Physics of Complex Systems (Fisica dei Sistemi Complessi)

The student will have to manage advanced concepts like: quantum problems for identical particles, the basics of the path-integral formulation of quantum mechanics, the basics of second quantization.

The course aims to discuss advanced concepts of quantum mechanics and several applications relevant for the future study of condensed matter physics, many-body theory and statistical mechanics.

1. Definition of the Propagator and Formulation of Quantum Mechanics Using Path Integrals 2. Time-Dependent Perturbation Theories 3. Identical Particles in Quantum Mechanics 4. Symmetries in Quantum Mechanics 5. The density matrix operator 6. Fermi and Bose distributions 7. Introduction to second quantization

Modalita di esame: Prova scritta (in aula);
Exam: Written test;
... The exam is written only, the student will be asked to solve exercises. The maximum score of the exam is 30/30, as per the Italian system.

Dirac, P. A. M.; The Principles of Quantum Mechanics (Oxford University Press) Merzbacher, E.; Quantum Mechanics (J. Wiley & Sons, Inc.) Messiah, A.; Quantum Mechanics (Dover) Picasso, L. E.; Lectures in Quantum Mechanics (Springer) d’Emilio, E., Picasso, L. E.; Problems in Quantum Mechanics: with Solutions (Springer) Sakurai, J. J.; Modern Quantum Mechanics (Addison-Wesley)

The knowledge transmitted by the course to students involves: • the formalisms of classical and quantum field theory and of the second quantization • the application to various models and the derivation of their significant properties. • the approach of statistical mechanics to quantum many-body systems • the application of the statistical-mechanics methods to several systems and the derivation of their thermodynamcal properties The transmitted abilities include: • understanding the quantum-field theory in providing an effective description of complex systems of identical particles and solving related problems; • applying quantum statistical mechanics to many-body systems; • applying a correct quantum-field formalism to the study of quantum many-body systems; • understanding how general is the formalism of statistical mechanics and how broad is the range of applicability of the techniques introduced in the course.

The course (first semester, first year) essentially represents the continuation of the BSc-program course ``Advanced methods for Physics and Quantum Physics". The teaching objectives aim to introduce advanced topics in quantum mechanics, the basis of quantum field theory, its application to quantum fluids, and the statistical-physics approach to quantum many-body systems. These will support the subsequent study of condensed-matter and solid-state physics. The course is essentially divided into three sections. The first part is focused on discussing the development of the canonical-ensemble description within the most-probable distribution method. This approach, particularly useful for quantum many-body systems, is then extended to the grand-canonical ensemble. In addition to study quantum systems typically considered in statistical-physics program, several applications concern more recent models and/or effects such as quantum phase transitions, ultracold atoms, and Bose condensates realized and investigated in the last two decades. The second part, is devoted to discuss time-dependent Hamiltonian systems, the interaction picture, the use of coherent-state approach in various applications, and the adiabatic approximation. The third part, in addition to a short introduction to Special Relativity, discusses the fundamental aspects of quantum field theory, and, after introducing both the Lagrangian and Hamiltonian formulation of classical field theory, presents the second quantization of both relativistic and non relativistic field theories. Various models are considered which, together with the electromagnetic field, the Klein Gordon and Dirac models, and the second-quantized nonlinear Schroedinger equation, will include the derivation of both bosonic and fermionic lattice models and the discussion of their properties. The knowledge transmitted by this course provides the background necessary to better approach subsequent first- and second-year courses of advanced Physics (such as Condensed matter theory, Statistical physics and Field theory and critical phenomena).

Statistical mechanics of quantum many-body systems. Canonical-ensemble description within the most-probable distribution method. Extension to the grand-canonical ensemble [6 h]. Paramagnetic materials, Brillouin function and magnetization. Photon gas, black-body radiation and Planck formula. The Debye phonon model: heat capacity and other low-temperature properties [8 h]. Ideal Bose-Einstein gas. The mechanism of the BE condensation. Derivation of the critical properties at the transition point. Fermi-Dirac non-interacting gas: Fermi sphere, state equation and pressure [6 h]. Vortex field theory, derivation of the point-like vortex model. Mean-field approach to the Kosterlitz-Thouless transition [4 h]. The Lagrange formulation of classical field theory, least-action principle and Lagrange equations. The Hamilton formulation of classical field theory, and the quantization of classical fields in the canonical scheme [4 h]. Second quantization of nonrelativistic field theories: microscopic mode operators, commutator and anti-commutator algebra, Fock states. Properties of bosonic and fermionic fields; Bose-Einstein and Fermi-Dirac statistics [6 h]. Quantum field theory of the electromagnetic field and of the Klein-Gordon model. Fermionic field and Dirac equation [8 h]. Weakly interactiong Bose gases: Bogoliubov approach, low-energy spectrum, phonon excitations, vacuum fluctuations. Derivation of the Gross-Pitaevskii equation [4 h]. Lattice quantum models: derivation of the Bose-Hubbard and Fermi-Hubbard models. Superfluid-Mott transition: properties of the superfluid and insulating phases. Perturbative approach and mean-field scheme [8 h]. Time-dependent systems, adiabatic approximation, Berry phase. Coherent-state approach to the Dicke model for the matter-radiation interaction. Fermi golden rule [6h]. Gauge transformations and quantum Mechanics. The case of a uniform magnetic field: Spectrum, Landau gauge, symmetric gauge, applications of symmetries and related wavefunctions [8 h]. Elements of Relativistic Quantum Mechanics. Introduction to Special Relativity. Relativistic aspects of Klein-Gordon and Dirac equations. The non-relativistic limit of the Dirac equation. Spin-orbit interaction. The massless limit of the Dirac equation. Effects of uniform magnetic field and scattering problems in relativistic quantum mechanics [12 h].

Modalita di esame: Prova orale obbligatoria;
Exam: Compulsory oral exam;
... The exam is oral and consists of three questions concerning different topics of the course. Each question corresponds to a score of 10 out of 30 marks. The goal, in general, is to test the knowledge of the course program. These questions aim to evaluate the understanding of students concerning 1) the methodology and the mathematical tools used to quantize classical fields and to develop the ensemble theory for quantum systems, and 2) the significant physical properties and effects characterizing many-body quantum systems described within the statistical physics and the second-quantization formalism. To this end one of the three question is focused on discussing the methods and the mathematical formalism presented in the course, while the other questions are devoted to discuss quantum effects, their derivation, and physical properties of interest emerging from the applications of the theory to quantum-field models and many-body systems.

The PDF file containing the notes relevant to the course lectures are available in the web page of this course accessible through the Teaching Portal of Politecnico. - W. Greiner, J. Reinhardt, Field Quantization, Springer Berlin, 1993 - W. Greiner, Quantum Mechanics, Special Chapters, Springer Berlin, 1998 - Kerson Huang, “Statistical Mechanics�, Wiley 1987; - Frederick Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill - Richard C Tolman, “The Principles of Statistical Mechanics� Courier Corporation, 1938.

To become familiar with techniques for the exact solution of computational problems, including dynamic programming and recursion with emphasis on classical combinatorial optimization problems such as convolution ones, minimum weight paths, spanning trees and matchings and their generalizations. To learn the fundamental concepts of computational complexity theory, the techniques for analyzing the computation time of an algorithm, and some approximation techniques for NP-complete problems. To be able to apply practically such algorithms to complex statistical inference and machine learning problems. To be able to classify new problems into complexity classes.

Mandatory course for the Master in Physics of Complex Systems, 1st year, 2nd term. The course introduces fundamental aspects of computation, including recursive computational tecniques such as dynamic programming and their application to optimization on graphs and networks, and the classification of problems into computational complexity classes. Important aspects of information theory and statistical inference are introduced, with applications to machine learning.

Recursion and dynamic programming (20h) - Introduction to graph theory, Trees and data structures, Algorithms on graphs (20h) - Algorithmic complexity, polynomial reductions and NP-completeness (15h) - Information theory and statistical inference: maximum entropy, maximum likelihood and Boltzmann learning, Belief Propagation and inference on trees, Inference of Trees: Chow-Liu theorem, Expectation Propagation, Hidden Markov models (25h)

E' possibile sostenere l?esame in anticipo rispetto all?acquisizione della frequenza

• Course handouts - Introduction to Algorithms, T.H. Cormen, C.E. Leiserson, R.L. Rivest, MIT Press, 2000. - Elements of the theory of computation, R. Lewis and C. H. Papadimitriou. Prentice-Hall. - Computer and Intractability. A Guide to NP-Completeness. M. R. Garey and D. S. Johnson. Publisher W. H. Freeman, 1979. - Information Theory, Inference, and Learning Algorithms, D. J. C. MacKay, Cambridge University Press, 2003. - The nature of computation. C. Moore, S. Mertens. Oxford University Press, 2011 - Information, Physics and Computation, M. Mezard, A. Montanari, Oxford University Press, 2009. - Biological Sequence Analysis, Durbin, Eddy, Krogh, Mitchison, Cambridge University Press, 1998

The student must acquire some basic elements of molecular biology and must learn to apply the techniques of statistical physics to some problems from the equilibrium and nonequilibrium physics of biological systems, mainly in the field of biopolymers.

