How useful is topology in physics?

[V0ODO0CH1LD]

Whenever I try to understand deeper aspects of the higher maths involved in physics I keep hearing about topology related stuff. How useful is it to learn topology in order to get a deeper understanding on the maths behind physics? Also, what other maths should I look into? Functional analysis? Calculus of variations?

 

[WannabeNewton]

If you want a rigorous understanding of the math behind many different physical branches then yes topology is quite important. In general relativity for example, differential geometry is the natural setting which is built heavily on topology. The natural setting for QM is functional analysis which is also heavy on topology, as well as real analysis.

It is of course not necessary to know the abstract mathematics in order to understand the actual physics so if your main goal is to know the physics then understanding the abstract mathematics would be secondary (for example you can very easily understand the physical content of Noether's theorem and the link between Hamiltonian and Lgrangian mechanics in the usual language presented in second year mechanics texts - you don't need to understand the underlying machinery of legendre transforms as maps between the cotangent and tangent bundles of the configuration space and know about the relationship between lie group actions on the cotangent bundle and the resulting exp induced vector fields relating to conserved quantities etc.)

 

[espen180]

As for the question "what good does formalism do?", while it may not cast significant light on the physics in and of itself (but I think this is subjective), or be useful in computations, it may provide clues about how to generalize a theory, or hints toward unification.

However, often formalism goes a long way. For example, some basic things about qm, like the impossibility of duplicating states, follows quite easily from the Hilbert space formalism.

For the "basic" stuff, e.g. general relativity and nonrelativistic qm, you would be well off with elementary differential geometry and functional analysis. Classical mechanics basically studies flows on symplectic manifolds, etc.

I think this article: http://ncatlab.org/nlab/show/physics and the ones it links to, documents most of the mathematics used in modern physics.

For the really heavy duty stuff at the end of the spectrum, e.g. axiomatic qft, you need just about everything. There is a kind of duality going on, where you have equivalent algebraic and geometric formalisms. Check out the "Related consepts" at ncatlab.org/nlab/show/AQFT

 

[V0ODO0CH1LD]

so is it the case that if I want to understand the generalizations and unifications of "all" principles of physics I should probably get used to the fact that I will have to look into "all" of mathematics?

 

[Vanadium 50]

I disagree. The vast majority of physics does not require topology. Studying it won't hurt you, of course, but it is not something on the standard curriculum.

 

[espen180]

so is it the case that if I want to understand the generalizations and unifications of "all" principles of physics I should probably get used to the fact that I will have to look into "all" of mathematics?

Who knows? We haven't seen all unifications and generalizations yet. But for the ones people are trying to accomplish, namely quantum gravity and TOE's, you already need a fair bit. I found this text a while ago. Maybe it can give you an idea of the scope.
http://people.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf

However, as Vanadium 50 says, for "conventional" physics you rarely need more than basic diff. geometry. Sometimes you need more, like very basic algebraic topology, as when discussing the Bohm-Aharanov effect or other phenomena where topological quantum numbers arise.

There is a very nice explanation of why there are only bosons and fermions in 3 dimensions, and why anyons with fractional spin can exist in 1 and 2 dimensional systems which uses a fair bit of topology:
www.ifi.unicamp.br/~cabrera/teaching/referencia.pdf