Joint IAS/PU Number Theory Seminar

I will talk about a program aimed at exploiting randomness of certain "graphs of symbols" in determining the precise structure of the pro-nilpotent Galois groups of number fields, and how this has been successfully implemented in nilpotency class 2. I will overview how some recent progresses in arithmetic statistics informs us of the difficulties in extending this, starting from nilpotency class 4 onward. Finally, I will explain  the strong ties that this problem has with two variants of the inverse Galois problem, namely the minimal ramification problem and Malle's conjecture: I will present some recent progress on the latter in the nilpotent case. This is based on joint works with Peter Koymans.