Character methods and probabilistic methods in groups

Character methods, probabilistic methods, and their combination, have been instrumental in the solution of various problems in group theory over the years.

In the talk I will provide a brief background and then focus on some brand new results and conjectures, related to random walks and to word maps.

In particular I will discuss a solution (joint work with Larsen and Tiep) of the so-called Probabilistic Waring Problem for finite simple groups, which applies strong new character bounds.

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