Applications in combinatorial number theory of iterated nonstandard extensions and idempotent ultrafilters
Mauro Di Nasso
Universita di Pisa
This talk will be part of CANT 2013, the Combinatorial and Additive Number Theory Conference, on May 21-24, 2013 at the Graduate Center.
Abstract: By using nonstandard analysis, and in particular iterated elementary (nonstandard) extensions, we give foundations to a peculiar way of manipulating idempotent ultrafilters. The resulting formalism is suitable for applications in Ram- sey theory of numbers. To illustrate the use of this technique, we give (rather) short proofs of two important results in combinatorial number theory, namely Milliken- Taylor’s Theorem (a generalization of Hindman’s theorem), and Rado’s theorem about partition regularity of diophantine equations, in a new version formulated in terms of idempotent ultrafilters.
Some familiarity with the notion of elementary extension will be assumed in the first part of the talk, but in the second part about applications I will not assume any specific prerequisite (also the notions of ultrafilter and of partition regularity will be recalled).
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.