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!