Title: The Connection between the von Neumann Kernel and the Zariski Topology
by Dikran Dikranjan (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract
Every group G carries a natural topology Z_G defined by taking as a pre-base of the family of all closed sets the solution sets of all one-variable equations in the group of the form (a_1)x^{ε_1}(a2)x^{ε_2}…(a_n)x^{ε_n} = 1, where a_i ∈ G, ε_i = ±1 for i = 1,2,…,n, n ∈ N. The topology was explicitly introduced by Roger Bryant in 1978, who named it the verbal topology, but the name Zariski topology was universally applied subsequently. As a matter of fact, this topology implicitly appeared in a series of papers by Markov in the 1940’s in connection to his celebrated problem concerning unconditionally closed sets: sets which are closed in any Hausdorff group topology on G. These are the closed sets in the topology M_G obtained as the intersection of all Hausdorff group topologies on G, which we call the Markov topology, although this topology did not explicitly appear in Markov’s papers. Both Z_G and M_G are T1 topologies and M_G ≥ Z_G, but they need not be group topologies. One can use these topologies to formulate Markov’s problem: does the equality M_G = Z_G hold? Markov proved that M_G = Z_G if the group is countable and mentioned that the equality holds also for arbitrary abelian groups (so one can speak about the Markov-Zariski topology of an abelian group). The aim of the presentation is to expose this history, to describe some problems of Markov related to these topologies, and to apply the theory to give a solution to the Comfort-Protasov-Remus problem on minimally almost periodic topologies of abelian groups. This problem is associated to a more general problem of Gabriyelyan concerning the realisation of the von Neumann kernel n(G) of a topological group; that is, the intersection of the kernels of the continuous homomorphisms G → T into the circle group. More precisely, given a pair consisting of an abelian group G and a subgroup H, one asks whether there is a Hausdorff group topology τ on G such that n(G,τ) = H. Since (G,τ) is minimally almost periodic precisely when n(G) = G, the solution of this more general problem also gives a solution to the Comfort-Protasov-Remus problem.