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.
(Some words to my background: In my university, there is no research group working in the mathematics of QFT. I did my thesis in some quantum gravity related stuff. On the physical side, the topic is related to holographic dualities in 3d quantum gravity approaches such as spin foam models. However, my main tasks were more on the mathematical side, especially in geometric and low-dimensional topology. I absolutely loved my project and I definitely can also imagine working more in that direction, but still I really would like to try out working on some project in the area of mathematical QFT, since this was always a topic I wanted to dive into and I guess, a Phd is the perfect way to do so. Hence, I unfortunately haven't research experience in things like algebraich QFT yet, however, since I did my Masters in mathematical physics, I did of course a lot of courses in the relevant directions of both theoretical physics and pure mathematics.)
Thanks a lot!