What’s my thesis actually about?
The working title of my thesis is Hopf-Galois Extensions and Descent Data in Quasicategories. When people ask me what my dissertation is on, I often recite this title in the hope that they’ll be so horrified that we won’t have to continue the conversation. Sometimes it works, but it’s been working infrequently enough that I decided to try to explain some of it. For those of you who are scared to continue, I’m pretty sure all you’ll really need to know is what the number line looks like. Actually, you know what, this is what the number line looks like:
Remember, it’s infinitely long. If you’re already familiar with the number line, as well as a lot of other mathematics, you may want to skip the next sections on space and spheres (or you may want to read them just to spend more time with me).
People get stressed out about space. First of all, I don’t mean outer-space. I’m not that kind of scientist. I mean idealized space, like an empty universe. The other thing I’m going to assume about “space” is that it has a center point, what mathematicians sometimes call the origin (which sounds pretty severe, out of context). And now the big leap: I don’t want to only talk about three dimensional space. You can imagine one and two dimensional space already. They’re just an infinite line (remember, the number line up there?) and an infinite plane (not an airplane) respectively. On an infinite plane we can determine any point by giving two numbers. If, for instance, we give the numbers 2 and 3, then this determines the point on the plane gotten to by going 2 steps East from the origin, and then moving three steps North (you can do it in the other order too, if you’d like). The point given by the pair of numbers (-2,3) would say go 2 steps West from the origin, and then 3 steps North, etc. etc. So, for all intents and purposes, all the information contained in two dimensional space (i.e. all possible locations) is the same thing as pairs of numbers (you can thank Descartes for that realization, by the way).
Three dimensional space is like an infinite box, and you can determine any point in three dimensional space by giving three numbers. So, again, the crucial thing to realize is that we can either describe three dimensional space by drawing a little box and trying to pick out a point in there somewhere, or we can just give three numbers, like (11.84, π, -600). Mathematicians have developed tons of ways of algebraically manipulating triplets of numbers to determine geometric facts (this is covered in Linear Algebra and Calculus III in college). The point is that it’s a lot easier to juggle collections of numbers than it is to try to think about really complex structures in three dimensional space.
Now, given all this, what do you think we mean when we’re talking about n-dimensional space? That’s right, just a list of n numbers! It’s okay if you didn’t get that on your own, it’s kind of a weird conceptual jump. But anyway, using the methods that we came up with for doing geometry in three dimensional space, we can now do geometry in four dimensional, five dimensional and 695-dimensional space (as well as some other dimensions). So now we know what space is. Let’s talk about some of the things that live inside of it.
For us, the only thing we’re really going to care about will be a sphere. Or, to be precise, a lot of spheres. An infinite number of them. And to make it even more complicated, I mean spheres in an arbitrary number of dimensions.
When I say sphere, you probably think of something like this, a ball in three dimensional space:
And, of course, you’re right. That is a sphere. It’s a sphere that lives in three dimensional space. But you can also have a sphere that lives in two dimensional space (i.e. on a plane). It looks something like this:
And you can even have a sphere in one dimensional space (i.e. a line). That just looks like two points sitting on a number line (remember from the beginning?), equidistant from zero. I’m not going to draw it, use your imagination.
What’s the common theme here? And, see, this is what mathematicians do, they take some basic ideas (like circles and spheres) and try to determine the essence of those ideas. You could argue that there are a lot of answers to this question, and you’d be right. But, for historical reasons, what mathematicians have decided makes something a sphere, in any number of dimensions, is that it has a center. All of the spheres we’ve looked at so far have something really special about them: every single piece of them is the same distance from a center point. Mess around with this for a while. Try to draw a triangle so that every point is the same distance from the center. You can’t. If you could, it’d be a sphere (or you might be living in some warped-ass universe, I don’t know). So a sphere in four dimensions is all the points (remember, these would just be quadruples of numbers) in four dimensional space that are the same distance from the origin. And we can do the same thing in any number of dimensions.
The other thing to notice is that if you were an ant living on the surface of the sphere in three dimensions, it would look like a plane. You’d think you were living in two dimensions. Same thing with a circle. An ant on a circle would think it was just on a line. In other words, the circle looks one dimensional, and a ball looks two dimensional, at least up close. For this reason, the circle is called the 1-sphere, the ball is called the 2-sphere, and in general a sphere living in n-dimensional space is called an (n-1)-sphere.
Putting Spheres on Other Spheres
For reasons we’ll have to get into at a later date, mathematicians are interested in putting spheres on the surfaces of different spheres. If you’re familiar with the terminology, what I’m talking about are functions between spheres. The way to think about this is to imagine taking a sharpie and drawing a circle on the side of a beach ball. You’ve now put a 1-sphere on the surface of a 2-sphere. Obviously there are an infinite number of ways to draw a circle on a beach ball. You can do similar things in higher dimensions, but to be honest, nobody can visualize higher dimensions, so mostly we just think about sharpies and beach balls and do algebra for the rest. We’d like to know all the possible ways of drawing an n-sphere on an m-sphere, for m and n possibly being different numbers. Ideally for every pair of n and m. It turns out that this is impossible (or at least, practically impossible). So we introduce a simplification. We say that two ways of drawing one sphere on another are the same if one can be continuously deformed into another. In other words, if we can push a sphere around on the surface until it looks like the other way of doing it.
