Explain the Gromov-Witten Theory and the Invariant

Question

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 🙂

In: 1

Answer 1

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.