Algebraic Entropy in Combinatorial Dynamical Systems

Explore a 53-minute lecture from the Asymptotic Algebraic Combinatorics 2020 conference, delving into algebraic entropy in combinatorial dynamical systems. Discover how the search for algebraic entropy leads to intriguing questions and answers in bipartite T-systems and R-systems. Examine the classification of Zamolodchikov periodic quivers, including 5 infinite families and 11 exceptional quivers. Investigate the master conjecture, toric quivers, and the affine affine classification. Learn about arborescence formulas, flow descriptions, and toric digraphs through various examples and motivations presented by Pavlo Pylyavskyy from the University of Minnesota, Twin Cities.

Intro
Cluster algebras: quiver mutations
Cluster algebras: variable dynamics
General T-systems (Nakanishi, 2011)
Bipartite recurrent quivers
Bipartite T-system
Tensor product
Zamolodchikov periodicity
The result
Fixed point
Strictly subadditive labeling
Finite finite quivers
The classification of Zamolodchikov periodic quivers
5 infinite families and 11 exceptional quivers
Four classes of quivers
Example: wild
ADE Dynkin diagrams
Algebraic entropy
Master conjecture
Toric quivers
affine affine classification: 41 infinite, 13 exceptional
A system of equations
Solution
Arborescence formula
Flow description
Flow example
Motivation
Examples: toric digraphs