Invariant of words from random matrices
Mathematics Colloquium
Let w be a word in a free group and let G be a finite (or more generally compact) group. A w-random element of G is obtained by substituting the letters of w with uniform random elements from G. For example, if w=xyxy^{-2}, the random element is ghg^{-2}, with g and h independent uniformly random elements of G. Composing with linear representations of G, we get w-random matrices. A series of works over the last decade has revealed many intriguing phenomena around w-random elements in nice families of groups, such as the symmetric groups or the unitary groups. In particular, many invariants of words, some new and some well-known, play significant roles in this theory.
This story involves probability, topology, algebra, combinatorics and representation theory. In the talk, which is aimed at graduate students, I will try to give a flavor of this interesting theory.