Gibbs measures, a term we will generically use to refer to probability distributions described via exponentials of Hamiltonians, appear in extremely diverse contexts ranging from statistical physics, probability, statistical inference/ML, and more recently, in the theory of deep learning and generative models. These distributions are often high- (or infinite-) dimensional, and thereby exhibit a fascinating interaction between entropy aspects derived from the ambient space, and energy aspects derived from the defining Hamiltonian. Since this Hamiltonian may include nonlinear interactions, for instance in spin glasses, the so-called "energy landscape" is often highly complex. Due to this intractable nature, various approximation and sampling techniques have been devised over the years, such as, MCMC techniques, mean-field approximations, message-passing heuristics, variational inference and score modelling to name a few. Viewed as a whole, the field represents one of the most important and far-reaching topics of modern research, borrowing ideas from and contributing back to an ever-growing range of disciplines. The purpose of this seminar would be to broadly visit some of these aspects, as well as to discuss various tools, both rigorous and non-rigorous, employed to dissect these objects.

References (please feel free to suggest more!):

1. Gibbs measures on lattices from a rigorous, statistical-physics viewpoint: https://www.unige.ch/math/folks/velenik/smbook/
2. Low-complexity Gibbs measures: https://arxiv.org/abs/1810.07278, https://arxiv.org/pdf/1612.04346
3. A collection of lectures and useful references to the stochastic localization/sampling literature: https://metaphor.ethz.ch/x/2024/fs/401-4634-24L/
4. A broad survey on the "physics" approach to statistical problems: https://arxiv.org/pdf/1511.02476
5. A more book-style presentation of these topics: https://sphinxteam.github.io/EPFLDoctoralLecture2021/Notes.pdf
6. A short collection of lectures describing tools from physics and their applications to statistical problems: https://brloureiro.github.io/assets/pdf/NotesPrinceton_BL.pdf
7. A book on graphical models and their inference: https://www.cs.columbia.edu/~blei/fogm/2023F/readings/WainwrightJordan2008.pdf
8. A survey on variational inference, and some intuitive and algorithmic aspects: https://arxiv.org/pdf/1601.00670
9. Energy-based models: https://arxiv.org/pdf/2101.03288, a tutorial: https://energy-based-models.github.io/ijcai2022-tutorial
10. A tutorial on normalizing flows: https://www.youtube.com/watch?v=7TOvhz93G9o
11. A tutorial on diffusion models: https://arxiv.org/pdf/2208.11970