Explore the fundamental theoretical obstacles encountered when attempting to compute and analyze dynamical systems in this 50-minute conference talk. Delve into the mathematical complexities that arise at the intersection of computation theory and dynamical systems, examining why certain aspects of these systems remain computationally intractable or present unique challenges for numerical analysis. Investigate the theoretical foundations that underpin our understanding of what can and cannot be effectively computed in the realm of dynamical systems, including discussions of algorithmic limitations, computational complexity, and the mathematical structures that create barriers to computation. Gain insights into cutting-edge research addressing these computational challenges and discover how theoretical computer science approaches can illuminate the fundamental nature of dynamical systems analysis.