This module discusses applications of statistical physics to biological systems, in particular biopolymers. To this end, basic elements of kinetics of phase transitions and molecular biology are also introduced.

Kinetics of phase transitions: thermodynamics of interfaces, metastability and classical nucleation theory, domain coarsening, phase ordering with or without conservation laws (12 hours). Introduction to molecular biology: the cell; small molecules; proteins and nucleic acids. (4 hours). Stretching a single DNA molecule: experiments, the Freely Jointed Chain, the one-dimensional cooperative chain, the worm-like chain (10 hours). DNA melting: experiments, zipper model, Poland-Scheraga model (7 hours). The helix-coil transition. Polymer collapse: Flory's theory. Collapse of semiflexible polymers: lattice models and the tube model. The self-avoiding walk and the O(n) model. (6 hours). An introduction to protein folding and design. RNA folding and secondary structure. Protein and RNA mechanical unfolding (15 hours). Molecular motors (6 hours).

E' possibile sostenere l?esame in anticipo rispetto all?acquisizione della frequenza

K. Sneppen and G. Zocchi, Physics in molecular biology, Cambridge P. Nelson, Biological Physics, Freeman B. Alberts et al, Molecular biology of the cell, Garland Lecture notes and slides will be provided.

By the end of the course, students will be able to identify and classify Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) underlying mathematical models of engineering problems, and they will acquire foundational knowledge of finite difference and finite element methods for solving such equations. In addition, students will develop the skills to design and implement efficient numerical methods for ODE/PDE-based engineering problems using Matlab.

Il corso è tenuto in inglese. Il corso introduce i metodi numerici per la risoluzione di equazioni differenziali ordinarie e alle derivate parziali, affrontando sia gli aspetti teorici sia quelli applicativi. L'obiettivo principale è fornire agli studenti le conoscenze e le competenze pratiche necessarie per eseguire simulazioni numeriche efficienti di problemi classici dell’ingegneria, una competenza essenziale per corsi avanzati e per progetti di laurea magistrale. Un aspetto centrale del corso è quello di fornire agli studenti una solida conoscenza delle metodologie nel calcolo numerico e addestrarli all’uso del software Matlab, affinché siano in grado di risolvere numericamente, con l'ausilio del calcolatore, alcuni problemi che comunemente si presentano nelle applicazioni dell’ingegneria. In particolare, tali conoscenze consentono agli studenti di individuare tra i metodi a loro disposizione quello più efficiente, sia in termini di complessità computazionale che di occupazione di memoria, per la risoluzione di un determinato problema, e di fare un’analisi critica della soluzione ottenuta. Questa capacità è particolarmente importante per gli ingegneri chimici che si troveranno a utilizzare codici commerciali per risolvere problemi complessi, come ad esempio quelli connessi con i processi di trasformazione dei materiali. Infatti, grazie a questo insegnamento, gli studenti acquisiscono le competenze matematiche necessarie per evitare risultati non attendibili o comunque poco accurati, che potrebbero derivare da scelte non adeguate degli algoritmi.

INTRODUCTION (4 hours) Lectures: Generalities on ordinary and partial differential equations. Examples of classical differential problems, in particular with reference to the chemical engineering ones. Stationary problems, eigenvalue problems and evolution problems. Classification of second order equations. Initial-value and boundary-value problems. Strong and weak solutions. FEM FOR LINEAR STATIONARY PROBLEMS (40 hours) Lectures and problem sessions (28 hours): • The Finite Element Method (FEM) for second order linear elliptic problems defined on bounded intervals or on bounded plane domains. • Sobolev spaces and weak formulations of differential problems. Integration by parts and Green’s formulas. • Problem space domain discretization. Finite elements and finite element spaces. Definitions and examples. Some main results on approximation theory. • Application of the FEM to linear elliptic problems, first defined on bounded intervals, and then on bounded plane domains, with different types of boundary conditions (Dirichlet, Neumann, Robin, mixed-type, of periodicity). • Some basic convergence results. • Eigenvalue problems. • Construction of the FEM stiffness matrix and solution of the associated linear system. • Numerical integration formulas. Conditioning of a linear system. Basic methods for the numerical solution of linear systems. Computer Lab sessions (12 hours) FEM FOR LINEAR TIME-DEPENDENT PROBLEMS (16 hours) Lectures and problem sessions (10 hours): • Weak formulations in the space domain of parabolic and hyperbolic problems, in particular of transport and wave propagation problems. Application of the FEM. • Basic on initial value ODE problems. • Finite difference schemes for the time discretization of ODE systems generated after performing the (space) FEM discretization. Computer Lab sessions (6 hours) NUMERICAL METHODS FOR ODE INITIAL-VALUE PROBLEMS (IVP) (10 hours) Lectures and problem sessions (7 hours): • One-step methods and convergence results for the solution of IVP-ODEs. • Stiff problems and numerical methods for their resolution. Computer Lab sessions (3 hours) FINITE DIFFERENCE METHODS FOR PDE BOUNDARY-VALUE PROBLEMS (BVP) (10 hours) Lectures and problem sessions (7 hours): • Finite difference methods for elliptic, parabolic and hyperbolic BVP-PDEs. • Stability and convergence results. • Reaction-diffusion-transport problems and numerical stabilization for transport dominating ones. Computer Lab sessions (3 hours)

Modalita di esame: Test informatizzato in laboratorio; Prova scritta (in aula);
Exam: Computer lab-based test; Written test;
... The exam consists of two tests: the first is an computer test, the second is a written test. The first test, to be held in the computer Lab, consists of 4 multiple-choice questions concerning the course topics. In this test students must show to be able to solve, by applying the numerical methods described in the lectures and by preparing short Matlab scripts, problems similar to those that have been solved during the Lab course sections. This to verify the students’ knowledge of the algorithms presented during the course and their programming and computing skills. Students must use the Matlab software installed in the Lab PC, and they are not allowed to use books, notes, personal PC and any electronic device. The maximum grade is 16 points: each correct answer is worth 4 points, an unanswered question is worth 0 points, and an incorrect answer results in a 20% penalty. The Lab test is scheduled for about an hour and a half. The second written test, to be held in a lecture room the same day of the Matlab test, consists of two/three exercises that include both practical and theoretical questions, covering the entire course content. The written test is scheduled for about an hour and a half and is worth up to 16 points. During the exam, it is forbidden to use books, personal material such as notes and handouts and any electronic device. The final grade is the total of the two test scores. A minimum score of 18 is required to pass, while full marks with all correct answers are awarded with honors (cum laude).

Lecture notes, as well as notes on the computer lab sessions, can be downloaded by the students from the Portale della Didattica. Further reading: A. Quarteroni, Numerical Models for Differential Problems, Springer Verlag, 2009. A. Quarteroni, F. Saleri, P. Gervasio, Scientific Computing with MATLAB and OCTAVE, Springer, 2014.

Introduction to data science After this course, students will Understand the fundamental ingredients of a data science problems Be able to apply the basics of Bayesian inference problems to inference problems Know the basics of random matrix theory, including the Wigner and Wishart ensembles as well as their spectra Know basic unsupervised learning techniques Know the basics of neural network architectures and neural learning dynamics in supervised learning. Advanced numerical methods The student will be able to understand the basic concepts of statistical mechanics, and will be capable of executing molecular dynamics simulations in different ensembles.

The course consists of two modules Introduction to data science [4 credits] This short course will give an introduction to data science for physicists. The course will introduce the basics of data science problems, and introduce students to Bayesian techniques for inference, unsupervised learning methods for exploratory data analysis, and neural networks. On the theoretical side, students will learn fundamentals of random matrix theory and of the theory of neural networks. Advanced numerical methods [4 credits] Introduction to numerical methods applied to molecular dynamics simulations. The course covers the theoretical foundations of molecular dynamics and Monte Carlo simulations, starting from fundamental concepts from Newtonian dynamics, statistical mechanics, and variational principles. The course includes practical sessions where students implementing and applying the algorithms.

Introduction to data science I Basics of Bayesian inference [R. Trotta] Introduction to inference: what is it? Why do we need it? Frequentist probability vs bayesian probability Confidence levels, posterior distributions, priors and the difference between all those If time allows: classical hypothesis testing vs Bayesian model comparison II An introduction to random matrix theory [J. Barbier] The Wigner ensemble and the semi-circular law Wishart Ensemble and Marchenko–Pastur law Spiked matrix models and the BBP transition III Introduction to unsupervised learning and dimensional reduction [A. Laio] Principal Component Analysis Multidimensional scaling and kernel methods Intrinsic dimension estimates IV Neural networks [S. Goldt] Neural networks 101: types and applications + some open theoretical problems Learning dynamics: empirical phenomena and theoretical predictions The impact of data structure Unsupervised learning: from Hebbian learning to independent components Advanced numerical methods (G.Bussi and C. Micheletti) Basic concepts of Newtonian dynamics and Statistical Mechanics: energy conservation, time reversibility and phase-space incompressibility, Liouville Theorem, Ergodicity. Derivation of equilibrium statistical mechanics from variational principles (definition of entropy, ideal gas, microcanonical, canonical and grandcanonical statistical ensembles, law of mass acrtion) [10h] Integration schemes for molecular dynamics: Verlet, Trotter splitting, Velocity Verlet. Dependence of the results on the time step. [10h] Sampling the canonical ensemble with Monte Carlo: Metropolis-Hastings rule, balance and detailed balance, hybrid Monte Carlo. [10h] Sampling the canonical ensemble with molecular dynamics: velocity rescaling, Berendsen thermostat, Andersen thermostat, Langevin dynamics, stochastic velocity rescaling. Stochastic equations: Itoh rule, Fokker-Planck equation. [10h]

Modalita di esame: Prova scritta (in aula); Prova orale obbligatoria;
Exam: Written test; Compulsory oral exam;
... Introduction to data science The course will be assessed by a written exam of two hours, covering all four topics of the course. Advanced numerical methods Oral exam.

