My Little 2021 Mathematics A-to-Z: Atlas

I owe Elkement thanks again for a topic. They’re author of the Theory and Practice of Trying to Combine Just Anything blog. And the subject lets me circle back around topology.


Mathematics is like every field in having jargon. Some jargon is unique to the field; there is no lay meaning of a “homeomorphism”. Some jargon is words plucked from the common language, such as “smooth”. The common meaning may guide you to what mathematicians want in it. A smooth function has a graph with no gaps, no discontinuities, no sharp corners; you can see smoothness in it. Sometimes the common meaning is an ambiguous help. A “series” is the sum of a sequence of numbers, that is, it is one number. Mathematicians study the series, but by looking at properties of the sequence.

So what sort of jargon is “atlas”? In common English, an atlas is a book of maps. Each map represents something different. Perhaps a different region of space. Perhaps a different scale, or a different projection altogether. The maps may show different features, or show them at different times. The maps must be about the same sort of thing. No slipping a map of Narnia in with the map of an amusement park, unless you warn of that in the title. The maps must not contradict one another. (So far as human-made things can be consistent, anyway.) And that’s the important stuff.

Atlas is the first kind of common-word jargon. Mathematicians use it to mean a collection of things. Those collected things aren’t mathematical maps. “Map” is the second type of jargon. The collected things are coordinate charts. “Coordinate chart” is a pairing of words not likely to appear in common English. But if you did encounter them? The meaning you might guess from their common use is not far off their mathematical use.

A coordinate chart is a matching of the points in an open set to normal coordinates. Euclidean coordinates, to be precise. But, you know, latitude and longitude, if it’s two dimensional. Add in the altitude if it’s three dimensions. Your x-y-z coordinates. It still counts if this is one dimension, or four dimensions, or sixteen dimensions. You’re less likely to draw a sketch of those. (In practice, you draw a sketch of a three-dimensional blob, and put N = 16 off in the corner, maybe in a box.)

These coordinate charts are on a manifold. That’s the second type of common-language jargon. Manifold, to pick the least bad of its manifold common definitions, is a “complicated object or subject”. The mathematical manifold is a surface. The things on that surface are connected by relationships that could be complicated. But the shape can be as simple as a plane or a sphere or a torus.

Every point on a coordinate chart needs some unique set of coordinates. And if a point appears on two coordinate charts, they have to be consistent. Consistent here is the matching between charts being a homeomorphism. A homeomorphism is a map, in the jargon sense. So it’s a function matching open sets on one chart to ope sets in the other chart. There’s more to it (there always is). But the important thing is that, away from the edges of the chart, we don’t create any new gaps or punctures or missing sections.

Some manifolds are easy to spot. The surface of the Earth, for example. Many are easy to come up with charts for. Think of any map of the Earth. Each point on the surface of the Earth matches some point on the sheet of paper. The coordinate chart is … let’s say how far your point is from the upper left corner of the page. (Pretend that you can measure those points precisely enough to match them to, like, the town you’re in.) Could be how far you are from the center, or the lower right corner, or whatever. These are all as good, and even count as other coordinate charts.

It’s easy to imagine that as latitude and longitude. We see maps of the world arranged by latitude and longitude so often. And that’s fine; latitude and longitude makes a good chart. But we have a problem in giving coordinates to the north and south pole. The latitude is easy but the longitude? So we have two points that can’t be covered on the map. We can save our atlas by having a couple charts. For the Earth this can be a map of most of the world arranged by latitude and longitude, and then two insets showing a disc around the north and the south poles. Thus we have an atlas of three charts.

We can make this a little tighter, reducing this to two charts. Have one that’s your normal sort of wall map, centered on the equator. Have the other be a transverse Mercator map. Make its center the great circle going through the prime meridian and the 180-degree antimeridian. Then every point on the planet, including the poles, has a neat unambiguous coordinate in at least one chart. A good chunk of the world will be on both charts. We can throw in more charts if we like, but two is enough.

