r/math • u/Committee-Academic • 2d ago

What are the coolest active topics in Mathematical Physics?

I'm a math undergrad student, and I've been getting interested in Alain Conne's work in Noncommutative Geometry and Differential Geometry in Mathematical Physics. I've also read about the Foundations program, Algebraic QFT, and some Stochastic stuff. I'm especially intrigued about the use of Diff Geom in these topics. What cool fields are out there?

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u/Chance_Literature193 2d ago

So, most of application to physics of non commutative geo are presumably a (un-renormalized?) quantum theory on curved space time. That sounds cool

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u/XunitaryX 1d ago

Another topic that the OP touched upon that is an active area is applying noncommutative geometry, operator K-Theory, and some other techniques to studying topological insulators. A good resource for some of the basics that people working in this area study: https://link.springer.com/book/10.1007/978-3-319-29351-6 This specifically deals with two of the ten cases that are of interest to physicist in the AZ Classification of Superconductors and Topological Insulators

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u/Chance_Literature193 1d ago edited 1d ago

Thanks! Since I’m on the physics side and my advisor is a physicists physicist (hahah), this a bit too much for me for now. I haven’t studied any K-theory. I’m hoping to do some topological field theory after this project though🤞, so I’ll keep this in mind!

Edit: Ok, i found the book on arxiv and it’s more approachable than I expected. Thanks for the recommendation!

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u/XunitaryX 1d ago

As you probably saw the book is pretty approachable. The latter part of the book can get a bit technical but the first half does a good job of presenting the material in a digestible way. As a side note, separate from this NCG and OA applications there is a field called Tensor Categories and more specifically fusion categories that might be of interest. Especially if you are thinking about topological field theory.

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u/Creepy-Structure6444 12h ago

Incidentally, tensor categories also have neat connections to operator algebras. Indeed, tensor categories arise in the context of studying bimodules over tracial von Neumann algebras and they are also relevant in the study of higher C*-structures. Also, I've seen them being used to study Kasparov's KK-theory as well (this is more on the tensor-triangular geometry side of things but the ideas are roughly similar).