Archive for the ‘Work’ Category

A Math Prezi

November 21, 2010

Recently I had to give a math talk about my work. Previous talks I’ve given I just did on the chalkboard, but this being my last math talk for a while, I thought I might finally try Beamer (nice quickstart). And then I realized I should try Prezi. I thought I’d share my exploration with you, in case you try to decide about something similar. If you want to just cut to the chase and see the final Beamer work (pdf or source tarball), or the final Prezi, go for it. If for some strange reason you actually care about the math, you’re welcome to read the paper (pdf or source tarball) my presentation was based on. My final recommendation: stick with Beamer (until Prezi starts handling TeX).

I started off making a Prezi just to see how it worked, how easy it was to use and such. It’s pretty simple to use, which is nice. Of course, as one comes to expect, it doesn’t handle LaTeX. Ok, so, no worries, I’ll just do some TeX to make pictures I need, and insert pictures. Prezi will actually let you put in pdfs. So what you could do is make a beamer presentation, with a very basic template and no titles, and then use “pdftk” to “burst” the pdf, making a single pdf for each frame, and then use ImageMagick‘s “convert” to change the file type, do some cropping and trimming and things (if desired), and then load all those little pictures where you want them. You probably won’t want to use the actual pictures as path steps in your prezi, because it’ll zoom all the way in, but that’s ok, because you can just wrap a hidden frame where you want it.

So I then made another prezi, putting text where I wanted, thinking I’d then go in and make lots of little pictures, and put them where I wanted. This prezi did lots more zooming and twisting, so it was sorta fun, but I did worry a little if it might turn some people off (or make them motion sick :)). And then I realized I didn’t really want to make all of those little pictures, that the fonts wouldn’t match, and that I’d probably still have resolution issues (I couldn’t find an easy way to make, say, svgs, from tex).

Ok, so, maybe I could cheat a little bit. You can get away with a lot if you just use italics for math. I thought maybe I could use this for most of the math, and then rely on fewer pictures to have to insert. Sadly, prezi won’t do italics (or underline, or bold!). That was fairly surprising to me. No LaTeX I basically expect, but no italics? Well, ok, maybe I can cheat another way. Surely there’s Unicode characters for most of what I want, I could just type those in. But no, prezi (I’d happily blame flash here, I don’t know what wasn’t really working) wouldn’t do that either. I’d type unicode characters, and nothing would show up. I’d copy a unicode character typed elsewhere, and try to paste it in, and nothing. Sigh.

I pretty much gave up at that point, and made a beamer presentation. But the prezi urge just wouldn’t die. I decided that if I took whole frames from my beamer presentation, and added those to my prezi, I would (a) have consistent fonts, (b) wouldn’t have lots of tiny pictures to upload, and (c) probably wouldn’t do as much twisting and spinning and zooming, and would maybe, then, end up with a better presentation. One could pdftk burst and convert like I mentioned before, but I think I was having some issues getting good resolution that way (looking back, I question this, so you may want to play around). So I decided I could take screenshots of every frame, when it was in full-screen mode, and use those as my pictures. ImageMagick’s “import” takes screenshots, and with the ‘window -root’ option, it grabs full-screen. I don’t know how to force xpdf to turn the page from outside the program, so I set up a quick little bash script that would beep, sleep 2 seconds, and then import a screenshot. Switch workspace to my full-screened xpdf (put ‘xpdf*FullScreenMatteColor: white’ in .Xdefaults and do ‘xrdb -merge .Xdefaults’ before running xpdf), and just press the spacebar after every beep. Badabing. 2-3 minutes later, and I’ve got a 1280×800 image of every frame from my presentation. Upload to prezi, twist, zoom, and you’re done.

Except, no. Prezi has the dis-fortune of having to work on any screen resolution. I don’t know what they’re magic zoomer does to decide how to zoom, but things don’t go great if you try to present my prezi fullscreen at a different resolution. And, unfortunately, I ended up in a room with a computer whose screen was at a different resolution, and that I wasn’t allowed to change. So I fell back on my beamer talk. :-/ People said it was good anyway.

According to the support forums and associated prezi, I maybe should have been able to figure this out. Perhaps converting to pngs was my downfall. I really thought I tried keeping things as pdfs. I’ve been wrong before. Oh well, it’s over now. And I did learn other fun stuff with all this fiddling.

