- Docente: Pierluigi Contucci
- Credits: 6
- SSD: MAT/07
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 8208)
Learning outcomes
At the end of the course the student is familiar with the advanced notions on statistical mechanics topics. He is able to investigate independently the novel topics of the discipline from research papers.
Course contents
- Introduction.
- Probability spaces, Entropy and its property.
- Probability spaces as simplexes.
- Interacting particle systems and Ising models. Ising model in d=1 with free and periodic boundary conditions.
- Dichotomic functions and their Fourier expansion.
- Simple notions for Ising models at d>1
- The thermodynamic limit and correlation inequalities.
- Mean field models. Curie-Weiss model.
- Thermodynamic limit for mean field models.
- The solution of the Curie Weiss model with bounds from above and below.
- Large number theorem and central limit theorem for the magnetization. Free model and model with interaction.
- Large deviation theory and solution of the Curie Weiss model.
- Theoretical inverse problem: from thermodynamic quantities to parameters.
- Phenomenological inverse problem: from real data to thermodynamic quantities.
- Maximum likelihood.
- The Sherrington and Kirkpatrick model. Parisi solution. Guerra Talagrand proof for the free energy. Panchenko proof for ultrametricity.
- Associative memory and the Hopfield model.
- The multi-specie generalization of the SK model.
- Deep-learning architecture with cilindrical and conical boundary conditions. Bounds for the free energy.
- Learning and retrieval equivalence.
- Seminars on recent advances.
Readings/Bibliography
- "Perspectives on Spin Glasses", P.Contucci e C.Giardinà,CUP, 2012
- "Spin Glasses and Complexity (Primers in Complex Systems)" D.Stein and C.Newman, OUP, 2013
- "Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction", S.Friedli and Y.Velenik,CUP, 2017
Teaching methods
Class lectures, examples, and exercises. Computer simulations illustrations. In case international students will be attending the course the lectures will be in english.
Assessment methods
The exam test is of oral type and possibly, if the student is interested, can take the form of a simple research seminar.
Teaching tools
Some lectures will be supported by illustrations of computer simulations or data elaboration.
Office hours
See the website of Pierluigi Contucci