# topology

### Nobel Prize Notes

On Pharyngula, PZ Myers examines the work of Yoshinori Ohsumi, who was awarded the prize in Physiology for his studies of autophagy in yeast. Autophagy, or self-consumption, is a strategy used by all cells to recycle malfunctioning bits of themselves, or to survive during times of starvation. But autophagy is also involved in cancer metastasis and may play a role in other diseases such as Parkinson's. Meanwhile, the Nobel prize in Physics did not go to LIGO and the observation of gravitational waves as widely expected. Instead it was divided between three individuals for "theoretical…

### Chaos

One mathematical topic that I find fascinating, but which I've never had a chance to study formally is chaos. I've been sort of non-motivated about blog-writing lately due to so many demands on my time, which has left me feeling somewhat guilty towards those of you who follow this blog. So I decided to take this topic about which I know very little, and use the blog as an excuse to learn something about it. That gives you something interesting to read, and it gives me something to motivate me to write. I'll start off with a non-mathematical reason for why it interests me. Chaos is a very…

### Understanding Non-Euclidean Hyperbolic Spaces - With Yarn!

One of my fellow ScienceBloggers, Andrew Bleiman from Zooilogix, sent me an amusing link. If you've done things like study topology, then you'll know about non-euclidean spaces. Non-euclidean spaces are often very strange, and with the exception of a few simple cases (like the surface of a sphere), getting a handle on just what a non-euclidean space looks like can be extremely difficult. One of the simple to define but hard to understand examples is called a hyperbolic space. The simplest definition of a hyperbolic space is a space where if you take open spheres of increasing radius around…

### Geometric L-systems

As I alluded to yesterday, there's an analogue of L-systems for things more complicated than curves. In fact, there are a variety of them. I'm going to show you one simple example, called a geometric L-system, which is useful for generating a certain kind of iterated function fractal; other variants work in a similar way. I'm going to show it to you in the context of an interesting fractal figure, called the T-square fractal. The T-square starts with a colored square centered inside of a white square twice its size. In each iteration, for each of the outermost squares, you create a square…

### Wonderful Mobius Transformation Video

Via The Art of Problem-Solving, a great video on Mobius transformations. I never really got how the inversion transformation fit in with the others before seeing this!

### The Mapping of the E8 Lie Group (Minor Update)

I've been getting tons of mail from people in response to the announcement of the mapping of the E8 Lie group, asking what a Lie group is, what E8 is, and why the mapping of E8 is such a big deal? Let me start by saying that this is way outside of my area of expertise. So I fully expect that I'll manage to screw something up as I try to figure it out and explain it - so do follow the comments, where I'm sure people who know this better than I do will correct whatever errors I make. Let's start with the easy part. What's a Lie group? Informally, it's a group whose objects form a manifold,…

### Simplices and Simplicial Complexes

One thing that comes up a lot in homology is the idea of simplices and simplicial complexes. They're interesting in their own right, and they're one more thing that we can talk about that will help make understanding the homology and the homological chain complexes easier when we get to them. A simplex is a member of an interesting family of filled geometric figures. Basically, a simplex is an N-dimensional analogue of a triangle. So a 1-simplex is a line-segment; a 2-simplex is a triangle; a three simplex is a tetrahedron; a four-simplex is a pentachoron. (That cool image to the right is…

### Homotopy

I've been working on a couple of articles talking about homology, which is an interesting (but difficult) topic in algebraic topology. While I was writing, I used a metaphor with a technique that's used in homotopy, and realized that while I've referred to it obliquely, I've never actually talked about homotopy. When we talked about homeomorphisms, we talked about how two spaces are homeomorphic (aka topologically equivalent) if and only if one can be continuously deformed into the other - that is, roughly speaking, transformed by bending, twisting, stretching, or squashing, so long as…

### Building Towards Homology: Vector Spaces and Modules

One of the more advanced topics in topology that I'd like to get to is homology. Homology is a major topic that goes beyond just algebraic topology, and it's really very interesting. But to understand it, it's useful to have some understandings of some basics that I've never written about. In particular, homology uses chains of modules. Modules, in turn, are a generalization of the idea of a vector space. I've said a little bit about vector spaces when I was writing about the gluing axiom, but I wasn't complete or formal in my description of them. (Not to mention the amount of confusion that…

### Twisted Spaces: Fiber Bundles

It's been a while since I've written a topology post. Rest assured - there's plenty more topology to come. For instance, today, I'm going to talk about something called a fiber bundle. I like to say that a fiber bundle is a cross between a product and a manifold. (There's a bit of a geeky pun in there, but it's too pathetic to explain.) The idea of a fiber bundle is very similar to the idea of a manifold. Remember, a manifold is a topological space where every point is inside of a neighborhood that appears to be euclidean, but the space as a whole may be very non-euclidean. There are all…

