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96764 - Mathematics for Complex Systems

Academic Year 2026/2027

                
                        Docente:
                        Emanuele Mingione
                    
                        Credits:
                        6
                    
                        SSD:
                        MATH-04/A
                    
                        Language:
                        English
                    
                        Moduli:
                        
                            Emanuele Mingione
                            (Modulo 1)
                        
                            Daniele Tantari
                            (Modulo 2)
                        
                        Teaching Mode:
                        
                                    In-person learning (entirely or partially) (Modulo 1); 
                                
                                    In-person learning (entirely or partially) (Modulo 2)
                                
                            Campus:
                            Bologna
                        
                            Corso:
                            Second cycle degree programme (LM) in
                            Mathematics (cod. 6730)

                                        Course Timetable
                                    
from Sep 14, 2026 to Dec 14, 2026

Learning outcomes

At the end of the course the student: - Has an in‑depth understanding of advanced concepts in statistical mechanics and their application to complex systems; Is familiar with the use of statistical mechanics methods in the study of disordered Ising models and neural networks; Is capable of developing a rigorous analysis of simple mean‑field models, both from static and dynamical perspectives.

Course contents

Review of probability;
Markov chains, ergodicity and convergence to equilibrium, Metropolis-Hastings, Gibbs measure;
Statistical mechanics formalism;
Neural networks and associative memory: the Hopfield model. Free energy, thermodynamic states, phase diagram;
Wigner-spiked model and SK model, the Replica method;
Factor graphs and belief propagation: application to community detection via the Stochastic Block Model.

Readings/Bibliography

Main references:

M.Mézard, A.Montanari - Information, Physics, and Computation - Oxford University Press, USA
(2009);
Nishimori, H.: Statistical Physics of Spin Glasses and Information processing. An Introduction. Oxford
Science Publications 2001;
Coolen, Kuhn, Sollich, Theory of Neural Information Processing Systems, Oxford University Press;
Talagrand,Michel, Spin glasses. Vol1-2

Furter readings:

Mark Newman - Networks_ An Introduction - Oxford University Press (2010);
Decelle, A., Krzakala, F., Moore, C., & Zdeborová, L. (2011). Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Physical Review E, 84(6), 066106;
Zdeborová, L., Krzakala, F. (2016). Statistical physics of inference: Thresholds and algorithms. Advances in Physics, 65(5), 453-552;

Teaching methods

Lectures.

Assessment methods

Assessment consists of an oral examination designed to verify the acquisition of the knowledge outlined in the course syllabus and to evaluate the skills attained in accordance with the learning objectives, such as:

Advanced concepts in Statistical Mechanics;
The ability to independently explore the most recent developments regarding the aforementioned topics and their most significant issues.

Teaching tools

Tools supporting teaching will be available on the Virtuale platform

Office hours

See the website of Emanuele Mingione

See the website of Daniele Tantari