Learn about renormalization theory in higher dimensions through this mathematics lecture that explores the extension of one-dimensional techniques to two-dimensional dynamical systems. Discover how combining renormalization approaches with non-uniformly partially hyperbolic systems theory enables the analysis of diffeomorphisms in dimension two. Examine the key concepts of critical points in higher dimensions, understand two-dimensional a priori bounds, and explore uniform control of dynamics at small scales. Progress through topics including unimodal interval maps in one dimension, quantitative Pesin theory, critical orbits in 2D, linear ordering on renormalization limit sets, and the regularity of first return maps. Master the mathematical foundations needed to understand this joint research work that bridges the gap between one-dimensional and higher-dimensional dynamical systems analysis.