topology:
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...
Posted on April 1, 2008 11:56 AM • 13 Comments •
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...
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Posted on August 1, 2007 10:15 PM • 7 Comments •
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!...
Posted on June 25, 2007 8:43 PM • 15 Comments •
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...
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Posted on March 19, 2007 3:03 PM • 38 Comments •
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...
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Posted on March 7, 2007 11:03 AM • 7 Comments •
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...
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Posted on March 4, 2007 9:32 PM • 13 Comments •
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...
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Posted on February 19, 2007 8:30 AM • 10 Comments •
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...
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Posted on January 28, 2007 10:32 AM • 14 Comments •
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...
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Posted on January 10, 2007 8:23 PM • 3 Comments •
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...
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Posted on January 2, 2007 4:04 PM • 2 Comments •
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...
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Posted on December 29, 2006 5:06 PM • 11 Comments •
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...
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Posted on December 21, 2006 8:06 PM • 7 Comments •
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...
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Posted on December 13, 2006 2:07 PM • 11 Comments •
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....
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Posted on December 12, 2006 5:19 PM • 13 Comments •
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...
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Posted on December 11, 2006 8:39 AM • 5 Comments •
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....
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Posted on November 27, 2006 8:20 AM • 6 Comments •
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...
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Posted on November 13, 2006 8:25 AM • 10 Comments •
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...
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Posted on November 7, 2006 8:04 AM • 5 Comments •
After my initial post about manifolds, I wanted to say a bit more about gluing. You can form manifolds by gluing manifolds with an arbitrarily small overlap - as little as a single point along the point of contact between...
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Posted on November 6, 2006 1:40 PM • 0 Comments •
Back in the early days of Good Math/Bad Math, when it was still at blogger, one of the most widely linked posts was one about the idea of dimension. At the time, I said that the easiest way to describe...
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Posted on October 31, 2006 4:09 PM • 8 Comments •
Time to get back to some topology, with the new computer. Short post this morning, but at least it's something. (I had a few posts queued up, just needing diagrams, but they got burned with the old computer. I had...
Posted on October 26, 2006 8:56 AM • 6 Comments •
So, after the last topology post, we know what a manifold is - it's a structure where the neighborhoods of points are locally homeomorphic to open spheres in some ℜn. We also talked a bit about the idea of gluing,...
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Posted on October 17, 2006 9:01 PM • 5 Comments •
Manifolds So far, we've been talking about topologies in the most general sense: point-set topology. As we've seen, there are a lot of really fascinating things that you can do using just the bare structure of topologies as families of...
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Posted on October 4, 2006 8:19 AM • 28 Comments •
If you've got a connected topology, there are some neat things you can show about it. One of the interesting ones involves fixed points. Today I'm going to show you a few of the relatively simple fixed point properties of...
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Posted on October 2, 2006 8:30 AM • 8 Comments •
Next stop on our tour of topology is the idea of connectedness. It's an important concept that defines a lot of useful and interesting properties of topological spaces. The basic idea of connectedness is very simple and intuitive. If you...
Posted on September 27, 2006 8:51 PM • 13 Comments •
This is going to be a short but sweet post on topology. Remember way back when I started writing about category theory? I said that the reason for doing that was because it's such a useful tool for talking about...
Posted on September 25, 2006 4:43 PM • 2 Comments •
One of the really neat things you can do in topology is play games with dimensions. Topology can give you ways of measuring dimensions, and projecting structures with many dimensions into lower-dimensional spaces. One of the keys to doing...
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Posted on September 21, 2006 11:39 AM • 7 Comments •
Just like you can define a sub-set of a set, or a sub-object of an object in a category, you can define a sub-space of a topological space. It's a pretty easy thing to understand; interestingly, a sub-space of a...
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Posted on September 18, 2006 3:21 PM • 1 Comments •
When we talk about topology, in general, the way we talk about it is in terms of shapes: geometric objects and spaces, surfaces, bodies that enclose things, etc. We talk about the topology of a torus, or a coffee mug,...
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Posted on September 11, 2006 8:30 AM • 13 Comments •
The past couple of posts on continuity and homeomorphism actually glossed over one really important point. I'm actually surprised no one called me on it; either you guys have learned to trust me, or else no one is reading this....
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Posted on September 6, 2006 10:15 PM • 12 Comments •
With continuity under our belts (albeit with some bumps along the way), we can look at something that many people consider the central concept of topology: homeomorphisms. A homeomorphism is what defines the topological concept of equivalence. Remember the clay...
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Posted on September 5, 2006 4:49 PM • 11 Comments •
(Note: in the original version of this, I made an absolutely huge error. One of my faults in discussing topology is scrambling when to use forward functions, and when to use inverse functions. Continuity is dependent on properties defined in...
Posted on September 4, 2006 4:22 PM • 19 Comments •
Yesterday, I introduced the idea of a metric space, and then used it to define open and closed sets in the space. (And of course, being a bozo, I managed to include a typo that made the definition of open...
Posted on August 24, 2006 8:20 PM • 7 Comments •
Topology usually starts with the idea of a metric space. A metric space is a set of values with some concept of distance. We need to define that first, before we can get into anything really interesting. Metric Spaces and...
Posted on August 23, 2006 7:46 PM • 20 Comments •
Back when GM/BM first moved to ScienceBlogs, we were in the middle of a poll about the next goodmath topic for me to write about. At the time, the vote was narrowly in favor of topology, with graph theory as...
Posted on August 22, 2006 7:11 PM • 34 Comments •
The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I've been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it's a big...
Posted on August 15, 2006 8:48 PM • 14 Comments •
