The appropriate language for describing the pure Yang-Mills theories is introduced. An elementary but precise presentation of mathematical tools which are necessary a geometrical description gauge fields given. After recalling basic notions differential geometry, it shown in what sense potential connection some fiber bundle, and corresponding field associated curvature. It also how global aspects theory (e.g., boundary conditions) coded into structure bundle. Gauge transformations equations...

We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but nevertheless are non-integrable. As more sensitive integrability test, we propose analysis complexity (``algebraic entropy'') map using growth degree its iterates: is associated with polynomial while generic exponential for chaotic systems.

We analyze the factorization process for lattice maps, searching integrable cases. The maps were assumed to be at most quadratic in dependent variables, and we required minimal (one linear factor) after 2 steps of iteration. results then classified using algebraic entropy. Some new models with polynomial growth (strongly associated integrability) found. One them is a nonsymmetric generalization homogeneous KdV (modified Schwarzian), this model have also verified "consistency around cube".

Abstract We describe a family of integrable lattice maps related to the known quad Q 4 . The integrability criterion we use is vanishing algebraic entropy. has advantage being parametrized rationally: all its parameters are unconstrained.

We give the basic definition of algebraic entropy for lattice equations. The is a canonical measure complexity dynamics they define. Its vanishing signal integrability, and can be used as powerful integrability detector. It also conjectured to take remarkable values (algebraic integers).

We illustrate the use of notion derived recurrences introduced earlier to evaluate algebraic entropy self-maps projective spaces. in particular give an example, where a complete proof is still awaited, but different approaches are such perfect agreement that we can trust get exact result. This instructive example experimental mathematics.

We consider various 2D lattice equations and their integrability, from the point of view 3D consistency, Lax pairs Bäcklund transformations. show that these concepts, which are associated with not strictly equivalent. In course our analysis, we introduce a number black white models, as well variants functional Yang-Baxter equation.

Using three different approaches, we analyse the complexity of various birational maps constructed from simple operations (inversions) on square matrices arbitrary size. The first approach comprises study images lines, and relies mainly univariate polynomial algebra, second is a singularity analysis third method more numerical, using integer arithmetics. These methods have their own domain application, but they give corroborating results, lead us to conjecture algebraic entropy class matrix...

By Johannes J. Duistermaat: 627 pp., £90.00, isbn 978-1-4419-7116-6 (Springer, New York, 2010).