While I was doing all this, I finally figured out how to use pstricks to do text in a circle (or along other paths). I think I’ve tried before, but never quite figured out what was going on with \PSforPDF, even if I was able to put text on a path. But this time I got it, thanks to this presentation [pdf] (which I probably looked at before, too). If you’re working on project.tex, put all the pstricks stuff in \PSforPDF blocks, run latex, dvips, and ps2pdf, eventually outputting project-pics.pdf. Then when you run pdflatex project.tex, since you’re doing pdflatex instead of latex, \PSforPDF probably expands to some sort of \includegraphics[project-pics], and imports the *-pics.pdf (making that -pics assumption about the filename) you just made. Good stuff. LaTeX will probably be one of the things I miss the most about getting out of mathematics in academia.

Approximating Functions of Spaces

February 25, 2010

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.

—-

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

Kan Extensions

April 1, 2009

Let’s set some notation up. Let \mathscr{A,B,C} be categories, and suppose that F:\mathscr{A}\to \mathscr{B} and G:\mathscr{A}\to \mathscr{C} are functors. Our goal is to define a functor H:\mathscr{B}\to\mathscr{C} that in some reasonable sense extends F (like, maybe we could make a commuting triangle?).

In particular, I’d like my H to also come with a natural transformation \eta:HF\to G (this is what justifies the term “extension”). And I’d like H to be the “best” such functor. That is, if K:\mathscr{B}\to\mathscr{C} comes with a natural transformation KF\to G, then I want to have a natural transformation K\to H (presumably making appropriate diagrams commute). If I can find an H with this property, I will call it the (right) Kan extension of G along F.

Calling this the right extension has always confused me. I always, in fact, had it backwards – I would have called this the left extension. Since HF\to G, I think of H as being on the left of G, so I thought it was the left extension. I guess instead I should be thinking about how it shows up on the right of any other extension (there’s a natural transformation to it).

Since I had my extension on the wrong side, I’ve always been confused by the pointwise construction of the extension. Let’s have H be the right extension, in the “correct” sense above, so that HF\to G. In several places (references at the bottom) I’ve seen that one can compute H(b) as a limit over an arrow (slice) category:

H(b)=\lim\limits_{\substack{Fa\to b\\ \in F\downarrow b}}G(a).

For convenience, let me suppose F:\mathscr{A}\to\mathscr{B} is the inclusion of a sub-category. Then I can suppress it from the notation, and the above becomes

H(b)=\lim\limits_{\substack{a\to b\\ \in \mathscr{A}\downarrow b}}G(a).

But here’s where I always got confused – I’m supposed to get a natural transformation H\to G. But this means I’m looking for a family of maps, each of which is a map out of a limit. I don’t like mapping out of limits. That’s not what they are for. Limits are for mapping to.

Today, I finally realized that you do, actually, map “out” of this limit, in a sense. Taking a\in \mathscr{A}, we’re supposed to get a map H(a)\to G(a) – that is, a map

\left(\lim\limits_{\substack{a'\to a\\ \in \mathscr{A}\to a}}G(a')\right)\to G(a).

The map we want actually comes from part of the definition of a limit. Since id:a\to a\in \mathscr{A}\to a, this arrow is an object that the limit is taken over. And the value of the functor whose limit we are taking, evaluated at this object, is just G(a). And so, by definition of limit, we always have a “projection” from the limit to the functor evaluated at any of the objects we are taking the limit over, and so we’ve got our map.

Of course, you’re supposed to check this is “works” – that’s its natural and universal and probably other things as well. But I’m happy now, because I finally have things on the standard side, and see where the maps come from.

While I’m on the subject, I should point out that if \mathscr{A} is a full subcategory of \mathscr{C}, then H(a)=G(a) for all a\in\mathscr{A}, because id:a\to a is an initial object in the arrow category you take the limit over. If the inclusion isn’t full, though, this need not happen.

As another (final) note, I should mention why I’m looking at Kan extensions. Possibly after changing limits to homotopy limits in the above, Kan extensions are useful because they preserve homotopy limits. With the notation above, \text{holim }H\simeq \text{holim }G. So if you’ve got a functor out of some big category, but show that it’s equivalent to the extension of a functor on a subcategory, you can work with the smaller category to think about homotopy limits.

