Mean-Field Spin Glasses: From Parisi PDE to Machine Learning Landscapes

Speaker: Anton Klimovsky, Senior Lecturer @ Würzburg University, Germany

Abstract: Mean-field spin glasses, often epitomized by the Sherrington-Kirkpatrick(SK) model, are paradigmatic disordered systems characterized by complexenergy landscapes and non-trivial phase transitions. This presentationelucidates the theoretical framework pioneered by Giorgio Parisi --recipient of the 2021 Nobel Prize in Physics for his revolutionaryreplica symmetry breaking ansatz. We examine the Parisi variationalformula for the limiting free energy and its profound analyticalimplications, focusing on the Hamilton-Jacobi-Bellman equation (the"Parisi PDE") and its stochastic control interpretations. Themathematical rigour underlying these developments is furthered by MichelTalagrand’s seminal contributions to probability theory and functionalanalysis, recognized by the 2024 Abel Prize.Furthermore, we explore the remarkable connections between spin glasstheory and contemporary machine learning. Drawing inspiration from JohnHopfield’s influential neural network models and the foundational workof Geoffrey Hinton on Boltzmann machines -- both of whom were honouredwith the 2024 Nobel Prize in Physics -- we discuss how spin glassconcepts inform energy-based models and the challenges of optimizingnon-convex, high-dimensional landscapes. This presentation underscoresthe interdisciplinary nature of mean-field spin glass theory and itsenduring relevance to rigorous mathematical analysis of complex systemsacross diverse scientific domains.