The requirements to be an atlas aren’t hard to meet. So a lot of geometric structures end up being atlases. Theodore Frankel’s wonderful The Geometry of Physics introduces them on page 15. But that’s also the last appearance of “atlas”, at least in the index. The idea gets upstaged. The manifolds that the atlas charts end up being more interesting. Many problems about things in motion are easy to describe as paths traced out on manifolds. A large chunk of mathematical physics is then looking at this problem and figuring out what the space of possible behaviors looks like. What its topology is.

In a sense, the mathematical physicist might survey a problem, like a scout exploring new territory, more than solve it. This exploration brings us to directional derivatives. To tangent bundles. To other terms, jargon only partially informed by the common meanings.

And we draw to the final weeks of 2021, and of the Little 2021 Mathematics A-to-Z. All this year’s essays should be at this link. And all my glossary essays from every year should be at this link. Thank you for reading!

The Summer 2017 Mathematics A To Z: Morse Theory

Today’s A To Z entry is a change of pace. It dives deeper into analysis than this round has been. The term comes from Mr Wu, of the Singapore Maths Tuition blog, whom I thank for the request.

Summer 2017 Mathematics A to Z, featuring a coati (it's kind of the Latin American raccoon) looking over alphabet blocks, with a lot of equations in the background.
Art courtesy of Thomas K Dye, creator of the web comic Newshounds. He has a Patreon for those able to support his work. He’s also open for commissions, starting from US$10.

Morse Theory.

An old joke, as most of my academia-related ones are. The young scholar says to his teacher how amazing it was in the old days, when people were foolish, and thought the Sun and the Stars moved around the Earth. How fortunate we are to know better. The elder says, ah yes, but what would it look like if it were the other way around?

There are many things to ponder packed into that joke. For one, the elder scholar’s awareness that our ancestors were no less smart or perceptive or clever than we are. For another, the awareness that there is a problem. We want to know about the universe. But we can only know what we perceive now, where we are at this moment. Even a note we’ve written in the past, or a message from a trusted friend, we can’t take uncritically. What we know is that we perceive this information in this way, now. When we pay attention to our friends in the philosophy department we learn that knowledge is even harder than we imagine. But I’ll stop there. The problem is hard enough already.

We can put it in a mathematical form, one that seems immune to many of the worst problems of knowledge. In this form it looks something like this: if what can we know about the universe, if all we really know is what things in that universe are doing near us? The things that we look at are functions. The universe we’re hoping to understand is the domain of the functions. One filter we use to see the universe is Morse Theory.

We don’t look at every possible function. Functions are too varied and weird for that. We look at functions whose range is the real numbers. And they must be smooth. This is a term of art. It means the function has derivatives. It has to be continuous. It can’t have sharp corners. And it has to have lots of derivatives. The first derivative of a smooth function has to also be continuous, and has to also lack corners. And the derivative of that first derivative has to be continuous, and to lack corners. And the derivative of that derivative has to be the same. A smooth function can can differentiate over and over again, infinitely many times. None of those derivatives can have corners or jumps or missing patches or anything. This is what makes it smooth.

Most functions are not smooth, in much the same way most shapes are not circles. That’s all right. There are many smooth functions anyway, and they describe things we find interesting. Or we think they’re interesting, anyway. Smooth functions are easy for us to work with, and to know things about. There’s plenty of smooth functions. If you’re interested in something else there’s probably a smooth function that’s close enough for practical use.

Morse Theory builds on the “critical points” of these smooth functions. A critical point, in this context, is one where the derivative is zero. Derivatives being zero usually signal something interesting going on. Often they show where the function changes behavior. In freshman calculus they signal where a function changes from increasing to decreasing, so the critical point is a maximum. In physics they show where a moving body no longer has an acceleration, so the critical point is an equilibrium. Or where a system changes from one kind of behavior to another. And here — well, many things can happen.