References: When Wikipedia isn’t enough (Kan extension), I look at MacLane’s “Categories for the Working Mathematician”. I’m also a big fan of Borceux’s 3 volume “Handbook of Categorical Algebra”. Perhaps with my new understanding, I should go see what I can make of MacLane’s statement that “All Concepts Are Kan Extensions”…

Non-Functorial

January 31, 2009

In other words, I’m hosed.

I’ve been thinking about a category of ‘abstract locally complete partitions’. Fix M=m\times \mathbb{R}^n, a disjoint union of copies of \mathbb{R}^n indexed by elements of the finite set m. My category is then supposed to consist of pairs (\rho,f) where \rho is a finite collection of affine spaces, A_{\rho}=\coprod_{i\in s}A_i, along with a partition, \Lambda, of the indexing set s, and f:A_{\rho}\rightarrow M an affine (on each component) map which is non-locally-constant. I interchangeably let \rho refer to the collection of spaces, or the partition on that collection of spaces.

To describe non-locally-constant, I must first remind you that I let my partition \Lambda on s induce a partition on A_{\rho} where x\sim y iff the component containing x is equivalent to the component containing y, mod \Lambda. That is, all points in a component A_i are considered equivalent, and two components are equivalent as determined by \Lambda. Now f being non-locally-constant means that there exists x,y equivalent mod \Lambda but with f(x)\neq f(y).

Now, given such an object (\rho,f) in my category, I would like to reduce it to a more restricted type of complete locally affine partition. In particular, I would like to reduce it to the case where

  • There are no more than m zero-dimensional affine spaces in A_{\rho}, and no more than m non-zero-dimensional affine spaces in A_{\rho}.
  • If A_i is a component of A_{\rho} that is not zero-dimensional, then the \Lambda-equivalence class of i is just i itself.
  • My map f takes distinct zero-dimensional spaces in A_{\rho} to distinct components of M. Similarly for non-zero-dimensional spaces.

To complete the description of my big category, I need to describe the morphisms. A map \alpha from (\rho,f) to (\rho',f') will be an affine map \alpha:A_{\rho}\rightarrow A_{\rho'} such that f'\circ \alpha=f and \overline{\alpha(\rho)}\leq \rho' – a property I will now further clarify. My \rho consists of a partition of the space A_{\rho}. By \alpha(\rho), I mean the transitive closure of the relation on A_{\rho'} where whenever x\sim y in A_{\rho} then f(x)\sim f(y) in A_{\rho'}. This process gives me an equivalence relation \alpha(\rho) on A_{\rho'}. Any time I have an equivalence relation \sigma on a (disjoint union of) affine space(s), I let \overline{\sigma} denote the finest coarsening of \sigma that is “locally affine” – meaning equivalence classes are (disjoint unions of) affine subspaces, and all equivalence classes in a given component are parallel (technically, there’s probably a little more than that, but it’s good enough for now I guess). So now we have the meaning of \overline{\alpha(\rho)}, and by \overline{\alpha(\rho)}\leq \rho', I simply mean that \overline{\alpha(\rho)} is coarser than \rho' (by which I mean the equivalence relation on A_{\rho'}.

So allow me to recap. My \alpha is an affine map with a property saying that an appropriate triangle commutes (f'\circ \alpha=f), and the affine closure of the image of the partition for \rho is coarser than the partition for \rho'. Since \rho' is a “complete” locally affine partition (any two points in the same component are in the same equivalence class), this also forces \overline{\alpha(\rho)} to be “complete”.

Of course, I’m not mentioning that this category is really “a category object in the category of topological spaces”. So really I have a space of objects, a space of morphisms, and enough maps between them to make sense of things. I’ll continue not mentioning that, saving it for another day.

Now, like I said, I want to be reducing any (\rho,f) in my big category to get it down to a particularly nice form. One of the main steps I had been relying on turns out to not be allowed, because it isn’t functorial. I hardly knew one could write down “obvious” maps that weren’t functorial, but I’ve apparently done so rather frequently lately.

