Spin Glasses, Learning, and Optimisation in High Dimension

This project aims at tackling problems in statistical mechanics, with a primary focus on spin glasses, followed by an exploration of machine learning related questions. Spin glasses, initially studied in the 1980s using non-rigorous methods by theoretical physicists, revealed complex behaviours which were previously unknown: They conjectured that the phenomenon they depicted should be present in many more models such as neuronal networks. From a mathematical perspective, spin glasses are disordered systems where each ""spin"" can be seen as a random variable, interacting with others through a network of complex, often conflicting, interactions. Recent recognition in the field, with Giorgio Parisi winning the Nobel Prize in 2021 and Michel Talagrand receiving the Abel Prize in 2024, highlights the importance of spin glass studies. However, many questions remain unanswered, particularly regarding the mixed SherringtonKirkpatrick model: the phase transition between the high and the low-temperature regime is not known rigorously. One key objective of the fellowship is to explore this phase transition mainly through a dynamical approach by viewing the disorder in the Hamiltonian as different Brownian motions, with temperature playing the role of time. Beyond spin glasses, the project will also investigate neural networks (specifically, one-hidden-layer models with a large but finite number of neurons): the algorithms underlying their training are still not fully understood, and we aim to elucidate why, despite the non-convexity of their objective, do they converge to a desirable solution? Although spin glasses and machine learning may seem unrelated at first glance, it is believed that many probabilistic techniques from spin glass theory can be applied to understanding machine learning algorithms, especially in high-dimensional settings. By bridging these two areas, the fellowship aims to deepen our theoretical understanding of both domains. ""