Topology and physics

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quasar987

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What is the motivation for a physicist to learn topology?

Are there fields of physics that make explicit use of the concept of topology? (which ones)

Do the ideas of topology give any insights into any topic of physics?

etc.
 

Danger

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My math education is essentially non-existent, and topology is a branch of math, but one application that I know of is in such fields as magnetic confinement for fusion experiments (Tokomak type reactors).
It also comes into play when trying to determine the shape of space-time, but that is so far beyond me that I'll leave SpaceTiger to clear it up.
 

quasar987

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I was under the impression that only differential geometry was involved in calculating the shape of space time. Can anyone confirm what Danger said?
 

George Jones

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quasar987 said:
I was under the impression that only differential geometry was involved in calculating the shape of space time. Can anyone confirm what Danger said?
Danger is right - topology is important for global properties of spacetime.

Penrose, Hawking, Geroch and made tremendous use of topological concepts in their work on spacetime.

Any differential manifold is also a topological manifold.

Einstein's equation is a local equation, which doesn't completely the global properties of spacetime. For example, is spacetime simply connected or multiply connected? There have been searches for mutiply connectness, but so far no evidence for this has been found.
 

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quasar987 said:
What is the motivation for a physicist to learn topology?

Are there fields of physics that make explicit use of the concept of topology? (which ones)

Do the ideas of topology give any insights into any topic of physics?

etc.
This intervention is not a professional one and many other people could give a better answer to your question than me. But if you think to the notion of parallel transport and to its consequence, the Berry's phase for example, you get a first concrete application of the indirect effect of the topology on physical phenomenon. In my non-specialist mind, geometry and topology are notions very closed together even if any specialist will immediately contradicts my point of view. Hope I could help you!
 
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I've never got the hang on topology, but i have tried many times lol. Topology actually speaks about very fundamental concepts.It mainly speaks about sets. It treats on how things in a set are connected to each other, no mather how you deform it. Sometimes people speaks about topology as "rubber physics". The role of topology in physics is to make assesments on global properties of systems. Most of what is taught in university speaks of local properties, equations are studied in the neighbourhood of... and stuff like that. To give an example, relativity teaches us that space-time is LOCALLY minkowskian, but it's global structure can't be directly extracted from the behaviour of the equations in a neighbourhood of a point. Differential topology deals with such matters. It is not an easy branch, at least for me, because it makes so little assumptions that demonstrating anything is very difficult
 

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