Insegnamento a.a. 2024-2025

1. Introduction To Statistical Physics. Spin systems, thermodynamic limit, phase transitions and critical phenomena, ergodicity breaking.
2. Networks and Random Graphs. Intro to Graph theory, Percolation and Giant Component, K-core, leaf removal, ODE analysis, Belief Propagation on Factor Graphs, Ising on sparse graph, Error Correcting Codes, Stochastic Block Model, Population Dynamics algorithm.
3. Statistical Physics of Learning. Perceptron Models,  Teacher-Student scenario, Storage case, Average Case, intro to the Replica Method, Phase diagram for the Perceptron, Binary Perceptron Inference: Impossible-Hard-Easy transitions, Algorithmic implementations using Message Passing.
4. Dynamical Systems. Ordinary Differential Equations, Perceptron Online Learning, mean field epidemic spreading, Stochastic Differential Equations, Fokker Plank, Denoising Diffusion Models.

• Explain relevant features of statistical physics systems and critical phenomena.
• Describe and model complex networks.
• Illustrate algorithmic approaches to tackle problems on graphs.
• Explain theoretical approaches to learning problems and characterize average-case hardness of learning, inference, and optimization problems.
• Describe message-passing algorithms and the replica method.
• List the main features of stochastic and deterministic dynamical systems and the corresponding theory.

• Identify the basic features allowing simple modeling of complex phenomena.
• Analyze complex systems using random graph theory and identify key structural properties relevant to real-world networks.
• Implement belief propagation algorithms to solve inference problems in probabilistic graphical models.
• Apply the principles of statistical physics to model and interpret the behavior of complex systems.
• Develop models for spin glasses and utilize them to understand disordered systems and optimization problems.
• Construct and solve differential equations, both ordinary and stochastic, to model dynamic processes in various applications.

• Lectures
• Practical Exercises

• Lectures will cover in detail each topic of the course.
• Practical Exercises will focus on specific instances of analytical or computational problem-solving. 

• 20% of the final grade given by individual assignments. The students will be required to derive and implement in a specific setting the generic algorithms discussed during the course.
• 80% of the final grade given by oral exam. Understanding of all the topics covered in the course will be evaluated. Besides verbally answering questions, students will be asked to reproduce part of calculations seen during the course using pen and paper or a whiteboard.

Reference textbooks, lecture notes, and lecture slides will be provided throughout the course.