It’s so complicated that I feel it should be the new benchmark of complicado. All i know it’s about Topology, i.e mathematics of “rubber sheet geometry”.
1. Tell me what it is and an example of it if applicable
2. why it’s important
2. Tell me the real life applications of this theory OR how it helps other fields of math/science
I came across this while generally exploring out-of-syllabus math stuff in my school years on Wikipedia. I’m just curious about what this is and how this is even useful, just like any curious five year old.
I guess asking about topics which have confusing Wikipedia articles will definitely enrich and grow the community in a way no other platform could 🙂
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Consider the following problem: You are given a 5 x 5 square. Is it possible to use straight cuts to cut the square into pieces that can be reassembled into a 6 x 4 rectangle?
If you try it, it will seem impossible, but how do you *prove* it’s impossible?
An answer is that you observe that
>1) Cutting any one piece by a straight line doesn’t change the total area of all the pieces.
>
>2) The areas of the given square and rectangle are different.
and thus conclude the task is impossible.
In general, there are many problems where we ask if there is a transformation of a given type between object A and object B. It’s very useful to find some quantifiable property that is unchanged by this type of transformation. That’s called *the invariant*. With that we can easily prove nonexistence of such transformation between two objects by computing their invariants and showing they are different. There are whole fields of math devoted to the study of invariants, e.g. algebraic topology.
That’s presumably what Gromov-Witten invariant is as well since the wiki states
>These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable.
I would strongly recommend to first learn topology, especially fundamental groups, (co)homology and basic stuff about manifolds. For this one in particular, symplectic geometry is also essential, which adds quite a bunch. Without those as basics, truly understanding something as Gromov-Witten invariants is pointless; it is like trying to understand how to build and work a steam engine, but you don’t even know what fire is.
However, to give some quick answers without really making any precise statements:
1. It provides a measure to compare certain types of things, called _symplectic manifolds_, by looking at (co)homology classes, which are somewhat corresponding to sub-manifolds up to “stuff”.
2. It helps to prove results and distinguish objects. As another post quoted from Wikipedia, it can distinguish certain previously undistinguishable symplectic manifolds, but that is still around topology, just with extra data.
3. Such applications usually do not exist (yet) for mathematical results, so you should not expect them. I am unaware of any application of this one in particular that is completely outside of topology/geometry, but I am also not working around symplectic things at all. It supposedly might be useful in string theory, but calling that a “real world application” is only done when postdocs desperately fish for grant money.
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