Mini-course by Anton Dzhamay (University of Northern Colorado)

Аннотация: In addition to being one of the current active research areas, the theory of discrete Painlevé equations makes it possible to see and learn some of the basic notions of algebra and algebraic geometry "in action” while working with very concrete examples and computations; this is one of the main goals of the course.
We give an introduction to some geometric ideas and techniques used to study discrete integrable systems, such as how to regularize a (birational) two-dimensional discrete dynamical system by changing the geometry of the configuration space (the blowup procedure), how to linearize the mapping on the Picard lattice of the resulting algebraic surface, and, in the discrete Painlevé case, how to recover the mapping from this linear action using the birational representation of its affine Weyl symmetry group.
This course assumes very basic prerequisites of Linear Algebra and Group Theory.

Lecture 1: A QRT Mapping: an example of a discrete integrable system.
Lecture 2: From QRT to Painlevé: a geometric deautonomization approach
Lecture 3: Point configurations and birational representations of some affine Weyl groups
Lecture 4: Discrete Painlevé equations
Lecture 5: Selected extra topics (optional)

References:
M. Noumi. Painlevé equations through symmetry, AMS Translations of Mathematical Monographs, 223 (2004).
K. Kajiwara, M. Noumi, Y. Yamada, Geometric Aspects of Painlevé Equations, J. Phys. A: Math. Theor. 50 (2017)
073001 (164pp), arXiv:1509.08186 [nlin.SI]
H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations,
Comm. Math. Phys. 220 (2001),
H., Duistermaat, Discrete integrable systems, Springer (2010).