Integrability and Number Theory
Integer sequences and integrable maps.
When does a nonlinear recurrence relation produce a sequence of integers? When can a nonlinear map be said to be integrable, in some sense? These two questions turn out to be related to each another, and one set of possible answers involves both the Laurent phenomenon and the arithmetic of integer or rational points on algebraic varieties. For illustration, a classical example related to Pell's equation will be discussed, as well as Somos sequences related to elliptic curves. Some higher-dimensional examples and open problems will also be presented.
Hurwitz numbers and the Infinite Wedge.
Hurwitz numbers count maps between complex curves with specified ramification. They can be viewed as a degenerate form of string theory, which led to predictions that certain generating functions for Hurwitz numbers have nice properties: taking the target to be a torus gives quasimodular forms, while taking the target to be a sphere leads to a solution to the 2-D Toda hierarchy. We'll give an overview of how these results can be explained mathematically by means of the infinite wedge, a useful tool for computing the representation theory of the symmetric groups.
Spectra of Sol-manifolds, or can one hear an indefinite binary quadratic form ?
I will explain how the quantisation of the integrable geodesic flows on Sol-manifolds leads to the classical number-theoretic problem about the values of a given indefinite binary quadratic form.
Integrable vs. non-integrable when the space is discrete.
In dynamical systems with discrete space, the distinction between regular and irregular motions must be re-considered. I articulate the main questions using examples, with emphasis on arithmetical phenomena.