonder when things are going to make sense (which incidentally happens the next day), so I give up for now. After not understanding something, I typically get a little tired and take a break. So I check my email---why do I check my email fifty times a day when no one ever emails me? Somehow, unable to learn from experience, I still always hope. Now, not only crushed by my mathematical shortcomings, I lament my unpopularity.
Thinking I won't make much progress on distributions now, I go to Urban's office to ask him for a copy of his paper. He informs me that he hasn't actually written it yet, though he has no doubt that his proof is correct. So he explains the basic idea to me (which I don't entirely follow) and asks again how I would hope to use it. I explain this question about Galois representations that I was thinking about in bed this morning, and we conclude that I can't put my problem into a context where I can use a similar approach. I go away somewhat disheartened.
I set up a gmail account, and think about distributions until the point at which I am completely disheartened. Fortunately, it's now a little after 4pm, which means it's tea time in the lounge. I go to take a break, mingle and eat some dried cranberries (I typically don't eat the cookies they have). Jacquet is there today, as he often is, though we only usually talk math once or twice each month. I talk to him about this paper of Raggy. I mention one of the results he proves and ask if he thinks we can get a similar result in other cases. He tells me perhaps some cases, though in general it will be more complicated. As our conversation deteriorates into unending silence, I go across the room to ask one of the undergrads if he decided where to go to grad school. We talk about grad schools, and he asks me a couple of math questions. Then, thinking I should get back to work, I return to the office around 5pm.
Since at this point I am still void of great ideas, I try to figure out how to get my Columbia email to forward to gmail so that everything forwarded gets appropriately filtered. Dinakar, my advisor from Caltech, is supposed to give a talk at 7:30pm to the math undergrads here, which I also plan to attend. He's visiting for the week and I want to talk to him about one of the projects he suggested to me a few months ago, though I haven't been thinking about it lately. So I start to look at the problem again. I'm trying to work out an explicit example to understand what's going on; to do this I need to know the solution to the following classical problem. Let n > 0 be an integer. For which integers k does the equation
have a solution? (Here a and b are also integers.) Then given such a k, what do all solutions for a and b look like? Number theorists have worked on such problems since at least the 18th century. I'm sure my question has been solved before (at least more or less), but I doubt there are more than a handful of number theorists today who could answer such a question off the top of their head. I dislike questions like this---I mean, I actually quite like the questions, but sometimes it seems to take an incongruous amount of effort to search through the literature, figure out what's being said, and go through tedious calculations just to find out something that every mathematician already knew 50 or 100 years ago. (Doing messy calculations is the part I really dislike, and I typically try to avoid it. Unfortunately, I'm not really that smart, so it's usually the only thing I can do.) Certainly if k = 1 or k = -1, then the solutions are worked out in many standard undergraduate number theory texts. However, to see which k have solutions and finding the solutions in a suitable form will require looking for the right references then working out specific details which I'm not looking forward to. But I don't have any more time to think about it now, because I have to go to Dinakar's talk/free pizza.
a2 - nb2 = k
Dinakar's talk was good. I think I'd seen all the stuff before, but I'd forgotten some of it so it was good to see again. Afterwards, I grabbed my undergraduate number theory text to take home (where I didn't find anything I didn't already remember on the above problem) and chatted with Dinakar while he waited for his brother to pick him up. We didn't really talk any serious math, mostly just math gossip (which is often more fun). Then I set off for home, later than I usually do, but it was a good day. I was happy. I ate three meals. I enjoyed working on these problems. I understood a little more about distributions. I got to talk to Jacquet and Dinakar today. And I should be able to work out the solution to this classical number theory problem however much trouble it may be.
Postscript: In case you're interested, here's what a partial answer to the abovementioned classical number theory problem looks like. I'll just state things in the special case of n=3 to make things simpler, though one could do arbitrary n > 0. Using some number theory due to Gauss and possibly Kronecker, one can show that
has a solution in positive integers exactly when the only primes dividing k are either 2 or primes of the form 12x+1 for x an integer, e.g., k=1, 2, 4, 8, 13, 16, 26, .... I will denote the square root of a number by sqrt( ) since I don't know how to get the square root sign in HTML. When k=1, the set of solutions are obtained from the numbers t= (2+sqrt(3))m by letting a be the integer term of t and b be the coefficient of sqrt(3) in t. For example, when m=1, t is just 2 + sqrt(3), so take a=2 and b=1. When m=2, t = (2 + sqrt(3))2 = 4 + 4*sqrt(3) + 3 = 7 + 4*sqrt(3), so a=7 and b=4. You can check both of these: 4 - 3*1 = 1 and 49 - 3*16 = 1. To find all solutions for a general k, it suffices to find one solution for k prime (which as mentioned above means k=2 or k is a prime of the form 12x+1). Then using all the solutions for k=1, we can construct the rest of the solutions. Similarly, for k a product of primes, we can construct all solutions from the solutions in the prime cases. For any given prime I can find a solution (there are some known algorithms to do this), but I want to write down a formula for a solution that works for all the primes (of the above type). This seems harder and I haven't done it yet, but as I don't think I actually need to know it, I probably won't.
a2 - 3b2 = k
Update (8/2013): If you have some more questions about what it's like to be a mathematician, check out the Q & A Session.
Sat Apr 23 17:06:17 EDT 2005