- Docente: Pierluigi Contucci
- Credits: 6
- SSD: MAT/07
- Language: Italian
- Teaching Mode: In-person learning (entirely or partially)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 6061)
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from Feb 16, 2026 to May 27, 2026
Learning outcomes
At the end of the course, the student has been introduced to the concept of emergent phenomena as a unifying principle of complexity theory. Within this framework, they have studied, as exemplary cases, thermodynamics and phase transitions, information theory and statistical inference, the analysis of discrete dynamics, and the modern theory of machine learning in both low- and high-dimensional settings. The student is able to interpret these tools as descriptions of collective behaviors that cannot be reduced to the simple sum of their parts, and to apply them to the study of complex systems in both the natural and the social sciences.
Course contents
Introduction. What is an emergent phenomenon.
Description of systems composed of many variables. Collective states, averages, and fluctuations. The role of probability.
Thermodynamics and phase transitions
Thermodynamic systems.
The First and Second Laws of Thermodynamics.
Entropy.
Thermodynamic potentials and equilibrium.
Phase transitions as the appearance of collective order.
Entropy, information, and inference
Relation between thermodynamic entropy and informational entropy.
The maximum entropy principle and statistical inference.
Boltzmann distributions as probabilistic models.
Measures of information and predictive capability.
Information and communication
Transmission of information in the presence of noise.
Error-correcting codes.
Machine learning and energy-based models
Principles of probabilistic machine learning.
Learning in low- and high-dimensional settings.
Energy-based models and neural networks.
Boltzmann machines as learning statistical systems.
Readings/Bibliography
Notes.
Thermodynamics, Enrico Fermi
The Mathematical Theory of Communication, C. E. Shannon, W. Weaver
Teaching methods
Lectures including theory, exercises, and computer simulations.
Assessment methods
Oral exam
Teaching tools
Some lectures will be accompanied by computer simulations related to the topics covered.
Office hours
See the website of Pierluigi Contucci