Introduction to data science As an introduction to the course, chapters 1-3, 20-21, 38-39, 41, and 44 of the book “Information Theory, Inference, and Learning Algorithms� by David MacKay are recommended reading. Advanced numerical methods Kerson Huang; Statistical mechanics Julia Yeomans; Statistical mechanics of Phase transitions Frenkel – Smith; Understanding Molecular Simulation: From Algorithms to Applications Tuckerman; Statistical Mechanics: Theory and Molecular Simulation

A) Introduction to Quantum Mechanics At the end of the course, the student is expected to be able to manage concepts like Hilbert spaces, solutions of Schrödinger equation with various methods, including approximate ones. B) Introduction to systems and computational neuroscience Students will be familiar with the core ideas in the neuroscience of vision and touch, and in the Bayesian theory of perception. They will be able to navigate and interpret the modern research literature on vision, touch, and Bayesian models of perception

The course consists of two modules A) Introduction to Quantum Mechanics The module will cover standard topics for an introduction to Quantum Mechanics. Starting from the definitions of states, operators and observables, and the relative operations between them, the student will be introduced to the Schrödinger equations and to the most important systems in one- and three-dimensions. The course will be concluded with an introduction to various approximate methods and the study of particles in a magnetic field. B) Introduction to systems and computational neuroscience The module covers a range of fundamental ideas in systems and computational neuroscience, with a particular focus on sensory perception.

A) Introduction to Quantum Mechanics [A. Rosa] Topics are divided in two groups. The first group contains very basic concepts, that the student will read from the notes, while frontal lectures will mainly focus on applications and exercises. The second group will be more in the form of “traditional� frontal lectures, with theory and exercises presented back-to-back. GROUP 1 1.1) The Rise of Quantum Mechanics • The Black Body Problem • The Hypothesis of Photons and the Photoelectric Effect • Optical Spectra of Atomic Elements • The Quantization of Angular Momentum: the Stern-Gerlach Experiment 1.2) Mathematical Foundations: Vector Spaces • bra|’s and ket’s: Definitions • Operators in Quantum Mechanics 1.3) Hermitian Operators • Definitions and Results • Eigenvectors and Eigenvalues 1.4) Physical Observable • Hermitian Operators and Physical Observables, Commuting and Non-commuting Observables • Measuring Observables and Quantum averages: A physical Interpretation 1.5) Representations in Quantum Mechanics • Representations of States and Operators • Representations and Probabilities in QM • Change of Representations 1.6) Eigenvectors and Eigenvalues of the Momentum; Uncertainty Principle 1.7) Schrödinger Equation, Conserved Quantities and Stationary States 1.8) Time Evolution in Quantum Mechanics GROUP 2 2.1) One-Dimensional Problems - A recap 2.2) Algebraic methods: harmonic oscillator and angular momentum 2.3) Schrödinger Equation in Three Dimensions; Central Potentials 2.4) Approximate methods I - Variational methods 2.5) Approximate methods II - Perturbative methods 2.6) Approximate methods III - Semi-classical (WKB) approximation 2.7) Motion of particles in a magnetic field B) Introduction to systems and computational neuroscience [D. Zoccolan, M. Diamond, E. Piasini] Part 1. Physiology and functions of the mammalian visual system Introduction to anatomy and physiology of the visual system A systems/computational approach to the study of the visual system; Anatomy of the visual system Classic findings about physiology of lower-level visual areas Data analysis approaches in Systems Neuroscience Classic findings about physiology of higher-level visual areas Descriptive models of visual neurons How to build models of visual neuronal responses (i.e., stimulus/response maps) Mechanistic models of the visual system Inferring the mechanisms underlying the response properties of visual neurons Functional models of the visual system Understanding neuronal population codes Part 2. Sensory Systems: Tactile Perception 1. Introduction to the study of the cerebral cortex 2. Sensory maps in the cerebral cortex 3. Transduction 4. Somatosensory system and pain 5. Methods for computational neuroscience of perception 6. Encoding and decoding 7. Perceptual memory 8. Neuroscience of perceptual knowledge Part 3: Bayesian modeling of perception Perception as Bayesian inference Bayesian inference under sensory noise Cue combination and evidence accumulation Discrimination, detection and classification

Modalita di esame: Prova scritta (in aula);
Exam: Written test;
... A) Introduction to Quantum Mechanics The exam is written only, the student will be asked to solve exercises. The maximum score of the exam is 30/30, as per the Italian system. B) Introduction to systems and computational neuroscience Written exam.

A) Introduction to Quantum Mechanics Merzbacher, E.; Quantum Mechanics (J. Wiley & Sons, Inc.) Messiah, A.; Quantum Mechanics (Dover) Picasso, L. E.; Lectures in Quantum Mechanics (Springer) d’Emilio, E., Picasso, L. E.; Problems in Quantum Mechanics: with Solutions (Springer) (6) Sakurai, J. J.; Modern Quantum Mechanics (Addison-Wesley) B) Introduction to systems and computational neuroscience No additional reading material is necessary beyond what is provided during lessons. However some of the teachers’ material comes from these sources and students are invited to seek additional background there: · Dayan, P. & Abbott, L. F. Theoretical Neuroscience. (MIT Press, 2001). · Martin, A. R., Brown, D. A., Diamond, M. E., Cattaneo, A. & De-Miguel, F. F. From Neuron to Brain, Sixth Edition. (Oxford University Press, 2021). · Rieke, F., Warland, D. & Bialek, W. Spikes: exploring the neural code. (The MIT Press, 1999). · Wichmann, F. A. & Hill, N. J. The psychometric function: I. Fitting, sampling, and goodness of fit. Perception & Psychophysics 63, 1293-1313, doi:10.3758/BF03194544 (2001). · Ma, Koerding and Goldreich (2022). Bayesian Models of Perception and Action (https://www.cns.nyu.edu/malab/bayesianbook.html) · Richard McElreath (2nd ed 2020). Statistical Rethinking. (https://xcelab.net/rm/ )

A) Introduction to Quantum Mechanics At the end of the course, the student is expected to be able to manage concepts like Hilbert spaces, solutions of Schrödinger equation with various methods, including approximate ones. B) Ecology and evolution At the end of the first part of the module, the students are expected to be able to quantitatively model community dynamics in the presence of different interaction types, identifying and justifying the important assumptions. They should also be able to perform stability analysis, identify the presence of bifurcations and have basic notions of limit cycles and chaos. At the end of the second part of the module, the student will know how to quantify the relative importance of the fundamental mechanisms of evolution (reproduction, mutation, selection, genetic drift and recombination) using the tools of stochastic processes.

The course consists of two modules A) Introduction to Quantum Mechanics The module will cover standard topics for an introduction to Quantum Mechanics. Starting from the definitions of states, operators and observables, and the relative operations between them, the student will be introduced to the Schrödinger equations and to the most important systems in one- and three-dimensions. The course will be concluded with an introduction to various approximate methods and the study of particles in a magnetic field. B) Ecology and evolution The goals of this module are both to teach students fundamental concepts in ecology and evolution and provide them with basic notions and tools in dynamical systems and stochastic processes. The module will be divided into two main parts. The first part will be devoted to introducing the students with classic models in population and community dynamics. The second part will focus on evolutionary theory and population genetics. We will discuss classic and modern experimental evidence of Darwinian (micro)evolution and the observational pieces of evidence of macroevolution. The students will study the effect of selection, mutations, and drift on the fixation of neutral, beneficial, and deleterious mutations in the context of simple stochastic models of population genetics.

