Topology...
Author
Dear all,
I know this is a bit OT, but I was wondering. I''ve been reading some topology recently, and while it''s pretty intresting stuff, I don''t really see it''s application. So far in the book I was reading, all the problems given are just to prove certian things about topology, in other words, learn topology to prove topology :-). Could anyone tell me what mathematictions use it all for? Do you need it if you study other fields of mathematics? Thanks,
Jesse
www.laeuchli.com/jesse/
Well, I can''t think of many things myself, but one springs to mind: some proofs in complex analysis, specifically the deformation theorem (and other related theorems), and to a small extent the identity theorem (maybe). Oh, and a sort of lemma, and the proof of it (to do with the identity theorem iirc) - the limit of a sequence is the same (or can be considered to be the same in some circumstances) as the limit point of a set.
I can''t think of much I''ve done other than in complex analysis that topology applies to. But then I''ve been doing fairly non-topology related courses.
Miles
I can''t think of much I''ve done other than in complex analysis that topology applies to. But then I''ve been doing fairly non-topology related courses.
Miles
Topology is very abstract so I wouldn''t expect a lot of "real world" usefulness.
However...
Euler''s rule is a topological theorem, Faces + Verts = Edges + 2 - 2 * (number of ''holes'' in polyhedra). Obviously, you can use this to detect holes in polyhedra.
Knot theory is also a branch of topology which has come on in leaps and bounds over the last 2 decades.
Is graph theory considered a part of topology or is it a part of combinatorics?
"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley
However...
Euler''s rule is a topological theorem, Faces + Verts = Edges + 2 - 2 * (number of ''holes'' in polyhedra). Obviously, you can use this to detect holes in polyhedra.
Knot theory is also a branch of topology which has come on in leaps and bounds over the last 2 decades.
Is graph theory considered a part of topology or is it a part of combinatorics?
"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley
A friend of mine did her PhD in Mathematics and studied Topology specifically. Her research had to do with unresolvable knots in DNA structures and how enzymes resolve those knots. Yep, OT.
-Kirk
-Kirk
There are two different branches that are called "Topology":
- Set topology. This is probably what you have been studying. It was born out of the need to add rigour to the concepts of limit and continuity, that had been used in the XIX century very freely, sometimes leading to false proofs.
- Algebraic topology. This is the most difficult part of Maths that I have studied. I am unable to explain what it is in a few lines. Some of the things studied in this field are bundles, shieves, homotopy, homology and cohomology theory.
Both branches are extremely useful for other areas of Mathematics. I used to study singularities of ordinary differential equations, and topology was one of the main tools used.
A big picture of modern Geometry is that we are interested in studying objects (mainly manifolds) that have some structure in them. We can study the same objects ignoring some of their structure. For instance, take a sphere:
- As a set, it's just the same as the real numbers, because you can make a one-to-one mapping between a sphere and a line.
- As a topologic manifold, its the same as a box, because there is a continuous one-to-one mapping between them.
- As a differential manifold, its the same as an ellipsoid, because there is a differentiable (?) one-to-one mapping between them.
Topology is just one possible level of precission when studying geometric objects, which gives us enough information to determine limits and continuity properties, but not enough to talk about differentiability.
[edited by - alvaro on October 16, 2002 10:24:52 AM]
- Set topology. This is probably what you have been studying. It was born out of the need to add rigour to the concepts of limit and continuity, that had been used in the XIX century very freely, sometimes leading to false proofs.
- Algebraic topology. This is the most difficult part of Maths that I have studied. I am unable to explain what it is in a few lines. Some of the things studied in this field are bundles, shieves, homotopy, homology and cohomology theory.
Both branches are extremely useful for other areas of Mathematics. I used to study singularities of ordinary differential equations, and topology was one of the main tools used.
A big picture of modern Geometry is that we are interested in studying objects (mainly manifolds) that have some structure in them. We can study the same objects ignoring some of their structure. For instance, take a sphere:
- As a set, it's just the same as the real numbers, because you can make a one-to-one mapping between a sphere and a line.
- As a topologic manifold, its the same as a box, because there is a continuous one-to-one mapping between them.
- As a differential manifold, its the same as an ellipsoid, because there is a differentiable (?) one-to-one mapping between them.
Topology is just one possible level of precission when studying geometric objects, which gives us enough information to determine limits and continuity properties, but not enough to talk about differentiability.
[edited by - alvaro on October 16, 2002 10:24:52 AM]
Topology is also used in some Physics fields, as in high-energy physics, when studying solitonic solutions resulting from field theory lagrangians. The example that comes to me is the Skyrme Model.
Not the most practical thing I agree.
Not the most practical thing I agree.
I don''t consider topology to be off-topic, and this thread is a welcome discussion, 
.
There are areas of game development where topological representation and analysis can be quite useful, even necessary.
For example it might be used for some pathfinding algorithms. (In, fact, pathfinding algorithms often use a navigation map that is really a sort of topology graph of a level.)
Topology is also quite important in some geometric algorithms for game development, such as subdivision surfaces or progressive meshes---important for displaying massive 3D game worlds, and for streaming 3D geometry over a network.
Even beyond things that are basically geometric in nature (navigation is usually geometric in nature, after all), topology can be used to represent human behavior and decision processes for AI, experience and health evolution, etc.
Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.
.There are areas of game development where topological representation and analysis can be quite useful, even necessary.
For example it might be used for some pathfinding algorithms. (In, fact, pathfinding algorithms often use a navigation map that is really a sort of topology graph of a level.)
Topology is also quite important in some geometric algorithms for game development, such as subdivision surfaces or progressive meshes---important for displaying massive 3D game worlds, and for streaming 3D geometry over a network.
Even beyond things that are basically geometric in nature (navigation is usually geometric in nature, after all), topology can be used to represent human behavior and decision processes for AI, experience and health evolution, etc.
Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.
There is a class of error correcting codes for quantum computers made possible by topological properties of quantum systems. These were inspired by the use of topology in studying exotic phenomena like the fractional quantum hall effect.
Modern topology is a relatively new field, maybe 80 years old or so. It''s often a long time before applications for new areas of pure maths are discovered.
Modern topology is a relatively new field, maybe 80 years old or so. It''s often a long time before applications for new areas of pure maths are discovered.
Author
btw, just wondering if Algebraic Topology and Set Topology where at all related...
Jesse
Jesse
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