I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics courses, it was also brought to my attention that operator theory is a very integral part of quantum mechanics. Are there applications of operator algebras in particular to quantum or relativity, or other fields of modern physics? What about applications of operator theory?
In a related vein, I've been trying to find some research problems at my level or only slightly higher so that I can get a flavor, either for these applications, or for the abstract subject itself, as it's studied today. I was wondering if anybody knows either of any such problems, or a source where I could find such problems. To be honest, being a first year graduate student going on second, I'm not even quite sure where I'd look to find research problems that I could definitely guarantee are open, let alone ones I could reach right now. In fact, I'm told that sometimes professors even mistake solved questions as being open. If you have good examples of solved problems that are recent and representative of what I might face in the future, those would be helpful too. Thanks.
Best Answer
The only applications to general relativity that I know of (my field!) is via Connes' noncommutative geometry, which is...complicated. Connes' work is freely (and legally!) available online. See also the math overflow thread Applications of Noncommutative Geometry, which may be interesting.
Operator algebras pop-up in Algebraic Quantum Field Theory too, which may be fun to look at. Actually, trying to discuss quantum fields on curved spacetime requires operator algebras.
The Physics.SX thread "Why are von Neumann Algebras important in quantum physics?" is also relevant.