We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures. Introduction. The Borel reducibility theory of Pol...

The k -dimensional Weisfeiler-Leman procedure ( - WL ), which colors -tuples of vertices in rounds based on the neighborhood structure graph, has proven to be immensely fruitful algorithmic study Graph Isomorphism. More generally, it is fundamental importance understanding and exploiting symmetries graphs various settings. Two are -equivalent if produces same final coloring both graphs. 1 -equi...

Relative self-linking and linking \numbers" for pairs of oriented knots and 2{component links in oriented 3{manifolds are de ned in terms of intersection invariants of immersed surfaces in 4{manifolds. The resulting concordance invariants generalize the usual homological notion of linking by taking into account the fundamental group of the ambient manifold and often map onto in nitely generated...

The fundamental dispersion relation of transverse electro-magnetic waves in a cold collisionless plasma is formally equivalent to the two dimensional inertio-gravity rotating shallow water system, where Coriolis frequency can be identified with frequency, and gravity wave phase speed plays role light. Here we examine this formal equivalence governing linearised equations, compare between propag...