The core of the project involves applications of nonstandard analysis to that part of combinatorial number theory that overlaps with infinite Ramsey theory on the natural numbers (more generally, on semirings). Results in this field usually have the form: “For every finite partition of a well organized infinite set S one of the cells of the partition is also well organized”. The solutions to such problems always involve the use of ideas and results coming from many different fields, including harmonic analysis, ergodic theory, topological dynamics, ultrafilters theory, and nonstandard analysis.

In this project we are more interested in new recent developments obtained with nonstandard methods, algebraic and topological properties of ultrafilters and ergodic and topological dynamics. 

Starting in the late 80's, but especially in the past few years, there has been an increasing use of nonstandard techniques in this area, because they can be used to reduce the complexity of the mathematical objects that one needs in a proof, therefore offering a much better intuition, which allows to obtain much simpler (and shorter) proofs. 

The main nonstandard technique that will be adopted in this project involves a peculiar nonstandard manipulation of ultrafilters, based on the notion of iterated hyperextensions of N. This allows to approach problems in this area by means of simple algebraic manipulations of nonstandard elements, which permits to obtain also certain qualitative informations on the ultrafilters to be used.