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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!

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  • $\begingroup$ You might be interested in an online course of mine on 2d chiral CFT (which is still work in progress) andreghenriques.com/Teaching/CFT2020.html. In the course notes, I mention a couple of important open problems (see p50, or p86). $\endgroup$ Dec 23, 2021 at 23:18
  • $\begingroup$ Thanks a lot! I will definitely check them out :-) $\endgroup$ Dec 23, 2021 at 23:40

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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.

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    $\begingroup$ Thank you very much for your answer and your advice. Indeed, the terminology "hot topic" was badly chosen and I agree that it is not an advisable criterion for a PhD. My question was more in the direction about some current topics in this area. :-) $\endgroup$ Dec 23, 2021 at 14:40
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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

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    $\begingroup$ Thanks a lot! I wasn't aware of these seminars, thats great! $\endgroup$ Dec 23, 2021 at 17:49
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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.

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  • $\begingroup$ Thank you very much for your answer. Indeed, I already stumbled over Hamburg while scanning the internet for suitable places. $\endgroup$ Dec 23, 2021 at 14:41
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    $\begingroup$ The information about Hamburg is somewhat out of date. Klaus Fredehagen retired already a few years ago, so this Hamburg group no longer exists. $\endgroup$ Dec 23, 2021 at 15:00
  • $\begingroup$ Sorry, I was not aware of this. Thanks for this information. $\endgroup$
    – Jon
    Dec 23, 2021 at 15:20
  • $\begingroup$ Ah okay, I see, thats unfortunate. Thanks for the information! $\endgroup$ Dec 23, 2021 at 17:51
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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.

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    $\begingroup$ Thank you very much for your advice, this is indeed a very usefull website! I know that finding a phd position takes some time, for that reason I already start looking yet. I will finish my Master's around April and my plan was to start with a phd sometimes after, so some time after finishing my Masters and the start of the new winterterm 2022/23 $\endgroup$ Dec 23, 2021 at 15:30
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    $\begingroup$ A similar cluster in the Netherlands is gqt.nl, which provides a good overview of events and people working in this area. $\endgroup$ Dec 28, 2021 at 13:12

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