IPAM Research Talk: Two Mathematical Perspectives on Intelligence: Spectral Learning and Adaptive Reservoir Computing

Artificial intelligence has renewed interest in a fundamental question: What mathematical principles give rise to learning and intelligent behavior? This talk presents two complementary mathematical perspectives on intelligence inspired by research conducted at the Institute for Pure and Applied Mathematics (IPAM). The first explores how spectral methods from dynamical systems use eigenvalues to reveal hidden structure in complex systems, leading to interpretable approaches for learning and prediction. The second examines adaptive reservoir computing through a nonlocal mathematical framework, where interactions occur across neighborhoods rather than only at individual points, allowing network connections to evolve through homeostatic mechanisms such as pruning and impairment to study how structure influences memory and computation. Together, these projects illustrate how modern mathematics provides new ways to understand intelligent behavior alongside traditional machine learning. The talk will emphasize the underlying mathematical ideas, computational experiments, and open research questions, with the goal of introducing undergraduate students to active areas of research at the intersection of dynamical systems, scientific computing, and artificial intelligence.