A) Introduction to Quantum Mechanics [A. Rosa] Topics are divided in two groups. The first group contains very basic concepts, that the student will read from the notes, while frontal lectures will mainly focus on applications and exercises. The second group will be more in the form of “traditional� frontal lectures, with theory and exercises presented back-to-back. GROUP 1 1.1) The Rise of Quantum Mechanics • The Black Body Problem • The Hypothesis of Photons and the Photoelectric Effect • Optical Spectra of Atomic Elements • The Quantization of Angular Momentum: the Stern-Gerlach Experiment 1.2) Mathematical Foundations: Vector Spaces • bra|’s and ket’s: Definitions • Operators in Quantum Mechanics 1.3) Hermitian Operators • Definitions and Results • Eigenvectors and Eigenvalues 1.4) Physical Observable • Hermitian Operators and Physical Observables, Commuting and Non-commuting Observables • Measuring Observables and Quantum averages: A physical Interpretation 1.5) Representations in Quantum Mechanics • Representations of States and Operators • Representations and Probabilities in QM • Change of Representations 1.6) Eigenvectors and Eigenvalues of the Momentum; Uncertainty Principle 1.7) Schrödinger Equation, Conserved Quantities and Stationary States 1.8) Time Evolution in Quantum Mechanics GROUP 2 2.1) One-Dimensional Problems - A recap 2.2) Algebraic methods: harmonic oscillator and angular momentum 2.3) Schrödinger Equation in Three Dimensions; Central Potentials 2.4) Approximate methods I - Variational methods 2.5) Approximate methods II - Perturbative methods 2.6) Approximate methods III - Semi-classical (WKB) approximation 2.7) Motion of particles in a magnetic field B) Ecology and evolution [J. Grilli] 1. Single population dynamics (exponential growth, logistic growth, Allee effect) 2. Two species dynamics. Fixed points and stability. Lotka-Volterra 3. Multispecies communities 4. Experimental and observational evidence of evolution 5. Genetic Drift 6. Mutation and Selection 7. Coevolution

Modalita di esame: Prova scritta (in aula); Prova orale obbligatoria;
Exam: Written test; Compulsory oral exam;
... A) Introduction to Quantum Mechanics The exam is written only, the student will be asked to solve exercises. The maximum score of the exam is 30/30, as per the Italian system. B) Ecology and evolution The exam is only oral. The students will be assigned homework that will be discussed during the oral exam.

A) Introduction to Quantum Mechanics Merzbacher, E.; Quantum Mechanics (J. Wiley & Sons, Inc.) Messiah, A.; Quantum Mechanics (Dover) Picasso, L. E.; Lectures in Quantum Mechanics (Springer) d’Emilio, E., Picasso, L. E.; Problems in Quantum Mechanics: with Solutions (Springer) (6) Sakurai, J. J.; Modern Quantum Mechanics (Addison-Wesley) B) Ecology and evolution Murray, Mathematical Biology vol I Strogatz, Non-linear dynamics and Chaos

The knowledge transmitted by the course to students involves: - Physical and chemical properties of functional materials involved in micro and nanotechnologies. - Characterization methods devoted to materials, micro/nanostructures and technological processes

The course is taught in English. Aim of the course is to provide the theoretical and the experimental tools concerning materials and characterization techniques involved in micro and nanotechnological processes. The role of this course is central for the development of the professional profile concerning Physics of Complex Systems, as it develops several skills required for learning the other courses in Master's Degree. The course is divided into two parts: in the first, the fundamental aspects of the physical and chemical properties of functional materials involved in micro and nanotechnological processes. In the second part, the student learns basic notions about the main characterization techniques dealing with materials (homogeneous and nanostructured) and synthesis processes.

Materials (3 credits) - Fundamentals of materials bonding and structure; materials properties inferred from chemical bonding; crystal defects. - Metals and Ceramics: Thermo-/Ferro-/Pyro-electrical properties, Piezoelectricity, Piezoresistivity - Materials for harsh environments - Shape memory materials - Nanostructured materials - Polymers Characterizations (2,5 credits) - Introduction to Materials Characterization: Composition, Structure and Morphology through fundamental interactions (photons-matter, electrons-matter, ions/particles-matter) - Optical microscopy (conventional [wide field], confocal/laser scanning) - Electron microscopies (SEM and TEM with Electron Probe Microanalysis) and e-beam lithography - Ionic microscopy (Focus Ion Beam) and related lithography - Optical Spectroscopy: Photodetectors/Optical Spectrometers, Reflectance/Transmittance spectroscopy, Photoluminescence spectroscopy, Raman spectroscopy Experimental Lab (0,5 credits) - Experimental demonstration of electron microscopies

Modalita di esame: Prova scritta (in aula);
Exam: Written test;
... The exam concerns a written test dealing with open questions on the developed theory and the experiments performed within the laboratory demonstrations, with a scheduled time of 1 h 30 min. Laptop, smarphones, book, pocket calculator, slides, notes are not allowed.

Selected chapters from the following texts: - Materials Science and Engineering: An Introduction, by William D. Callister, Ed. Wiley - Foundations of Materials Science and Engineering, by William F. Smith and Javad Hashem, Ed. McGraw-Hill - Solid State Chemistry and its Applications, by Anthony R. West, Ed. Wiley - Structural and chemical analysis of materials, by J. P. Eberhart, Ed. Wiley - Surfaces and interfaces of solid materials, by H. Luth, Ed. Springer - A guide to scanning microscope observation, by JEOL Inc. - Optical processes in semiconductors, by J. I. Pankove, Ed. Dover Publ. Inc. - Optical diagnostics for thin film processing, by I. P. Herman, Ed. Academic Press - Fundamentals of photonics, by B.E.A. Saleh and M. C. Teich, Ed. Wiley Learning material provided by teachers

With this course, students will acquire both theoretical knowledge and computational skills to develop agent-based models and analyze collective phenomena emerging in real-world systems involving rational and bounded-rational agents, with applications ranging from biological systems to socio-economic ones.

In recent decades, ideas and methods from statistical physics have become valuable tools for understanding self-organizing collective phenomena and the macroscopic statistical patterns emerging in large-scale socio-economic and biological systems composed of interacting agents. This course is designed to introduce students to the modeling of agent-based systems and their applications across interdisciplinary fields. A central focus is on examining how simple interactions between individual agents—using frameworks like game theory and basic behavioral rules—can give rise to non-trivial collective behaviors, including market trends, opinion polarization, social segregation, and animal flocking. The course will explore how micro-level interactions link to macro-level outcomes, providing students with a robust understanding of these processes.

1. Foundations of Individual Decision-Making (15h) - Preferences and choice; utility function representation - Decisions under risk: lotteries and Expected Utility Theory - Sequential decision problems, separability and information processing - Optimal Planning and its relation to Optimal Control, - Bellman equation and Dynamic Programming - Markov Decision Processes - Introduction to Reinforcement Learning 2. Multi-Agent Interactions (25h) - Normal-form games: pure strategies, utility function and payoff tables strategic dominance, best-response and Nash equilibria, mixed strategies, Nash theorem; special classes of games - Extensive form and repeated games: game tree, information sets, backward induction, equilibrium refinements, and Folks theorems - Bounded rationality, social learning and herding behavior - Population dynamics and evolutionary game theory 3. Agent-Based Modeling and Computational Applications (20h) Illustrative case studies carried out using computer simulations include: - Evolution of competition, cooperation and mutualism - Opinion dynamics, the emergence of consensus and social norms - Collective motion in swarms and flocks - Spatial segregation and pedestrian dynamics - Modeling financial markets: price formation, bubbles and crashes

E' possibile sostenere l?esame in anticipo rispetto all?acquisizione della frequenza

• Lecture notes and reading material provided by the teacher. - D.P. Bertsekas, “Dynamic Programming and Optimal Control, Vol. I�, Athena Scientific, 1995. - K. Leyton-Brown and Y. Shoham, “Essentials of Game Theory: A concise, multidisciplinary Introduction�, Morgan&Calypool Pubs, 2008. - A. Mas-Colell, M.D. Whinston, and J.R. Green, “Microeconomic theory, volume 1�. Oxford University Press New York, 1995. - M. Maschler, E. Solan, S. Zamir, “Game Theory�, Cambridge Univ. Press, 2013. - M. Broom, J. Rychtar, “Game-theoretical models in biology�, Chapman & Hall – CRC, 2013. - D. Easley and J. Kleinberg, “Networks, Crowds, and Markets�, Cambridge Univ. Press, 2010. - F. Slanina, “Essentials of Econophysics Modelling�, Oxford Univ. Press, 2014. - J.-P. Bouchaud, “Crises and collective socio-economic phenomena: simple models and challenges�, J. Stat. Physics, 2013. - C. Castellano, S. Fortunato, V. Loreto, “Statistical Physics of Social Dynamics�, Review of Modern Physics, 2009. - T. Vicsek, A. Zafeiris, “Collective Motion�, Physics Reports, 2012. - T. Lux, “Applications of Statistical Physics to Finance and Economics�, Handbook of research and complexity, 2009. - M. Jusup et al., “Social Physics�, Physics Reports, 2022. - R. Axtell, J.D. Farmer, “Agent-Based Modeling in Economics and Finance: Past, Present and Future�, J. Economic Literature, 2025.

Students are expected to be able to formalize problems in modeling and inference of complex systems and apply the suitable tools for their analysis.

The course aims at providing a conceptual framework, based on a modern approach to probability theory, that provides a unifying language for different branches of theoretical sciences, including information theory, coding theory, statistical inference and statistical mechanics. The course also aims at providing a basic working knowledge of these subjects, by developing analytical and computational skills.

Definitions of probability, Kolmogorov axioms, de Finetti and Jaynes (4 hours) Stochastic independence, conditional probability, Bayes theorem and inference (4 hours) Random variables (4 hours) Classical probability: Urn models, balls and boxes, random walks (6 hours) Generating functions: Integer random variables, branching process (6 hours) Borel-Cantelli lemmas. Laws of large numbers. Limits in probability. (4 hours) Limit laws for sums of independent random variables. (6 hours) Limit theorems for extremes: The Random Energy Model. (6 hours) Information, Shannon theorem and the Asymptotic Equipartition Property. (4 hours) Mutual and relative information. (6 hours) Large deviations: thin tails and fat tails (8 hours) Distributions of maximal entropy, generalised thermodynamics (6 hours) Examples of correlated variables: Phase transitions (8 hours) Information theory, statistics and Bayesian inference (8 hours)

Modalita di esame: Prova scritta (in aula); Prova orale obbligatoria;
Exam: Written test; Compulsory oral exam;
... Assessment is based on a mid-term written test, on exercises that cover the first part and a final oral exam.

