Dynamics in innite dimensions
Juliana Fernandes (UFRJ, Rio de Janeiro, Brasil)

Poincaré’s work in the 1890s laid the groundwork for dynamical systems, and together with Lyapunov and Birkho, he emerged as a co-founder of this eld. Their early developments laid the foundation for modern dynamical systems, including both nite and innite-dimensional problems. This talk addresses the "Innite Dimensional Challenge": the analysis of the longtime dynamics of solutions to partial dierential equations (PDEs). In the context of parabolic or hyperbolic PDEs, solutions can be seen as trajectories in Banach spaces, governed by linear or nonlinear evolutionary equations. However, even the proper denition of a solution in innite dimensions is more complex. This challenge will be illustrated by a reaction-diusion equation, whose solution semigroups exhibit dissipation and compactness properties, leading to global attractors that capture the system’s longtime behavior.