Roman Kossak

To find "criteria of simplicity" was the goal of David Hilbert's recently discovered twenty-fourth problem on his renowned list of open problems given at the 1900 International Congress of Mathematicians in Paris. At the same time, simplicity and economy of means are powerful impulses in the creation of artworks. This was an inspiration for a conference, titled the same as this volume, that took place at the Graduate Center of the City University of New York in April of 2013. This volume includes selected lectures presented at the conference, and additional contributions offering diverse perspectives from art and architecture, the philosophy and history of mathematics, and current mathematical practice.

These authors give an account of the present state of research on lattices of elementary substructures and automorphisms of nonstandard models of arithmetic. Major representation theorems are proved, and the important particular case of countable recursively saturated models is discussed in detail. All necessary technical tools are developed. The list includes: constructions of elementary simple extensions; a partial classification of arithmetic types, in particular Gaifman's theory of definable types; forcing in arithmetic; elements of the Kirby-Paris combinatorial theory of cuts; Lascar's generic automorphisms; and applications of Abramson and Harrington's generalization of Ramsey's theorem. There are also chapters discussing ?1-like models with interesting second order properties, and a chapter on order types of nonstandard models.