Mexico

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:

\item {\it 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 $M$.

\item {\it 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 {\it 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.