The perturbative expansions in weak coupling constant in quantum field theories are in general divergent, with the coefficients growing in factorial. One way to make a rigorous sense out of the divergent series is to resum the series using Borel transform. When the large order behavior of the expansion is sign-alternating Borel resummation may yield the full amplitude. But in QCD the series are...

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 study of definable equivalence relations on complete separable metric spaces (i.e., Polish spaces) has emerged as a new direction of research in descriptive set theory over the past twenty years. Quotients of such equivalence relations include the orbit space of an ergodic group action, the set of Turing degrees of subsets of natural numbers, and the space of isomorphism classes of countabl...

It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the 1D Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be conside...

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain

We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. We discuss the main open questions in the area.