W. Feller, An Introduction to Probability Theory and its Applications (J.Wiley & Sons 1968). Cover and Thomas, Elements of Information Theory (J. Wiley & Sons 2006). E. T. Jaynes, Probability Theory: the logic of science, (Cambridge U. Press 2003). M. Mezard, A. Montanari, Information, Physics and Computation (Oxford Univ. Press 2009). C.W. Gardiner, Handbook of stochastic methods (Springer-Verlag, 1985).

A) Ecology and evolution At the end of the first part of the module, the students are expected to be able to quantitatively model community dynamics in the presence of different interaction types, identifying and justifying the important assumptions. They should also be able to perform stability analysis, identify the presence of bifurcations and have basic notions of limit cycles and chaos. At the end of the second part of the module, the student will know how to quantify the relative importance of the fundamental mechanisms of evolution (reproduction, mutation, selection, genetic drift and recombination) using the tools of stochastic processes. B) Introduction to systems and computational neuroscience Students will be familiar with the core ideas in the neuroscience of vision and touch, and in the Bayesian theory of perception. They will be able to navigate and interpret the modern research literature on vision, touch, and Bayesian models of perception.

The course consists of two modules A) Ecology and evolution The goals of this module are both to teach students fundamental concepts in ecology and evolution and provide them with basic notions and tools in dynamical systems and stochastic processes. The module will be divided into two main parts. The first part will be devoted to introducing the students with classic models in population and community dynamics. The second part of the module will focus on evolutionary theory and population genetics. We will discuss classic and modern experimental evidence of Darwinian (micro)evolution and the observational pieces of evidence of macroevolution. The students will study the effect of selection, mutations, and drift on the fixation of neutral, beneficial, and deleterious mutations in the context of simple stochastic models of population genetics. B) Introduction to systems and computational neuroscience The module covers a range of fundamental ideas in systems and computational neuroscience, with a particular focus on sensory perception.

A) Ecology and evolution [J.Grilli] 1. Single population dynamics (exponential growth, logistic growth, Allee effect) 2. Two species dynamics. Fixed points and stability. Lotka-Volterra 3. Multispecies communities 4. Experimental and observational evidence of evolution 5. Genetic Drift 6. Mutation and Selection 7. Coevolution B) Introduction to systems and computational neuroscience [D. Zoccolan, M. Diamond, E. Piasini] Part 1. Physiology and functions of the mammalian visual system Introduction to anatomy and physiology of the visual system A systems/computational approach to the study of the visual system; Anatomy of the visual system Classic findings about physiology of lower-level visual areas Data analysis approaches in Systems Neuroscience Classic findings about physiology of higher-level visual areas Descriptive models of visual neurons How to build models of visual neuronal responses (i.e., stimulus/response maps) Mechanistic models of the visual system Inferring the mechanisms underlying the response properties of visual neurons Functional models of the visual system Understanding neuronal population codes Part 2. Sensory Systems: Tactile Perception 1. Introduction to the study of the cerebral cortex 2. Sensory maps in the cerebral cortex 3. Transduction 4. Somatosensory system and pain 5. Methods for computational neuroscience of perception 6. Encoding and decoding 7. Perceptual memory 8. Neuroscience of perceptual knowledge Part 3: Bayesian modeling of perception Perception as Bayesian inference Bayesian inference under sensory noise Cue combination and evidence accumulation Discrimination, detection and classification

Modalita di esame: Prova scritta (in aula); Prova orale obbligatoria;
Exam: Written test; Compulsory oral exam;
... A) Ecology and evolution The exam is only oral. The students will be assigned homework that will be discussed during the oral exam. B) Introduction to systems and computational neuroscience Written exam.

A) Ecology and evolution Murray, Mathematical Biology vol I Strogatz, Non-linear dynamics and Chaos B) Introduction to systems and computational neuroscience No additional reading material is necessary beyond what is provided during lessons. However some of the teachers’ material comes from these sources and students are invited to seek additional background there: · Dayan, P. & Abbott, L. F. Theoretical Neuroscience. (MIT Press, 2001). · Martin, A. R., Brown, D. A., Diamond, M. E., Cattaneo, A. & De-Miguel, F. F. From Neuron to Brain, Sixth Edition. (Oxford University Press, 2021). · Rieke, F., Warland, D. & Bialek, W. Spikes: exploring the neural code. (The MIT Press, 1999). · Wichmann, F. A. & Hill, N. J. The psychometric function: I. Fitting, sampling, and goodness of fit. Perception & Psychophysics 63, 1293-1313, doi:10.3758/BF03194544 (2001). · Ma, Koerding and Goldreich (2022). Bayesian Models of Perception and Action (https://www.cns.nyu.edu/malab/bayesianbook.html) · Richard McElreath (2nd ed 2020). Statistical Rethinking. (https://xcelab.net/rm/ )

Knowledge of the microscopic mechanisms, models and tools for describing the behavior of interacting ions, electrons and excitations in solids. Ability to apply the knowledges to the theoretical and numerical study of reference systems in condensed matter physics, from solids to ultra cold atomic gases.

The course provides theoretical tools for understanding the properties of aggregate quantum systems with a huge number of particles. In particular, the crystalline solid, characterized by the ion lattice and the electronic degrees of freedom, is described at a microscopic level, starting from the Schroedinger equation. The low temperature ordered phases, the response to external perturbations, and the transport and optical properties are investigated, with reference to more recent applications, such as high-Tc superconductivity, quantum Hall effect, nano and low dimensional physics, and atomic physics. An introduction to numerical techinques appropriate for their investigation is also provided.

1. From condensed matter to the crystalline solid. The fundamental Hamiltonian of a solid in first quantization; the Born Oppenheimer approximation. The crystalline solid; Bravais and reciprocal lattice. 2. Review of basic concepts of quantum statistical physics. Second quantization, density matrix, grand canonical ensamble, chemical potential. Free Fermi and Bose gases. 3. Single electron approximation. The Sommerfeld model; specific heat and effective mass. Bloch theorem; bands and Fermi surface; weak potential and tight binding approximations; graphene bands. 4. Lattice dynamics. The dynamical matrix; phonons; optical and acoustic modes; the Debye model and specific heat. 5. Electron-phonon interaction. The electron-phonon Froelich Hamiltonian; polarons; the Holstein model; second order processes and effective electronic Hamiltonian. 6. Transport properties: Drude conductivity, thermal conductivity and Wiedemann- Franz law. Classical and Quantum Hall effect. 7. Optical properties. Macroscopic formulation of electrodynamics in dispersive media: complex refraction index, absorption coefficient and dissipated power. Microscopic formulation: interaction of electrons with electromagnetic radiation; 8. Electron-electron interaction in momentum space. The Hartree-Fock approximation; direct and exchange interaction. The jellium model and ferromagneticm. Screening and Thomas Fermi semiclassical theory. Density Functional Theory. 9. Introduction to Fermi and Luttinger liquids. 10. Conventional superconductors. The Cooper instability. BCS microscopic theory. The gap equation. 11. Electron-electron interaction in Wannier basis. Ferromagnetism and the Heisenberg Hamiltonian. Mott insulator and the Hubbard model. Quantum phase transitions and mean field phase diagram. 12. Phenomenology and modelisation of high-Tc superconductors. 13. Numerical simulations: the Density Matrix Renormalization Group (DMRG) method 14. The concept of Nanostructures. K-dot-p theory, envelope function, quantum wells, wires and dots.

Modalita di esame: Prova scritta (in aula); Prova orale obbligatoria; Elaborato progettuale in gruppo;
Exam: Written test; Compulsory oral exam; Group project;
... The final mark will be obtained from the three following compulsory steps: -) Written exam (exercises and questions on the topics of the course): up to 20 points; -) Oral exam (discussion of the written exam and presentation of a topic chosen by the student): up to 8 points; -) Numerical project in groups: up to 4 points.

H. Bruus, and K. Flensberg, Introduction to many body quantum theory in condensed matter physics, 2002 C. Di Castro, R. Raimondi, Statistical mechanics and applications in condensed matter, Cambridge University Press, 2015 R.P. Feynman, Statistical mechanics: a set of lectures, Benjamin Cummings Publishing Company, 1972 N.W. Ashcroft, N.D. Mermin, Solid State Physics, Hartcourt Courtrige Pubiher, 1976 G. Grosso and G. Pastori Parravicini, Solid State Physics, Academic Press, 2000 P. Fazekas, Lecture Notes on Electron Correlation and Magnetism, World Scientific, 1999 U. Roessler, Solid state theory: an introduction, Physica Verlag, 2009 J. Solyom, Fundamentals of the physics of solids, vols 1,2,3, Springer, 2007-2010 A. Montorsi, Notes of the course, 2021

1. Knowledge of the basics of statistical learning. 2. Knowledge of the theory and methods of classification for continuous and discrete data using maximum likelihood and Bayesian methods. 3. Knowledge of dimensionality reduction techniques and kernel methods. 4. Knowledge of neural networks: training/testing and applications. 5. Knowledge of tracking filters. 6. Ability to use Python programming frameworks, as well as the TensorFlow environment.