So take a smooth function. And take a critical point that it’s got. (And, erg. Technical point. The derivative of your smooth function, at that critical point, shouldn’t be having its own critical point going on at the same spot. That makes stuff more complicated.) It’s possible to approximate your smooth function near that critical point with, of course, a polynomial. It’s always polynomials. The shape of these polynomials gives you an index for these points. And that can tell you something about the shape of the domain you’re on.

At least, it tells you something about what the shape is where you are. The universal model for this — based on skimming texts and papers and popularizations of this — is of a torus standing vertically. Like a doughnut that hasn’t tipped over, or like a tire on a car that’s working as normal. I suspect this is the best shape to use for teaching, as anyone can understand it while it still shows the different behaviors. I won’t resist.

Imagine slicing this tire horizontally. Slice it close to the bottom, below the central hole, and the part that drops down is a disc. At least, it could be flattened out tolerably well to a disc.

Slice it somewhere that intersects the hole, though, and you have a different shape. You can’t squash that down to a disc. You have a noodle shape. A cylinder at least. That’s different from what you got the first slice.

Slice the tire somewhere higher. Somewhere above the central hole, and you have … well, it’s still a tire. It’s got a hole in it, but you could imagine patching it and driving on. There’s another different shape that we’ve gotten from this.

Imagine we were confined to the surface of the tire, but did not know what surface it was. That we start at the lowest point on the tire and ascend it. From the way the smooth functions around us change we can tell how the surface we’re on has changed. We can see its change from “basically a disc” to “basically a noodle” to “basically a doughnut”. We could work out what the surface we’re on has to be, thanks to how these smooth functions around us change behavior.

Occasionally we mathematical-physics types want to act as though we’re not afraid of our friends in the philosophy department. So we deploy the second thing we know about Immanuel Kant. He observed that knowing the force of gravity falls off as the square of the distance between two things implies that the things should exist in a three-dimensional space. (Source: I dunno, I never read his paper or book or whatever and dunno I ever heard anyone say they did.) It’s a good observation. Geometry tells us what physics can happen, but what physics does happen tells us what geometry they happen in. And it tells the philosophy department that we’ve heard of Immanuel Kant. This impresses them greatly, we tell ourselves.

Morse Theory is a manifestation of how observable physics teaches us the geometry they happen on. And in an urgent way, too. Some of Edward Witten’s pioneering work in superstring theory was in bringing Morse Theory to quantum field theory. He showed a set of problems called the Morse Inequalities gave us insight into supersymmetric quantum mechanics. The link between physics and doughnut-shapes may seem vague. This is because you’re not remembering that mathematical physics sees “stuff happening” as curves drawn on shapes which represent the kind of problem you’re interested in. Learning what the shapes representing the problem look like is solving the problem.

If you’re interested in the substance of this, the universally-agreed reference is J Milnor’s 1963 text Morse Theory. I confess it’s hard going to read, because it’s a symbols-heavy textbook written before the existence of LaTeX. Each page reminds one why typesetters used to get hazard pay, and not enough of it.

Minimal Yet Interesting Surfaces

Some days you just run across a shape you never heard of before and that’s interesting. Matthias Weber of The Inner Frame gave me one last night. In a string of essays Weber shows a figure which comes up from minimal surface theory. This is a study of making a shape that fits to some given boundary while keeping a property called “mean curvature” equal to zero. This is how mathematicians make it sound all academic when they talk about soap bubbles in wire frames.

This is from a particular kind of surface developed in the 1860s by Alfred Enneper, whom I admit I never heard of before either. It’s just outside my specialty. But he was a student of Peter Gustav Lejeune Dirichlet, who’s just all over partial differential equations and Fourier series. Enneper and Karl Weierstrauss — whose name is all over analysis — described a way to describe these surfaces, using differential geometry. Once again I’m sad I don’t know that field more, as it produces such compelling pictures.

Here Weber introduces the surface, complete with a craft project! If you’d like you can cut out and fit together a wonderful exotic little surface. The second essay looking at some shapes with similar properties, and at what you get by stacking these surfaces. The third part extends this even farther, to the part of mathematics that’s just Googie architecture. I hope you enjoy.

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