## Approximating Functions of Spaces

The branch of mathematics known as topology is concerned with the study of shapes. Whereas shapes in geometry are fairly rigid objects, shapes in topology are much more flexible; topologists refer to them as “spaces.” If one space can be flexed and twisted and not-too-drastically mangled into another space, topology deems them to be the same. It becomes much more difficult, then, to tell if two spaces are different. A primary goal in topology is to find ways to distinguish spaces.

Another fundamental question in topology is concerned with the ways to put one space into another space – to understand the functions between spaces. Each space is a collection of points. A function from space $X$ to space $Y$ is a way to assign points in $X$ to points in $Y$. If $X$ is a collection of students, and $Y$ a collection of tables, then a function from $X$ to $Y$ is a way to assign each student to a table. In topology, we don’t allow just any function from $X$ to $Y$. While the spaces are flexible, we have to be careful not to separate points from $X$ that are close to each other. Using the students and tables example, we might think about two students holding hands as being close. These students could be placed at the same table, or perhaps neighboring tables, but cannot be separated across the room. A function that doesn’t separate points too much is called “continuous,” and these are the types of functions topologists consider; topologists tend to call them “maps.”

It turns out that these two primary questions of topology are actually related. If one wants to determine how similar shapes $X$ and $Y$ are, one might begin by introducing a third space, $Z$, and asking about the maps from $Z$ to $X$ and from $Z$ to $Y$. If the collection of maps are the same in both cases, one expects that $X$ and $Y$ are similar, at least somewhat. More information can be obtained by replacing $Z$ by another space $W$, and repeating the process. Typically the spaces $Z$ and $W$ are fairly well-understood spaces, like circles and spheres.

Spaces, and the maps between them, can be quite complicated in general. By restricting to various types of spaces, or types of maps, one is able to make significant progress. One important class of spaces consists of what are called “manifolds.” Intuitively, a manifold is a space which, when viewed from quite close, looks flat (like a line, or a plane), and has no corners. If you were a tiny ant, walking along on a mathematician’s idealized sphere, for example, you might get the impression that you were walking on a giant sheet of paper. Indeed, a similar viewpoint of our own world was common in the not too distant past.

Circles and spheres, and lines and planes themselves, make good examples of manifolds to keep in mind. In fact, lines and planes, and the higher dimensional “Euclidean” spaces, are the fundamental building blocks for manifolds. The defining property of a manifold is that when you get close enough, you are looking at a Euclidean space. Manifolds are essentially spaces obtained by gluing together Euclidean spaces. An interesting example, known as the Möbius strip, can be modeled by taking a strip of paper, introducing a half-twist, and taping the ends together. A tiny ant crawling along on the resulting object would have a hard time noticing that it isn’t just crawling along a strip of paper.

If one’s attention is restricted to studying manifolds, instead of more general spaces, it makes sense to also restrict the types of maps under consideration. General continuous maps need not respect the information about manifolds that makes manifolds a nice class of spaces (they are reasonably “smooth”). We replace, then, all continuous maps with a more restricted class of maps which preserve the structure of manifolds. A particularly nice such class consists of those maps known as “embeddings.” An embedding will not introduce corners in manifolds, and also will not send two points to the same point (embeddings would place only one student at each table, in the earlier example).

When studying manifolds, then, a topologist may be concerned with the collection of embeddings between two manifolds. If the manifolds are called $M$ and $N$, then we might denote the embeddings of $M$ into $N$ by $E(M,N)$. This is then a function itself – a function of two variables, $M$ and $N$. If we fix one of the variables, say we only think about $M$ being a circle, we still have a function of one variable, and have made our study somewhat easier.

Leaving $M$ fixed, how do the values $E(M,N)$ change as $N$ changes? Said another way, if we modify $N$ slightly, what is the effect on $E(M,N)$? If it is difficult to find $E(M,N)$, how can it be approximated? How can the function itself be approximated?

These questions are strikingly similar to questions asked in calculus. Given a function that takes numbers in and spits numbers out ($y=e^x$, for example) what happens to the output values ($y$) if the input value ($x$) is changed slightly? If we know about the value at a particular point ($e^0=1$), what can be said about values nearby ($e^{1/2}$, say)? The answers to these questions lie with the derivative, and its “higher” analogues (the derivative of the derivative, and so on). If one knows about the derivatives of a function at a point, one can create “polynomial” approximations to the function, near that point.

It turns out that something quite similar happens when studying the embedding function (and other functions like it). Some sense can be made of derivatives, polynomials, and best approximations, all in the context of functions of spaces (instead of functions of numbers).

I have been studying the embedding function, and its polynomial approximations, when $M$ is fixed. I let $M$ be a collection of disjoint Euclidean spaces of any dimension; so I might take $M$ to be 3 lines and 2 planes, all separate from each other. I also restrict my attention to $E(M,N)$ only when $N$ itself is a Euclidean space. Since any manifold is built out of Euclidean spaces, the cases I consider are important building blocks to understanding more general embedding functions.

Previous work has already covered some of the cases I consider. If $M$ is a finite collection of points, the collection of embeddings is called a “configuration space.” Loosely, this case covers the idea that embedding may not bring two points together, and is somewhat of a “global” situation. Another case is when $M$ only has one piece, say a single line. Here, one is exploring more the notion that embeddings may not introduce corners, a “local” situation. In both of these cases, the best polynomial approximations for the embedding functions have been identified. Moreover, useful descriptions of the approximations have been obtained.

In the more general situation I consider, I have been interacting with both aspects of embeddings. Since my spaces, $M$, may have many pieces, I am involved in global aspects of embeddings. Since my $M$ may have pieces of any dimension, I am involved in local aspects of embeddings. Unifying the description of the approximations in these two cases has been my task.

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A somewhat different, perhaps more elementary version of this is also available.