So, what is this construction? Given (\rho,f), one step I want to take is to replace any subset of the components of A_{\rho} that are (1) all related by the partition, and (2) all map to the same component of M. I want to replace such a subset by the affine span (direct sum in the category of affine spaces) of the spaces in it. This seems entirely reasonable. Given a bunch of affine spaces, and a map to an affine space, I get, for free, a map from the direct sum of the original affine spaces. That’s what direct sums do.

However, since I have disjoint unions as targets of maps \alpha, I run into trouble. Consider, for example, (\rho,f) where the space A_{\rho} consists of three disjoint points, the equivalence relation has them all equivalent, and f sends them all to the same component. Consider (\rho',f') the same three points, the same f, but only two of the points are equivalent. The obvious map \alpha has \overline{\alpha(\rho)}\leq \rho', and the triangle commutes by construction. Now, when I take affine spans as mentioned, I end up with a single space in \rho (a plane, the affine span of 3 points), and two spaces in \rho' (a line (the span of two points), and a point). That’s a problem, because I no longer know what to do with \alpha. As I mentioned before, given a map from a bunch of affine spaces to a single affine space, I get a map from the affine span to the single space. However, given a map from a bunch of affine spaces to a bunch of affine spaces, I no longer get a map from the affine span to the same bunch of spaces (the single affine span, being connected, can only end up in one of the target spaces).

So that’s upsetting. It’s not even the only thing I’ve written down recently that wasn’t a functor. Back to the drawing board, as a fella says.

My Problem with 0

January 28, 2009

In my research recently, I’ve been debating between two setups for a category. My category is supposed to have, as objects, a finite set s of spaces, a partition \Lambda of s, and a map from the disjoint union of those spaces to a space M. I tend to bundle all of this information up into \rho (for the finite set, it’s partition, and the collection of spaces) and f (the map to M). In my situation, M is a disjoint union of copies of \mathbb{R}^n. The spaces I have in \rho have, for a while, been affine spaces. But there’s also always been a question about maybe having them be vector spaces. The difference, of course, is the existence of 0.

There are a few ways to think about affine spaces. The least precise is to say it is a vector space that forgot where its 0 is. With this idea, a pointed affine space is (essentially) a vector space. Every affine space has an underlying vector space, and given two points in the affine space, you can find their difference, which will be a vector in the underlying vector space. Since differences are defined in a vector space, every vector space is (essentially) an affine space whose underlying vector space is the one you started with.

Now, I have this collection of spaces (either vector or affine) and an affine map f to M – that is, the map is affine (linear after a linear translation) on each space in \rho. Since I have an equivalence relation \Lambda (by abuse of notation) on my spaces in \rho, I can take the transitive closure of the image, f(\Lambda), and get an equivalence relation on M. I then have this process in mind where I convert this equivalence relation on M to one of a particularly nice form, which I have been calling ‘locally affine.’ For more, see my earlier writeup.

Part of the process to convert an equivalence relation to one that is locally affine involves looking at pairs of points that are parallel to some linear subspace of (a component of) M. For reasons that deserve to be called ‘continuity’, this is not a great procedure to do in M. If two points are parallel to some line, and you wiggle the two points a little, there’s no reason to assume they are still parallel. And that messes some things up (at least, can, and seems to with what I’ve been hoping to do), or, if nothing else, makes them uglier. So what I’ve been trying to accomplish, or approximate, is to do the similar operations on my original \rho in my category of ‘abstract’ locally affine partitions. I would like to convert the original \rho to something pretty similar, staying in the category I’m defining while not changing the (locally affine coarsening of the) image of the equivalence relation too much.

It’s tempting to assume that my spaces in \rho are affine spaces, because a big part of making the locally affine partitions above is taking affine spans of things. Taking the affine span of a vector space would give me an affine space, but that would mean I’ve changed categories (from a category using vector spaces to a category using affine spaces). But taking the affine space of affine spaces causes no such problem. The problem with using affine spaces is that sometimes I also want to take linear spans, which is not defined for affine spaces. Taking linear spans only works if you have a 0. Grr. I’m going to have to start being more clever, or more careless. I wish I knew which.

