topology/geometry in physics

electroweak

Which fields (and specific projects) of physics have the closest connections to topology and geometry? If your research involves a lot of math of this sort, can you tell me about it?

I took introductory QFT (P&S) and string theory (Polchinski) sequences this last academic year and touched on several such ideas, about which I know only the basics:
-- Holography and AdS/CFT
-- Compactifications and moduli spaces
-- Yang-Mills instantons
Math is obviously quite relevant to the formulation of these concepts, but to what extent does knowing topology and geometry aid one in modern research?

It seems like anything having to do with nonperturbative field theory requires cutting-edge math. This new focus on duality and QGP looks super neat. Is it math-intensive?

Does modern GR research involve much math? The class I took used a bit of general topology and tensor work but it was all fairly basic. Does cutting-edge GR work (what is cutting-edge GR work, anyway?) involve cool concepts in topology and geometry? I feel like I saw the phrase "fiber bundle" in a GR paper at one point...

I know little to nothing about condensed matter physics, but I hear it can be mathy. How so? Is CFT a big deal? Do physicists still think about VOAs? What is this knotted fields business? What about TQFT? Do any recent developments in quantum information involve interesting math?

My plan is to continue learning algebraic/differential topology/geometry as I have been for about a year now: it's interesting enough to keep me motivated even if I can't apply it. But it would really be great if I could work it into my research in physics. And since I'm nearing the point in my career where I need to begin developing more specialized interests, I'd like to know what my options are. Again, I'd love to hear about any type of theory research that involves what you consider to be beautiful math -- HEP, CMT, whatever. Thanks!

kloptok

You can perhaps take a look at the books:

Bernhard Schutz: "Geometrical methods in mathematical physics"
Frankel: "The geometry of physics: an introduction"
John Baez: "Gauge fields, knots and gravity"

I'm not an expert on this but from what I've seen I would say that geometry is widely used in String Theory, gauge theories and QFT in general, and also of course in General Relativity. Fiber bundle is definitely a word I recognize (although it may not be in the books above)! =) There are also fields in Condensed Matter theory I think which use topology, the thing that comes to mind is topological insulators.

Take a look at the books above and you will probably find loads of use of topology and geometry in physics.

cosmik debris

I think it was in the book "Gravitation" by MTW that one of them proposes that "all physical theories are geometry". This was not a new idea in 1974 or thereabouts but it is one reference that I remember.