Valery Miskin - Set ideals everywhere

Although set ideals are objects of algebraic nature they penetrate like comets many branches of mathematics from Set Theory and Combinatorics to Topology, Analysis, Measure, Ergodic and Number Theory. They appears naturally in these settings and adorn the tree of Mathematics just as the lights of Christmas lighting strings connecting old and new mathematical concepts and problems. As to the ideals arising in Set Theory and Infinite Combinatorics I concentrate on the classical Commutative Algebra concept of quotient of two set ideals (which is not yet a well known instrument in Set Theory and Topology) and related cardinal invariants of set ideals. This concept proved to be crucial in Boolean ring contexts while solving the problem of isomorphism of symmetry groups of set ideals and the problem of description of set ideals with complete or maximal symmetry groups (problem of H. Macpherson and P. Neumann) and in general while studding the automorphisms and normal subgroup lattice of their symmetry groups. I discuss the solutions to these problems and related open problems. As to the properties of set ideals of topological origin I discuss the concepts related to Stone-ech compactification of w and state a generalization of a Banach-Kuratowski theorem in terms of imprimitivity domains and provide some nice properties of the automorphism group of an absolutely homogeneous countably separated Borel space. I mention some resently posed open problems particularly related to meager and null sets on the real line.