Can someone explain some real world uses of topology (mathematical). I read about Perelman, and it is crazy.?

Question

Perelman and his topology discovery. Can someone put some meaning behind this?

Answer 1

Most of the real world applications of topology are in the "real world" of mathematics.

The Fundamental Theorem of Algebra states that every polynomial on degree greater than or equal to 1 has a complex root.

To prove it, you have to go into the field of complex analysis. To get to the point in complex analysis where you can do that proof requires you to go through topology on the way.

Topology allows you to prove things using abstract constructs and then apply them to concrete examples.

Suppose you have theorem X. You can prove that it's true for the real line, then prove it again for the plane, then prove it again for 3 dimensional space, and on and on. Or, you can prove that it's true for any topological space, which makes it true for 1-D, 2-D, 3-D,...n-D spaces. It also makes it true for topological spaces you've never imagined.

Answer 2

Topology is especially important because it analyzes the basic properties of geometric objects and number sets that are used in algebra.

A good example is the theorem that if a continuus real function takes both negative and positive values on a closed interval, it must have a zero in that interval. This is intuitively clear, but not very easy to prove. Topology proves it by deriving the more general proposition that the continuous image of a compact & connected set is compact & connected.

In physics, topological properties also play an important role. For instance, a coaxial cable (used with antennas etc.) can conduct an electromagnetic signal with little loss, whereas a normal copper wire cannot. This is caused by the topological structure of the wires.

Answer 3

I am taking a differential geometry course right now that is sort of an extension of topology. We are dealing with manifolds and surfaces. Differential geometry (along with the topology) is being used in string theory and in some Einstein problems.

Also, if you consider metric spaces in topology you can prove many theorems from analysis a little easier (as was already mentioned).

To me, topology seems to be a link between many of the higher level mathematics. So this means that topology is used in many areas of mathematical applications. Plus topology is just too cool.

Answer 4

Even the best mathematicians have already said that they don't know what the real world application of this discovery will be. They just said it's going to be a new way of looking at dimensions and topology after people have had time to chew on it for a while.

Look how long it took to apply E=mc^2!!

Answer 5

The basic ideas of topology surfaced in the mid-19th century as offshoots of algebra and ANALYTIC GEOMETRY. Now the field is a major mathematical pursuit, with applications ranging from cosmology and particle physics to the geometrical structure of proteins and other molecules of biological interest. http://www.chez.com/alcochet/toposi.htm

To check some uses :
http://en.wikipedia.org/wiki/Topology#See_also
Local geometry
Spatial curvature
network topology
Link Topology is the study of the linked structure of the World Wide Web.

Electronic boards as you can barely see at:

http://www.santafe.edu/research/publications/workingpapers/01-05-029.pdf

Digital cartography
http://www.dcs.hull.ac.uk/CISRG/publications/DPs/DP7/DP7.html
Any situation in the real world that needs to manage spatial structures is related to topology.

Answer 6

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