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August 30, Tuesday
12:00 – 13:30

Let a,b,c,… in S_n be random permutations on n elements, chosen at uniform distribution. What is the distribution of the permutation obtained by a fixed word in the letters a,b,c,…, such as ab,a^2, a^2bc^2b, or aba^(-2)b^(-1)? More concretely, do these new random permutations have uniform distribution? In general, a free word w in F_k is called uniform if for every finite group G, the word map $w: G^k to G$ induces uniform distribution on G (given uniform distribution on G^k). So which words are uniform? This question is strongly connected to the notion of primitive words in the free group $F_k$. The word w is called primitive is it belongs to some basis, i.e. a generating set of size $k$. It is an easy observation that a primitive word is uniform. It was conjectured that the converse is also true. We prove it for F_2. In a very recent joint work with O. Parzanchevski, we manage to prove the conjecture in full. Akey ingredient of the proofs is a new algorithm to detect primitive elements.