Arithmetic and Geometric Irrationality via Substructures of Nonstandard Models

Arithmetic and Geometric Irrationality via Substructures of Nonstandard Models

Thu, 28/05/2009
17:00 		Tim Gendron (Mexico) Logic Seminar Add to calendar L3
Arithmetic and Geometric Irrationality via Substructures of Nonstandard Models
This purpose of this talk will be to introduce the idea that the spectrum of nonstandard models of a “standard” algebraic object can be used much like a microscope with which one may perceive and codify irrationality invisible within the standard model. This will be done by examining the following three themes:
  • Algebraic topology of foliated spaces We define the fundamental germ, a generalization of fundamental group for foliations, and show that the fundamental germ of a foliation that covers a manifold $ M $ is detected (as a substructure) by a nonstandard model of the fundamental group of $ M $.
  • Real algebraic number theory. We introduce the group $ (r) $ of diophantine approximations of a real number $ r $, a subgroup of a nonstandard model of the integers, and show how $ (r) $ gives rise to a notion of principal ideal generated by $ r $. The general linear group $ GL(2, \mathbb{Z}) $ plays here the role of a Galois group, permuting the real ideals of equivalent real numbers.
    • \item Modular invariants of a Noncommutative Torus. We use the fundamental germ of the associated Kronecker foliation as a lattice and define the notion of Eisenstein series, Weierstrass function, Weierstrass equation and j-invariant.