If we go back to our sharpie-on-a-beach ball situation, let’s replace the sharpie with a piece of string tied in a loop. We’re allowed to lay the string on the beach ball any way we want (even crossing itself!). If you think hard about it, you’ll notice that every way of laying a piece of string on a beach ball can be pushed into every other way, especially if we drop all pretense and just admit that the string is an abstract entity and can be stretched or compressed as much as we like. So, with this simplification, that is, identifying ways of laying the string that can be deformed into one another, there’s only one way of putting the string on the beach ball (think about it!).
If we wanted to put a circle (a so-called 1-sphere) on a 0-sphere (a pair of points on the number line equidistant from 0), we’d have no choice to but to completely compress the circle and just stick it on one of those points (we’re not allowed to put some of it on one point, and some on the other, it has to stay in one piece). So there are two ways to “put” a 1-sphere on a 0-sphere and only one way to put a 1-sphere on a 2-sphere (allowing for this deformation stuff we mentioned before). For an example that is known, but that we can’t possibly visualize, it turns out that there are exactly 2 ways of putting a 4-sphere (that is, a sphere in five dimensions) on a 2-sphere (a ball). Weird right?? How can we even know that? Well, you can’t. I can. And I’ve spent an absurd amount of time in school just to know dumb facts like that.
For the record, these “ways of putting spheres on other spheres except they’re the same if we can deform them into one another” are known to mathematicians as the homotopy groups of spheres. If you’ve had the misfortune of listening to me talk at length about math before, you’ve probably heard these words. You can find more information here.
Color Between the Lines
Alright, so we have these things called the homotopy groups of spheres. And at least for the moment you’re willing to take on faith that, maybe just out of curiosity, we’d like to know what they are for arbitrary n and m. Turns out we still don’t know. We know a lot of them, but you may have noticed that there are an infinite number of numbers, and we still only know a finite collection of these groups. What’s worse, we don’t even have any good algorithm for computing them all. Maybe this in itself is a good reason to study them, simply because they’re so damned complicated.
One enormous step forward was made by my advisor, Jack Morava, and some of his colleagues (e.g. Doug Ravenel, Steve Wilson, J. Frank Adams, Daniel Quillen), back in the 70’s. What they realized was that all this data, all these functions between spheres, is following a really big organizational principal that divides it up into layers. We don’t have anything close to enough space to explain how this works, or how one would even know something like this, but suffice it to say that there’s a nice way of stratifying all of this data that wasn’t known before (and that doesn’t have anything explicitly to do with the numbers n and m). Why is this important? Well, for two reasons: the first is that it’s just kind of cool that this big mess of complicated information has any kind of large scale structure at all. The second is that if we know something about the structure of this information, we can try to anticipate what the next layer, or the next piece of data in some fixed layer, is going to look like. And this makes it a little bit easier to determine these data.
It was this structure, this stratification, that gave this area of mathematics the name chromatic homotopy theory. The idea is that all of the information at once, white light in this analogy, is hard to understand, but if we split it into its constituent wavelengths, we can understand it one piece at a time.
The Drop (you know, like in Dubstep?)
Here’s the thing that really blew everyone’s mind about the stratification described above. That stratification was familiar. People, e.g. my advisor, had seen it before. It has some very specific structure, and people had seen that structure before, not in homotopy theory, but in number theory. In other words, this entirely different branch of mathematics, mostly concerned with studying prime numbers and their structure, already had something that looked a lot like this picture. So, of course, the mathematicians I mentioned above, as well as many more, really dug into this. And it was the real deal! These two stratifications, as near as anyone can tell, are the same, and people have been able to use their knowledge of number theory to get better at computing the homotopy groups of spheres. Unfortunately, we still don’t know why these two things are the same. What, exactly, do spheres have to do with prime numbers? Why should they interact at all? There’s a lot that can be said on this topic, but as far as I know, there’s not really a solid answer to this question. This, in some sense, is the driving question of my thesis, and really of most of my research.
My thesis and research have a lot of moving parts, but the thing that’s got me most excited at the moment is the following: there’s a sort of mediator between the number theory picture and the homotopy groups picture, called MU (we just say the letters, i.e. em-you, but you can say it however you like). You can think of this as a sort of tunnel between these two worlds (that this MU thing did this, by the way, was worked out by the late Dan Quillen, who I mentioned above). And so far, MU has always been sort of built up from geometry. It’s kind of involved, but it happens to be something called a cobordism theory, which is something basically built from surfaces in space. So it may not be completely surprising that this has something to do with spheres (although it’s actually still pretty trippy). What my thesis does, among other things, is build up MU in a more number-theoretic way. In fact, there are some striking similarities between what I do to build MU, and what number theorists (like Michel Lazard) did in the 50’s when they built up the number theory side of what later became known as MU. What the goal is, then, is to give an intuitive reason behind this strange linkage between topology and number theory. I definitely don’t come even close to fully answering this question, but that’s what my thesis is actually about.