Meh

December 15, 2008

Progress has been, as usual, slow. And I haven’t been updating here. My advisor’s last advice was to try to avoid the category \mathscr{L}_M I mentioned previously (category of locally affine partitions on M=m\times \mathbb{R}^n). Ideally I can use something like the \mathscr{C}_M category and just find appropriate reductions. I should be looking to deformation retract \mathscr{C}_M down to a smaller category, and also deformation retract \mathscr{J}_M (which I mostly know has the right homotopy type, in the holim) down to the same category. Or something like that.

So I’ve had two sorts of thought strands going. This is not unexpected, as my overall project is really just the blending of two known cases (some of my advisor’s work). One thing I’ve been thinking about is when M=m\times \mathbb{R}^0 is just a finite set. And I’ve been thinking about a subcategory \mathscr{S}_M of finite sets with partitions, as opposed to the full category \mathscr{C}_M of abstract completely locally affine partitions (what a mouthful), with non-locally-constant maps to M. I think my goal is to deformation retract \mathscr{S}_M down to something along the lines of those pairs (\lambda,f), where \Lambda \vdash s\xrightarrow{f}M and f is injective on the components of \Lambda (or, perhaps even injective overall). To do this, I gotta get the right topology on \mathscr{S}_M, which I’m pretty sure I’ve not yet done.

Part of the reason I think I’ve not yet done this is based on my other line of thought, concerning the case M=1\times \mathbb{R}^n (frequently n=1 when I’m thinking about it). I’m still thinking about a subcategory of \mathscr{C}_M consisting of just finite sets (with partitions) and non-locally-constant maps to M. But I need to set things up so that “points coming together” corresponds to a different partition. That is, when I have two points in the same equivalence class of \Lambda and their image in M lie in the same component (which is easy when M=1\times \mathbb{R}^n), then I could make a sequence of functions that are non-locally-constant for this \Lambda and bring the two points together. The limiting point should be considered as a map from a smaller \Lambda. Another way to say this is that I should think of points in “thick diagonals” (some, but not all, points are the same, e.g. where x=y in \mathbb{R}^3) of M^n as coming from some “smaller partition”.

I’m not saying any of this well. Probably that’s a bad sign. Perhaps I’ll try again tomorrow. Or after I run some ideas by my advisor on Tuesday. Anyway, half the point of this post was to show off the pictures I was looking at today. I was trying to draw my category (or small portions of it) in the case M=m\times \mathbb{R}^0, and was, for some reason, pleased with the following picture:

Math Dec. 14I don’t think I’ll explain too much about it, since I’m not sure how useful it’ll be long-term. But if you’ve thought about set partitions of the set \underline{4}={1,2,3,4}, you might recognize the poset (ordered by refinement) in the diagram above. For any partition \Lambda, I have written down the possibilities for maps to M=\underline{2}\times \mathbb{R}^n such that the composite with the projection M\to \underline{2} is non-locally-constant on blocks of \Lambda. Those underlined in red are, furthermore, injective on blocks.

More Catchup

December 1, 2008

My last post started laying down some of the framework for what I’ve been thinking about, and I thought I’d continue on with that here.

Recall M=m\times \mathbb{R}^n, and that I have a category \mathscr{C} which I think of as a category of “complete locally affine partiitons”. From this I construct the category \mathscr{C}_M consisting of pairs (\rho,f), where \rho\in \mathscr{C} (so \rho=(\rho,A_{\rho},s,\Lambda)) and f:A_{\rho}\rightarrow M does not factor through \rho (that is, there are two points that are equivalent in the partition \rho, but their images are different in M). I’ve also taken up the habit of saying that these maps “disrespect” \rho. Out of this category I had a functor nlc(-,V), the non-locally-constant maps to a real vector space V. Of course, since this is supposed to a be “topological category” (I have a space of objects, and a space of morphisms…), a functor from the category is really a fiberwise space over the space of objects. Let me denote this space by \mathscr{N}_{\mathscr{C}_M}.

I have another category that I didn’t mention yesterday. I denote it by \mathscr{L}_M, and it consists of locally affine partitions of M. Given two locally affine partitions, \sigma,\sigma' in this category, there is an arrow \sigma\to\sigma' precisely if \sigma\leq \sigma' (that is, \sigma is coarser than \sigma' – if two elements are related by \sigma' then they are related by \sigma).