The medium of instruction is English. The course introduces statistical and machine learning methods for processing data and signals in order to extract information from them. Such methods are based on models that are typically learned from training data. In particular, the course covers statistical methods, where the data are modelled as a set of measured (or acquired) samples of a random process, and neural networks, where the model is a suitable combination of basic units (neurons) and the model parameters are found through numerical optimization. The first part of the course deals with statistical learning in maximum likelihood and Bayesian sense. The techniques presented during the course are applied to classification problems in several domains of ICT, including classification of continuous as well as discrete data, and dimensionality reduction using principal component analysis and kernel methods. In the second part of the course, feedforward and convolutional neural networks are introduced. Finally, statistical tracking filters are introduced. Throughout the course, the concepts above are illustrated with practical examples taken from ICT applications. A large part of the course takes place in the LAIB laboratories, where students implement and assess the methods discussed during the lectures. The lab activities are performed using the Python language, and possibly employing cloud-based environments such as Google CoLab. This course is designed jointly with the course “ICT for health�, with the objective to provide students with a coordinated “machine learning� approach that can be applied to several ICT problems; in particular, this course deals primarily with machine learning basics, classification and neural networks, while the “ICT for health� course addresses regression and clustering topics.

Discrete-time random signals and processes, probability distributions, model fitting (1.2CFU), including: • Introduction to machine learning; supervised vs. unsupervised learning, regression vs. classification, k-NN classifier, the curse of dimensionality, overfitting. • Review of probability, continuous vs. discrete random variables, Bayes’ rule, independence, common probability distributions, Gaussian multivariate distribution. Foundations of estimation theory (1.2CFU), including: • Maximum likelihood estimation • Bayesian and MAP estimation • Application to Gaussian processes Classification and latent linear models (0.6CFU), including: • Hypothesis testing • Bayesian decision theory and Bayesian classifiers • Performance assessment of classifiers • Principal component analysis and kernel methods Feedforward and convolutional deep neural networks (2.4CFU), including: • Perceptrons and architecture of a neural network • Backpropagation algorithm • Loss functions, overfitting and regularization • Convolutional networks and deep learning • Neural networks for classification, segmentation and object detection • Neural networks quantization • Transformers and generative adversarial networks Tracking filters (0.6CFU), including • Kalman filter

Modalita di esame: Prova orale obbligatoria;
Exam: Compulsory oral exam;
... The exam aims at assessing the knowledge and understanding of the topics described during the course, and the ability to critically discuss such topics. The exam is oral, and it may also use the report containing the description of the experiments carried out during the computer labs, including the obtained results. The exam typically starts from a discussion of one or more experiments done during the computer labs, and can extend to span the whole set of topics presented during the course. A lot of emphasis is given to the theory underlying each of the experiments. Possible topics addressed during the oral examination are: - a description of the considered system (or algorithm) and of its simulation model, - the theoretical background of the methods used in the Matlab-based experiments, - a discussion of the main parameters adopted in the experiments, - a clear presentation and discussion of the obtained results, - a discussion of the methods used to evaluate the performance of the algorithms, - a comparison with theoretical results (when applicable) The oral discussion is evaluated based on its correctness, the level of knowledge that the student has acquired on the topic, the ability to apply the acquired know-how, to clearly communicate the technical material with accurate terms and to correctly analyze, interpret, and comment the obtained results. During the oral examinations, the students may not consult any material,. The score of the oral examination is up to “30 e lode�, and the student passes the exam if they obtain at least 18/30.

The following texts will be used during the course: Regarding statistical models and machine learning, the reference book is: K.P. Murphy, “Machine learning – a probabilistic perspective�, MIT press, 2012. Regarding neural networks, there is no reference book.

The student must acquire a deep knowledge and understanding of the statistical physics of interacting systems and of its methodologies, and must become able to solve models in this domain, using exact or approximate analytical approaches.

In this module the knowledge of statistical physics is developed, with a particular attention to modern aspects involving interacting systems and critical phenomena.

1. Statistical physics Review: canonical and grand-canonical ensembles, non-interacting systems (10 hours). The Ising model: introduction and exact solutions in one dimension and on the fully connected graph (10 hours). Mean-field approximation (10 hours). Beyond mean-field approximation: Bethe-Peierls and belief propagation (5 hours). The two-dimensional Ising model: Peierls argument, low- and high-temperature expansions, free energy in zero field on a square lattice. The two-dimensional XY model at low temperatures (15 hours). An introduction to the real-space renormalization group (10 hours).

E' possibile sostenere l?esame in anticipo rispetto all?acquisizione della frequenza

M. Plischke and B. Bergersen, Equilibrium statistical physics, World scientific R.K. Pathria and P.D. Beale, Statistical mechanics, Academic Press L. Peliti, Statistical mechanics in a nutshell, Bollati Boringhieri J.P. Sethna, Entropy, order parameters and complexity, Clarendon Lecture notes and slides will be provided.

1. Learning the basic methodologies to simulate statistical systems. 2. Understanding the use of numerical methods to obtain approximate solutions to otherwise intractable problems. 3. Learn a high-level general-purpose scientific computing language (julia, jupyter notebook etc.) in a Linux/Unix environment. 4. Acquiring general-purpose data analysis and visualisation skills.

This course provides the students with tools in numerical analysis that are frequently used in modern theoretical physics and in the analysis of complex systems. The use of numerical methods, and statistical analysis, is fundamental for the research in theoretical physics. Numerically simulating a complex system is often the only way to get some intuition about its behavior. The course will provide first an introduction to multivariate statistics, then will be the theoretical background to build a theory of stochastic processes, such as Markov Process with discrete/continuous time and state evolution (e.g. Monte Carlo Markov Chains, Master Equation, Fokker-Plank Equation). The concepts acquired in the first part of the course will be then used in the second part of the course to develop a theory of numerical techniques to analyze concrete complex systems (e.g. spin models for phase transitions, epidemic modeling, etc.)

1) Concepts of probability and statistics (2 credits) Random variables. Statistical description of data. Introduction to multivariate statistics. Relevant distributions. Maximum entropy estimation of the parameters of relevant probability distributions. Bayes theorem. Large numbers law, central limit theorem. Finite size correction to central limit theorem. Large deviations. Experimental Data Analysis. Random Walks, Wiener process, Master Equations. Fokker-Planck equation. 2) . Numerical methods (2 credits). Introduction to basic Unix/Linux commands. Introduction to the Julia language. Simple programs, plotting data, input and output. Finding roots of equations: bisection, regula falsi, secant and Newton's methods. Pseudorandom number. Defining a custom pdf sampler. The case of discrete variables. Monte Carlo (MC) integration. 3) Simulation of discrete systems at equilibrium (2 credits). Sampling the canonical ensemble with Monte Carlo: Metropolis-Hastings rule, balance and detailed balance, hybrid Monte Carlo. Applications to phase transitions Critical phenomena. Finite-size scaling analysis. Simulated annealing. Simple ordinary differential equation: the harmonic oscillator. Euler-Cromer, mid-point, leap-frog. Stability analysis. Least square regression. Finite dimensional lattices. Ising model in finite dimensional lattice. Phase transitions. 4) Improved Monte Carlo methods (2 credits). Sampling efficiently from complicated distributions. Faster than-the-clock Monte Carlo schemes. Rejectionless Monte Carlo. Wolff's algorithm. Hit-and-run Monte Carlo. Introduction to the final project.

E' possibile sostenere l?esame in anticipo rispetto all?acquisizione della frequenza

• Frederick Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill - Crispin W. Gardiner - Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer Series in Synergetics 13) (1994); - C. Kittel, "Elementary statistical physics". Courier Corporation. (2004); - Kerson Huang, "Statistical Mechanics", Wiley (1987); - M. E. J. Newman G. T. Barkema - Monte Carlo Methods in Statistical Physics (1999); - Luciano M Barone, Enzo Marinari, Giovanni Organtini, Federico Ricci Tersenghi - Scientific Programming C-Language, Algorithms and Models in Science. World Scientific Publishing Company (2013); - Malvin H. Kalos, Paula A. Whitlock - Monte Carlo Methods , Wiley-VCH (2008); - Michael P. Allen, Dominic J. Tildesley - Computer Simulation of Liquids. Oxford University Press (2017).

Al termine dell’insegnamento, studenti e studentesse avranno acquisito una conoscenza e una comprensione approfondita delle principali definizioni nonché dei dibattiti relativi alla diplomazia tecno-scientifica, così come dei processi storici che hanno caratterizzato le relazioni tra progetti tecnico-scientifici e dinamiche internazionali. Saranno inoltre in grado di applicare tali conoscenze per sviluppare una visione critica delle motivazioni complesse che guidano la cooperazione tecno-scientifica a livello globale, e di individuare le problematiche specifiche del settore tecnologico nei processi di diplomazia e nelle relazioni internazionali contemporanee.

