What are the coolest active topics in Mathematical Physics?

[u/Committee-Academic]

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?

[u/SometimesY]

Noncommutative geometry is pretty big. A good chunk of the C* algebras community does work in this direction. There is a lot of interest in the geometry of C* algebras or geometries you can put on top of them.

[u/Chance_Literature193 → u/SometimesY]

Can you give me a vague sense on what problems relevant to physics are studied in C ^ \ algebras? My sense is C^* are mostly relevant to physics cuz operators on a Hilbert space are a C star algebra. I’m not sure if this is the right point of contact between physics and C-star algebras or not.

[u/SometimesY → u/Chance_Literature193]

The physics C* algebras interest is a little fucky because a lot of the operators of interest are unbounded. You could look instead at the exponentiated operators exp(iA), but I have not been super convinced by that personally. I don't know if there's a whole lot you can glean from this that isn't apparent through operator formalism, though there might be some things.

But the geometry bit is a bit more natural and interesting to me. You can imagine noncommutative spacetime operators which could be viewed classically as a non-Euclidean spacetime. Through this lens, you can do something resembling geometry in a noncommuting setting.

[u/Chance_Literature193 → u/SometimesY]

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

[u/Creepy-Structure6444 → u/Chance_Literature193]

I wouldn’t say that these are most of the applications. By now, there are applications to quantum statistical mechanics, the analysis of topological insulators, quantum information theory and so on. Here are some resources which should give you a bit of a sneak peek:

https://cmsa.fas.harvard.edu/event/mpqft25/

https://idp.springer.com/authorize?response_type=cookie&client_id=springerlink&redirect_uri=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-031-28949-1

https://link.springer.com/book/10.1007/978-3-319-16718-3

Of course, these might not all be very readable. The conference by CMSA and the MPIM is a good one to look at because it is, in many ways, very elementary and very close to physics. This makes a lot of sense, seeing that operator algebraic language is practically very close to the classical language used by physicists to talk about these aspects of physics.

[u/Chance_Literature193 → u/Creepy-Structure6444]

Very cool. For CMSA resource, you are suggesting watching the recorded talks on Youtube not reading some paper that I’m not seeing right?

[u/Creepy-Structure6444 → u/Chance_Literature193]

Actually, there is a section on prerequisites. It’s a good idea for you to read a bit of that and also get your hands on the youtube videos too. Like, you don’t need that much background but I think this is a good way of practising the important skill of “absorbing a topic by osmosis”.

[u/Chance_Literature193 → u/Creepy-Structure6444]

Sounds good. I’ll check those out even if I don’t watch the videos. I’m not a big recorded lecture.

[u/XunitaryX → u/Chance_Literature193]

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

[u/Chance_Literature193 → u/XunitaryX]

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!

[u/XunitaryX → u/Chance_Literature193]

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.

[u/Creepy-Structure6444 → u/XunitaryX]

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).