I’m being a little imprecise with the definition of this category for now, because I’m still playing with it and trying to figure out what makes it workable. The idea is that there is a functor im:\mathscr{C}_M\to \mathscr{L}_M given by taking the image. That is, a point in \mathscr{C}_M is a pair (\rho,f), and it makes sense to consider f(\rho), whose affine completion is a locally affine partition of M. Out of the category \mathscr{L}_M we have yet another “non-locally-constant” functor, nlc(-,V). Given a locally affine partition \sigma, nlc(\sigma,V) will be maps M\to V that disrespect \sigma. Of course, I’m again in the situation of topological categories, so really this functor is an object \mathscr{N}_{\mathscr{L}_M} over the space of objects of \mathscr{L}_M.

I can now take the pull-back of \mathscr{N}_{\mathscr{L}_M} along im, obtaining an object (functor) which I’ll denote \mathscr{N}_{\mathscr{L}_M}^* over \mathscr{C}_M. so, if you are counting along at home, that’s two functors over \mathscr{C}_M, and one over \mathscr{L}_M. With any luck I’ll be able to compare all the associated homotopy limits.

There’s one more category, which I’ve been calling \mathscr{J}_M. The letter ‘J’ comes from the fact that this category comes out of some join construction, based on a theorem of Thomason for manipulating (ho)limits (and the general brilliance of my advisor). I’ve got a functor out of it, and I know that the homotopy limit of this guy is the space of embeddings I want. When I get \mathscr{L}_M sorted out, there should be a nice obvious functor \mathscr{J}_M\to \mathscr{L}_M (essentially an inclusion), and that’ll give me the ways to tie together all of my homotopy limits, and tie them all to the space of embeddings. But that’ll have to wait for another day. I was tired when I started this post, and it’s only gotten worse.

Research Catchup

November 29, 2008

I’m debating about having this web space be a regular log of whatever mathematics I look at on a given day. Part of the goal will be to jot down notes on my research. I don’t know why that’s a goal, what it will accomplish, but nevertheless… Anyway, if that’s going to be the case, I thought I should get some notation going.

So, I have some understanding of \Sigma^{\infty} C(m,V), the (suspension spectrum of the) space of configuration of m points in a (real) vector space V (with inner product). Namely, I can write it as

\textrm{holim}_{\rho\in P} \Sigma^{\infty}(\hom(m,V)-\hom(m/\rho,V))

Here \hom just means set functions from the set m=\{1,\ldots,m\} to V. The category P that the homotopy limit is taken over is the poset of set partitions of m (besides the “discrete” partition, where nothing is related to anything else, besides itself). With this in mind, the notation \hom(m/\rho,V) stands for maps from m to V that factor through \rho. For shorthand, I will let \textrm{nlc}(\rho,V) be the set of maps m\to V that do not factor through \rho (that is, \{f|\exists x\equiv y\mod \rho\ni f(x)\neq f(y)\}). Here “nlc” stands for “non-locally constant” (a notational choice that might make more sense later, read on). Thus, I write

\Sigma^{\infty} C(m,V)\simeq \textrm{holim}_{\rho\in P} \Sigma^{\infty} \textrm{nlc}(\rho,V)

I also have some understanding of \Sigma^{\infty} \textrm{Mor}(\mathbb{R}^n,V), the (suspension spectrum of the) space of linear inclusions. It can be written tantalizingly similarly, as

\textrm{holim}_{0\neq E\leq \mathbb{R}^n} \Sigma^{\infty}(\hom(\mathbb{R}^n,V)-\hom(\mathbb{R}^n/E,V)

Here \hom stands for linear maps, and we are picking out the linear maps that are non-constant on linear subspaces E of \mathbb{R}^n. We might as well write this as \textrm{nlc}(E,V), to parallel our earlier notation.

The idea of my research is to blend these two approaches and be able to understand \textrm{Emb}(m\times \mathbb{R}^n,V) (well, its suspension spectrum…). I have one sort of easy way to understand it, as a product, but the understanding is not natural enough (in a strict sense). What I’m currently working on is the more natural understanding.