La diplomazia tecno-scientifica è un ambito in crescente rilievo nel dibattito accademico e politico, che riguarda l’interazione tra scienza, tecnologia e relazioni internazionali. Enti nazionali e sovranazionali, come l’Unione Europea, la includono tra le priorità strategiche per affrontare sfide globali come il cambiamento climatico, la sicurezza energetica e la regolazione delle tecnologie emergenti. Il ruolo della cooperazione tecnico-scientifica si rivela centrale anche in situazioni di crisi geopolitica, come dimostra il caso del progetto ITER, in cui la collaborazione tra Unione Europea, Stati Uniti e Russia è proseguita nonostante le tensioni legate alla guerra in Ucraina. L'insegnamento introduce studenti e studentesse alle pratiche storiche e contemporanee della diplomazia tecno-scientifica, evidenziando come oggetti, infrastrutture, standard, progetti e conoscenze tecnico-scientifiche influenzino le relazioni tra Stati. Attraverso l’analisi di casi studio ed esempi concreti, l’insegnamento fornisce strumenti per comprendere criticamente il ruolo della tecnica e della tecnologia nelle dinamiche globali e per formare ingegneri, architette, architetti, e designer in grado di operare in contesti internazionali complessi.

1. Che cos’è la diplomazia tecno-scientifica (9 ore) La parte introduttiva dell'insegnamento è dedicata alla presentazione delle principali nozioni, definizioni e concettualizzazioni della diplomazia tecno-scientifica, con l’obiettivo di fornire una base teorica per l’analisi dei casi e delle pratiche contemporanee. 2. Critiche alla retorica della diplomazia tecno-scientifica (11 ore) A partire dall’analisi di esperienze e istituzioni spesso citate come esempi di diplomazia tecno-scientifica, si esamineranno le principali critiche a una narrazione idealizzata e apolitica del ruolo della scienza e della tecnologia nelle relazioni internazionali. 3. Storia delle relazioni tra scienza, tecnologia e relazioni internazionali (20 ore) Questa sezione esplorerà lo sviluppo storico del ruolo della scienza e della tecnica nello Stato e nella diplomazia, dalla seconda metà del XIX secolo fino all’epoca contemporanea. Saranno analizzate le trasformazioni politiche, istituzionali e culturali che hanno ridefinito queste relazioni nel tempo. 4. Analisi di casi studio (20 ore) Questa parte dell'insegnamento sarà dedicata a un’analisi approfondita di casi concreti di diplomazia tecno-scientifica, attraverso progetti internazionali o strategie statali. Saranno discussi esempi come: - il ruolo della scienza e della tecnica nella politica estera di vari stati, tra cui Cina e Stati Uniti; - la geopolitica dello smartphone e la guerra dei chip; - il ruolo del CERN nell’integrazione europea nel passato e nel presente; - il progetto ITER e la fusione nucleare come strumenti di distensione politica alla fine della Guerra Fredda; - intelligenza artificiale, interessi nazionali e diplomazia.

Modalita di esame: Prova scritta in aula tramite PC con l'utilizzo della piattaforma di ateneo;
Exam: Computer-based written test in class using POLITO platform;
... L’esame consiste in una prova scritta della durata di 2 ore, composta da: 10 domande a risposta multipla (ognuna con quattro opzioni, una sola corretta); 2 domande a risposta aperta. Ogni risposta corretta vale +2 punti. Le risposte errate o non date valgono 0 punti. Le domande a scelta multipla sono progettate per verificare la conoscenza e la comprensione da parte di studenti e studentesse delle principali definizioni e concettualizzazioni di diplomazia tecno-scientifica, delle critiche ad essa rivolte, nonché dei processi storici analizzati durante l’insegnamento. Esse contribuiscono alla valutazione dei primi due risultati di apprendimento attesi, ovvero: la comprensione dei dibattiti e delle definizioni chiave; e la conoscenza dei principali sviluppi storici del rapporto tra scienza, tecnica e relazioni internazionali. Ciascuna risposta aperta può ottenere un massimo di 5 punti. Le domande aperte valutano la capacità critica di applicare le conoscenze acquisite, discutendo in autonomia due dei casi studio affrontati nell'insegnamento. In particolare, misurano la capacità di: sviluppare una riflessione autonoma sulle motivazioni complesse che guidano la cooperazione tecno-scientifica internazionale; individuare e analizzare le problematiche specifiche del settore tecnologico nei processi diplomatici. Tali aspetti corrispondono ai restanti risultati di apprendimento attesi, relativi alla capacità di analisi critica, autonomia di giudizio e applicazione contestuale delle conoscenze. Durante la prova scritta non è consentito consultare testi, appunti o materiali didattici, né utilizzare dispositivi digitali con accesso al web (smartphone, smartwatch, tablet, ecc.). L’esame è considerato superato con un punteggio minimo di 18/30. Il punteggio massimo conseguibile è 30/30. L’esito della prova scritta sarà comunicato su portale della didattica entro una settimana dallo svolgimento dell’esame.

Bibliografia essenziale: - Ruffini, Pierre-Bruno. Science and Diplomacy: A New Dimension of International Relations. New York, NY: Springer Berlin Heidelberg, 2017. - Royal Society/AAAS. New Frontiers in Science Diplomacy. London: Royal Society, 2010 https://royalsociety.org/topics-policy/publications/2010/new-frontiers-science-diplomacy. - Royal Society/AAAS. Science Diplomacy in an Era of Disruption. London: Royal Society, 2025. https://www.aaas.org/sites/default/files/2025-02/Final_Science%20diplomacy_15%20years%20on_report_WEB.pdf - Directorate-General for Research and Innovation (European Commission), Rasmus Gjedssø Bertelsen, Laurent Bochereau, Effrosyni Chelioti, Ã�gota Dávid, Dovilė Gailiūtė-JanuÅ¡onė, Martina Hartl, et al. A European Framework for Science Diplomacy: Recommendations of the EU Science Diplomacy Working Groups. Publications Office of the European Union, 2025. https://data.europa.eu/doi/10.2777/9235330. - Turchetti, Simone, Matthew Adamson, Giulia Rispoli, Doubravka Olšáková, and Sam Robinson. “Introduction: Just Needham to Nixon? On Writing the History of ‘Science Diplomacy.’â€� Historical Studies in the Natural Sciences 50, no. 4 (September 23, 2020): 323–39. https://doi.org/10.1525/hsns.2020.50.4.323.

the student is expected to acquire an overview of various computational aspects of complex systems, and to develop an understanding of several interdisciplinary applications

The course consists of 4 modules of 3 ECTS each. One module is compulsory, while the other 3 modules can be chosen by the student among the list given here below

