Ultrafilters and nonstandard methods in combinatorics of numbers
Universita di Pisa
In certain areas of Ramsey theory and combinatorics of numbers, diverse non-elementary methods are successfully applied, including ergodic theory, Fourier analysis, (discrete) topological dynamics, algebra in the space of ultrafilters. In this talk I will survey some recent results that have been obtained by using tools from mathematical logic, namely ultrafilters and nonstandard models of the integers.
On the side of Ramsey theory, I will show how the hypernatural numbers of nonstandard analysis can play the role of ultrafilters, and provide a convenient setting for the study of partition regularity problems of diophantine equations. About additive number theory, I will show how the methods of nonstandard analysis can be used to prove density-dependent properties of sets of integers. A recent example is the following theorem: If a set $A$ of natural numbers has positive upper asymptotic density then there exists infinite sets $B$, $C$ such that their sumset $C+B$ is contained in the union of $A$ and a shift of $A$. (This gives a partial answer to an old question by Erdős.)
The slides are here.
Mauro Di Nasso is a Ricercatore in the Math Department of the University of Pisa, studying mathematical logic and its applications. The main topics of his research are nonstandard methods and their foundations, infinite combinatorics (ultrafilters) and foundational theories of counting (numerosities). He received his doctorate in 1995 from the Universita di Siena.