What are the "hot" topics in mathematical QFT at the time? I am currently finishing my Master's studies in mathematical physics. One topic which always interested me a lot were modern mathematical approaches to Quantum Field Theory (QFT) as well as the mathematical structures arising from studying these theories. Hence, I would like to go into this direction for my Phd. For this, I would like to know, what are some "hot" topics in this area, in the sense of topics in which a lot of active research and effort is put into. Of course "mathematical QFT" is a little bit vague, but I am really open for everything, like algebraic QFT, axiomatic or constructive QFT, etc.
Also, I was looking through a lot of websites of universities in order to localize places where things like this is are discussed. However, I was not very sucessful yet, since it seems that mathematical QFT is not so common as many other topics in physics or mathematics. So, if anyone knows some places where one could try to apply in this direction (preferably in (middle) europe), I would be happy if he/she can share his knowledge with me in the comments or in answers.
Thanks a lot!
 A: Luckily for you, there is an online seminar where you can get a panorama of what is currently going on in the area of mathematical QFT. It is the webinar "Analysis, Quantum Fields, and Probability". To watch live upcoming seminars you will have to join their mailing list, so they can send you the relevant Zoom links. You can also watch the previous talks on their Youtube channel.
QFT is a broad subject with many aspects that give rise to specific problems for mathematicians to work on. If you are interested in the renormalization group aspect, you might want to look at my previous answers:
Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?
Formal mathematical definition of renormalization group flow
A: There is an excellent group at Hamburg University working on axiomatic quantum field theory. Most work is headed toward qft in curved backgrounds. Another good reference is at Trento University with similar areas of research.
My personal taste is that there are a lot of open questions to be understood like confinement in some qft or quantum gravity addressing renormalization and path integrals. For the latter, a proper rigorous understanding is missing and would be hugely helpful.
A: A good way to meet your future adviser (besides already being located at their institution) is to go to conferences or workshops on the topic that you are interested. Referring to the topics of interest that you have mentioned specifically, a list of relevant conferences (as well as some other information) can be found at the Local Quantum Physics Crossroads portal. It is not as active as it could be, and the Covid-19 pandemic has significantly reduced the number of meetings that are currently running. But it might still be helpful.
Of course, if you'd like to find a PhD position tomorrow, this method is not so useful. It is by nature opportunistic and does require some months of advanced planning.
A: I am not sure that "hot topic" is an advisable criterion for a Ph.D. research project, since this will typically mean that easy/doable questions have been done and only the hard/intractable questions remain. You might want to base your choice on a department that offers a broad range of topics in mathematical physics, including QFT. The offer of York University seems attractive, just by way of example.
