New Frontiers in Rigidity, Flexibility and Complexity in Dynamics and Related Applications.

Overview

Modern dynamics has evolved into a central pillar of mathematics, linking the study of chaos and complexity with deep structural rigidity phenomena. Originating from Poincaré’s foundational insights and formalized through Kolmogorov-Sinai entropy, the field has seen monumental progress, ranging from Ratner’s measure rigidity theorems to Ornstein’s isomorphism theory for Bernoulli shifts. Today, the discipline is defined by a fascinating dichotomy: the rigidity of algebraic actions, which enforces strict structural constraints, and the flexibility of smooth systems, where invariants like Lyapunov exponents exhibit vast malleability. These theories have notably transcended their dynamical roots, driving spectacular solutions to longstanding problems in number theory, such as the Oppenheim and Littlewood conjectures.

This program is structured around two parallel frontiers in modern dynamics. One major component focuses on homogeneous dynamics and number theory, highlighting the recent surge in quantitative methods—including effective equidistribution and random walks on homogeneous spaces—and their applications to Diophantine approximation on manifolds, fractals, and automorphic forms. Complementing this is a deep exploration of complexity, rigidity, and flexibility, with a primary emphasis on the subtle stochastic behaviors of parabolic dynamics. This stream will investigate renormalisation techniques, spectral theory, and mixing properties, while simultaneously addressing foundational challenges in abstract ergodic theory, including classification problems and symbolic coding. The event aims to provide a platform for participants to synthesise tools from harmonic analysis, logic, and representation theory, fostering new collaborations across these diverse branches.