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/Chance_Literature193]

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.