Anatoly M. Vershik, mathematician

I met Anatoly Vershik in 1994 at King's College London, where he gave a talk. He is president of the St. Petersberg Mathematical Society. With Gelfand and Graev, he found the first non-local representation of the current group of SL(2,C). He has wide interests in mathematics, including the Fock decomposition of Lévy processes. In a recent paper with N. V. Tsilevich, Fock factorisation and decompositions of the L2 spaces over general Lévy processes, the authors solve for this case a problem posed in my lectures of 1968. There, I showed that any infinitely divisible cyclic representation of a group can be embedded in a Fock space, a result that I termed the `Araki-Woods embedding theorem', as it looked similar to a general result of Araki and Woods of 1966. My result is not exactly a special case, as Araki and Woods deal only with the Type I case. I left open the question of whether the embedding was an isomorphism; that is, whether the action of the group on the vacuum spanned the Fock space, or generated a proper subspace. This question is solved (when the Lie group is R) in the paper by Tsilevich and Vershik.