Given a disjoint union of affine spaces, say A=\coprod_{i\in s} A_i, let us say that an equivalence relation \sim on A is locally affine if there are vector spaces E_i\leq V(A_i) (V(A_i) is the vector space underlying the affine space A_i) such that \sim factors through A/E=\coprod_i A_i/E_i. That is to say, if x\sim y (x\neq y) and x,y\in A_i, then whenever a,b\in A_i have a-b\in \langle x-y\rangle (that is, the line ab is parallel to the line xy), then a\sim b. We say that a locally affine partition is complete if each E_i=V(A_i) (that is, the partition factors through the underlying set s). Thus, complete partitions are in 1-1 correspondence with set partitions of the underlying set s. As another bit of notation, for any equivalence relation \rho on A, there is a unique finest coarsening of \rho that is locally affine, and we denote this locally affine coarsening by \overline{\rho}.

Now, we make a category, \mathscr{C}, of complete locally affine partitions. The objects are tuples \rho=(\rho,A_{\rho},s,\Lambda) where A_{\rho}=\coprod_{i\in s}A_i, \Lambda is a set partition of the finite set s, and \rho is the complete partition corresponding to the set partition \Lambda. This notation is overdetermined, of course. What you need to determine an object is… a finite set s with a partition \Lambda and a dimension function d:s\rightarrow \mathbb{N}_0. Intuitively, then, A_i will be \mathbb{R}^{d(i)}. Maps \rho\to \rho' in this category will be locally affine maps f:A_{\rho}\to A_{\rho'} such that \overline{f(\rho)}\leq \rho'. Perhaps a word on the notation… Given any old set partition, and a function out of that set, you can take the image of the partition. This gives you a relation (not necessarily transitive) on the target set, and you then take it’s transitive closure to get an equivalence relation on the target. For us, we then take the locally affine coarsening of this image relation. The inequality \leq means that \overline{f(\rho)} is a coarser partition than \rho'. This is equivalent to saying that the underlying set partition for \overline{f(\rho)} is coarser than the underlying set partition for \rho'.

Let M=M_{m,n}=m\times \mathbb{R}^n=\coprod_m \mathbb{R}^n. (So, as a reminder, my overall goal is to understand \textrm{Emb}(M,V).)

Next, we get a (contravariant) functor \textrm{nlc}(-,M) from this category \mathscr{C} to spaces. Intuitively, given a \rho\in \mathscr{C}, \textrm{nlc}(\rho,M) is the subspace of maps A_{\rho}\to M that are affine on each component, and do not factor through the relation \rho (as in our original two cases). Using this functor, we get the Grothendieck category (I think that’s the right word for it), which I denote by \mathscr{C}_M=\mathscr{C}\ltimes \textrm{nlc}(-,M). Objects are pairs, (\rho,f) where \rho\in \mathscr{C}, and f\in \textrm{nlc}(\rho,M). Morphisms… are what you’d guess (being careful about the contravariance of \textrm{nlc}).

We also have a functor \textrm{nlc}(-,V) from \mathscr{C} to spaces (it’s the same thing as above, essentially). By factoring first through the projection \mathscr{C}_M\to \mathscr{C} we consider \textrm{nlc}(-,V) as a functor from \mathscr{C}_M to spaces. It is now a standard (or so) result that \lim_{\mathscr{C}_M}\textrm{nlc}(-,V) is equivalent to \textrm{Nat}_{\mathscr{C}}(\textrm{nlc}(-,M),\textrm{nlc}(-,V)). Of course, being after (stable) homotopy types, I’m supposed to use \textrm{holim} instead of the normal limit, and end up with what my advisor calls the space of “homotopy natural transformations”. I’m also supposed to stick some \Sigma^{\infty}s in there, to get an equivalence with \Sigma^{\infty} \textrm{Emb}(M,V).

Of course, I’m only hopeful that I will get an equivalence. I have not yet shown it’s true. In the above, some details were swept under the rug (surprise, surprise). For example, nearly all of the categories above were “topological”. Meaning, I have a space of objects, and a space of morphisms, and various maps… I’ve got a category object (pair?) in the category of spaces. And so “functor” is something you have to be a little more careful with defining, and then “holim” as well. Nevertheless… consider yourself mostly caught up. If nothing else, on the notation.

More to come? Until then, you can entertain yourself with my advisor’s work, that I am hugely indebted to. It’s on the arxiv.