 Properties of the free group. (because of the stringology thing)
 Cayley Graphs. (because of the stringology thing)
 quadratic congruence (“square root in modulo”)
 algebraic probability

Probabilistic methods in algebra (not algebraic methods in probability; the other way around)

The number of cycles in a random permutation
Let be a natural number, and let be a permutation of , drawn uniformly at random. Using the cycle decomposition, one can view as the disjoint union of cycles of varying lengths (from to ). For each , let denote the number of cycles of of length ; thus the are natural numbervalued random variables with the constraint We let be the number of cycles (of arbitrary length); this is another natural numbervalued random variable, of size at most … for fixed converge in joint distribution to independent Poisson distributions of intensity respectively…

Define a partition of to be a finite or infinite multiset of real numbers in the interval (that is, an unordered set of real numbers in , possibly with multiplicity) whose total sum is . For instance, is a partition of … In the spirit of the universality phenomenon, one can ask what is the natural distribution for what a “typical” partition should look like; thus one seeks a natural probability distribution on the space of all partitions, analogous to (say) the gaussian distributions on the real line, or GUE distributions on point processes on the line, and so forth. It turns out that there is one natural such distribution which is related to all three examples above, known as the PoissonDirichlet distribution.