### Another Example Sheaf: Vector Fields on Manifolds

There's another classic example of sheaves; this one is restricted to manifolds, rather than general topological spaces. But it provides the key to why we can do calculus on a manifold. For any manifold, there is a sheaf of vector fields over the manifold. Let's start by explaining what a vector field is. A vector field over a topological space T is a mapping which associates each point in T with a vector in some euclidean space ℜn. What the vector field is useful for is defining the concept of differentiation in the manifold. In normal euclidean space, what a rate of change means is…

### Examples of Sheaves

Since the posts of sheaves have been more than a bit confusing, I'm going to take the time to go through a couple of examples of real sheaves that are used in algebraic topology and related fields. Todays example will be the most canonical one: a sheaf of continuous functions over a topological space. This can be done for *any* topological space, because a topological space *must* be continuous and gluable with other topological spaces. Let's quickly recall the definition of sheaves. A sheaf *F* is a mapping from open sets in a topological space **T** to objects in a category **C** that has a…

### A Second Stab at Sheaves

I've mostly been taking it easy this week, since readership is way down during the holidays, and I'm stuck at home with my kids, who don't generally give me a lot of time for sitting and reading math books. But I think I've finally got time to get back to the stuff I originally messed up about sheaves. I'll start by talking about the intuition behind the idea of sheaves. The basic idea of a sheave is to provide a way of taking some local property of a topological space, and demonstrating that it holds everywhere. The classic example of this is manifolds, where the *local* property of being…

### Stepping Back a Moment

The topology posts have been extremely abstract lately, and from some of the questions I've received, I think it's a good idea to take a moment and step back, to recall just what we're talking about. In particular, I keep saying "a topological space is just a set with some structure" in one form or another, but I don't think I've adequately maintained the *intuition* of what that means. The goal of today's post is to try to bring back at least some of the intuition. So let's recall just what a topological space is. Our definition from the [very beginning of the topology series was:][top-…

### Big to Small, Small to Big: Topological Properties through Sheaves (part 2)

Continuing from where we left off yesterday... Yesterday, I managed to describe what a *presheaf* was. Today, I'm going to continue on that line, and get to what a full sheaf is. A sheaf is a presheaf with two additional properties. The more interesting of those two properties is something called the *gluing axiom*. Remember when I was talking about manifolds, and described how you could describe manifolds by [*gluing*][glue] other manifolds together? The gluing axiom is the formal underpinnings of that gluing operation: it's the one that justifies *why* gluing manifolds together works. […

### Big to Small, Small to Big: Topological Properties through Sheaves (part 1)

Suppose we've got a topological space. So far, in our discussion of topology, we've tended to focus either very narrowly on local properties of **T** (as in manifolds, where locally, the space appears euclidean), or on global properties of **T**. We haven't done much to *connect* those two views. How do we get from local properties to global properties? One of the tools for doing that is a sheaf (plural "sheaves"). A sheaf is a very general kind of structure that provides ways of mapping or relating local information about a topological space to global information about that space. There are…

### Groupoids and Strange Definitions

In my last topology post, I started talking about the fundamental group of a topological space. What makes the fundamental group interesting is that it tells you interesting things about the structure of the space in terms of paths that circle around and end where they started. For example, if you're looking at a basic torus, you can go in loops staying in a euclidean-looking region; you can loop around the donut hole, or you can loop around the donut-body. Of course, in the comments, an astute reader (John Armstrong) leapt ahead of me, and mentioned the fundamental group*oid* of a…

### Walking in Circles: Fundamental Groups

In algebraic topology, one of the most basic ideas is *the fundamental group* of a point in the space. The fundamental group tells you a lot about the basic structure or shape of the group in a reasonably simple way. The easiest way to understand the fundamental group is to think of it as the answer to the question: "What kinds of different ways can I circle around part of the space?" To really understand what the fundamental group is, we need to think back to what I said about group theory. Group theory is most basically about defining *symmetry*; and symmetry in turn means a kind of *…

### Groups and Topology

I'm going to start moving the topology posts in the direction of algebraic topology, which is the part of topology that I'm most interested in. There's lots more that can be said about homology, homotopy, manifolds, etc., and I may come back to it as some point, but for now, I feel like moving on. There's some fun stuff in algebraic topology which comes from the intersection between group theory and topology. To be able to talk about that, you need the concept of a *topological group*. First, I'll run through a very quick review of groups. I wrote a series of posts on group theory for GM/BM…

### Building Interesting Shapes by Gluing

I thought it would be fun to do a couple of strange shapes to show you the interesting things that you can do with a a bit of glue in topology. There are a couple of standard *strange* manifolds, and I'm going to walk through some simple gluing constructions of them. Let's start by building a Torus. It's not strange, but it's useful as an example of interesting gluing. We can make a torus out of simple rectangular manifolds quite easily. We start by building a cylinder by doing the circle construction using rectangles instead of just line segments. So we take four squares; curve each one into…