Module 1: Nonlinear Physics and dynamical systems [B. Andreotti] (COMPULSORY MODULE) The physics of the so-called “complex” systems is based on a multitude of coexisting dynamic mechanisms, on a multiplicity of spatial or temporal scales, or on dynamics that are violently out of equilibrium. In this context, the linearization of problems, which makes powerful analytical mathematical tools available, becomes inoperative. Non-linear differential equations, for which there is no longer any basis for functional decomposition, since there is no longer any principle of superposition, require specific problem-solving techniques, either theoretically or experimentally. Nonlinear physics has no particular object of study. It brings together a culture and a set of theoretical and experimental tools that can be used in all disciplinary fields. For this reason, researchers with this background can be found in many different communities: biophysics, soft matter, hydrodynamics, economics, condensed matter, social theory, geosciences, atmospheric sciences, optics, acoustics, non-equilibrium statistical physics, etc. For this reason, there is no fixed corpus associated with ‘Nonlinear Physics’. I have chosen to approach it from the point of view of hydrodynamics, elasticity and soft matter, i.e. by working on partial differential equations, to complement the rest of the curriculum. We will start with… linear problems. Then we will consider weakly nonlinear developments and bifurcations. Then we will see the fundamental aspects of asymptotic calculus, singularities, self-similar solutions and their emergence. Finally, we will discuss the physics of dynamical systems, chaos and turbulence. This outline, organized according to mathematical tools, will be associated with a second layer organising the visit of various modern problems in continuum mechanics. Module 2: Advanced nonlinear physics [B.Andreotti] In 1917, the biologist and mathematician d’Arcy Thomson published a book that has inspired many contemporary scientists: On growth and form. The question posed is the mechanistic origin of forms. This question, which requires the articulation of different disciplines, lies at the heart of a branch of non-linear physics that is concerned with the natural sciences: morphogenesis. It’s a question that remains extremely topical in biophysics for understanding the origin of life and its diversity. In the advanced non-linear physics course, we will take up this question a century after d’Arcy Thomson. We’ll start by looking at some of the big questions on the existing Universe. How are planets formed? What forms the Earth’s relief? How can we describe volcanoes, mountains, rivers, hot and cold deserts and their sand and ice dunes? How are clouds formed? Why don’t most clouds cause rain? How do cyclones form? How does water circulate in the soil? How do animals move? How do living micro-organisms work? How do plants and animal embryos take shape? What is the dynamics of populations? Where does global warming and the ongoing collapse of life come from? And much more. Module 3: Disordered systems [V. Ros] Most physical systems in nature can be considered as disordered in the sense that either the relevant dynamical degrees of freedom interact heterogeneously, or because they are subjected to random forces or fields or because they are frozen into amorphous/glassy structures. In all these cases, disorder gives rise to a complex and rich phenomenology characterized for example by metastability, the emergence of rough energy landscapes, slow dynamics and avalanche-like responses. The study of disordered systems has become important not only because they are central in statistical and condensed matter physics but also because the theoretical concepts developed to study them have important ramifications in a large variety of fields including computer science, optimization, statistics, high-dimensional inference and learning problems. This course will introduce a broad class of classical disordered systems and will discuss the theoretical fundamental concepts and methods to understand their equilibrium and out of equilibrium physics. A particular attention will be devoted to discuss how such concepts and methods can be apply beyond physics. Module 4: Nonequilibrium and active systems [F. Van Wijland] Statistical mechanics has brought a fundamental change of paradigm in physics: rather than solving complex dynamics (e.g. Newton or Schroedinger equations), the study of matter can now be done using static, probabilistic approaches, provided the system under study is in (thermal) equilibrium. As statistical mechanics progresses towards new area of research (biophysics, geophysics, driven systems), new frameworks are needed to reproduce the successes of equilibrium statistical mechanics. In these lectures, we will study the new tools which have been developed over the past few decades to study non-equilibrium systems. The first part of the lectures will be dedicated to study these tools in the context of relaxations towards thermal equilibrium (derivation of Langevin equation, Ito calculus, Fokker-Planck equation & operator, Master equation). In the second part, we will illustrate and apply these tools to study a research field which has attracted a lot of interest recently: active matter. This field encompasses systems in which individual units are able, at the microscopic scale, to convert energy stored in the environment to self-propel (bacteria, active colloids, vibrated granular media, etc.). Module 5: Statistical field theory and soft matter [J.-B. Fournier] Statistical field theory (SFT) applies ordinarily to systems close to a critical phase transition, where large-scale fluctuations are observed. However, soft matter (polymers, biological membranes, interfaces, liquid crystals, etc.) often exhibits strong thermal fluctuations because larges deformations have excitations energies close to kT. As the number of symmetries in nature is limited, these systems are described by the same SFT tools as critical systems. They thus exhibit critical exponents and scaling laws. For instance, the partition function of a self-avoiding walk or polymer is a path integral that can be mapped to an O(n) model in the replica n→0 limit and studied with the renormalization group techniques. Thus polymers share the same exponents as magnets. In the O(n) models, the field are vectors, but what if they become tensorial? We may use again our SFT skills and end up describing the liquid crystal ordering that lies within your computer screen. This is what physics is about: describing a large variety of phenomena with a limited number of concepts and tools. Module 6: Biophysics [A. Callan-Jones] To motivate how physics can give insight into — and also get insight from — living systems, consider how an organism develops. Yes, genetics is important, but how a homogeneous embryo breaks symmetry to become a developed being involves collective motions from the sub-cellular to the organism scale. As a result, modern biophysics borrows and has helped to advance concepts from nonlinear physics, soft matter, and non-equilibrium statistical physics. In this course we will focus on cell (and some tissue) biophysics, but will also delve into how organisms respond and adapt to their surroundings. We will emphasize throughout how intrinsically driven chemical reactions and molecular changes can give rise to out of equilibrium macroscopic responses. The course will broken down into five chapters, as illustrated in the figure. Module 7: Quantum dynamics [L. Cugliandolo] In these lectures we will introduce a selection of theoretical methods used to study the out of equilibrium dynamics of quantum complex systems. We will start by recalling some essential features of quantum mechanics. Then we will enter the discussion of modern topics such as: 1- the definition and effects of a quantum bath, 2- the modelization and treatment of quenched randomness, 3- the description of chaos in quantum systems, 4- the dynamics following quenches. At all stages a parallel with the classical counterparts will be made. Detailed notes will support the lectures

Modalita di esame: Prova scritta (in aula);
Exam: Written test;
... written tests, according to the rules of the partner Universities Paris-Cité, Paris Saclay and Sorbonne Université

can be found at https://physics-complex-systems.fr/cursus.html

Machine learning (ML) is one of the most dynamic and exciting areas in modern data-driven research. Built upon inspiration from fields as different as statistics, computer science, neurosciences and physics, it allows for automatic learning from complex large-scale datasets. The lectures aim at introducing the core concepts and algorithms of ML in a way easily understood by physicists, both in the setting of supervised learning (linear and logistic regression, ensemble methods, deep neural networks…) and unsupervised learning (dimensional reduction, clustering, generative modelling…). The course is accompanied by Python Jupiter Notebooks, which allow for testing the main algorithms presented in the lectures, and introduces some highly used ML Python packages to the students.

Modalita di esame: Prova scritta (in aula);
Exam: Written test;

READING MATERIALS: P Mehta, M Bukov, CH Wang, AGR Day, C Richardson, CK Fisher, D Schwab, A high-bias, low-variance introduction to Machine Learning for physicists, Physics Reports (2019) 1-124. STUDY MATERIALS: can be found at https://physics-complex-systems.fr/cursus.html

Processi stocastici/Dinamiche su network (Dinamiche su Network)Conoscenze di: - elementi di base della teoria (topologica e algebrica) dei grafi diretti e indiretti con particolare enfasi sui concetti di centralità , connettività , periodo di un nodo, flussi su rete; - teoria delle matrici stocastiche e sub-stocastiche, delle loro proprietà spettrali e del loro uso nei modelli matematici deterministici e stocastici per dinamiche su reti; - nozioni fondamentali di ottimizzazione convessa di flussi su rete, della teoria dei giochi competitivi e delle relative dinamiche evolutive; - principali modelli di grafi aleatori e delle loro proprietà elementari. Capacità di: - costruire e comparare modelli matematici per sistemi interconnessi che nascono in reti sociali, economiche, biologiche e infrastrutturali; - applicare nozioni di base della teoria dei grafi topologica e algebrica, dei processi di Markov, della teoria dei giochi e dei sistemi dinamici per analizzare le reti ed i sistemi dinamici definiti su di esse; - individuare e valutare proprietà delle reti quali centralità , connettività , comportamenti emergenti, scalabilità ; - costruire algoritmi di tipo cooperativo su reti, di valutarne la velocità di convergenza e la complessità e di implementarli su moderne piattaforme per la simulazione numerica. Processi stocastici/Dinamiche su network (Processi stocastici)Al termine dell’insegnamento si chiederà allo studente di: - conoscere le definizioni di processi di conteggio, processi di Poisson omogenei e non omogenei, catene di Markov a tempo discreto e continuo, martingale, moto browniano. - conoscere le definizioni dei principali oggetti matematici associati ai processi stocastici elencati sopra, quali tempi di arresto, distribuzioni stazionarie, probabilita' di transizione, classificazione degli stati, etc; - conoscere gli enunciati dei teoremi base della teoria dei processi stocastici e comprenderne il significato; - conoscere le dimostrazioni di alcuni teoremi base della teoria dei processi stocastici e comprenderne i passaggi; - comprendere le modalità di sviluppo della teoria dei processi stocastici e relazionarla alle esigenze applicative: perché è stato enunciato un certo teorema? Come puo essere applicato ad un modello matematico concreto? - applicare la teoria base dei processi stocastici per risolvere esercizi; - applicare la teoria base dei processi stocastici in maniera critica per costruire, analizzare e confrontare modelli matematici di dinamiche aleatorie; - saper dimostrare in autonomia nuovi risultati pertinenti la teoria dei processi stocastici, ed essere eventualmente in grado di contribuire al suo sviluppo.

Processi stocastici/Dinamiche su network (Dinamiche su Network)L’insegnamento ha lo scopo di presentare gli elementi fondamentali della teoria dei grafi per arrivare allo studio di modelli matematici che descrivono dinamiche su network sia di tipo deterministico che aleatorio. Vengono descritti gli elementi di base della teoria dei grafi topologica e algebrica la teoria delle matrici stocastiche e alcuni elementi dei processi Markoviani Vengono poi considerati modelli di dinamiche su grafi sia deterministiche che probabilistiche. Sono infine discusse varie applicazioni: algoritmi su reti per stime e inferenze distribuite, modelli per la dinamica delle opinioni e modelli epidemici, modelli di teoria dei giochi.Processi stocastici/Dinamiche su network (Processi stocastici)I processi stocastici vengono utilizzati come modelli matematici negli scenari più disparati, dalla biologia alla finanza, dalla gestione di magazzino allo studio del traffico su internet, dalla medicina alla blockchain, dalla meteorologia all'ecologia. Questo corso, pur non potendo entrare nel dettaglio delle varie applicazioni, si pone l'obiettivo di fornire un'introduzione rigorosa alla teoria dei processi stocastici, con lo scopo di fornire le basi matematiche per la comprensione, la progettazione e l'analisi di tali modelli. Verranno definite ed esplorate formalmente le principali classi di processi stocastici, tra cui i processi di conteggio, i processi di Poisson, le catene di Markov, le martingale e il moto Browniano. Attraverso lo studio delle loro proprietà fondamentali, gli studenti acquisiranno sia una comprensione teorica che strumenti pratici per costruire e analizzare modelli di sistemi dinamici aleatori.

Processi stocastici/Dinamiche su network (Dinamiche su Network)1. Teoria dei grafi (topologica e algebrica) - Elementi di teoria dei grafi: cammini, cicli, connettività , distanza, diametro. Esempi. Grafi indiretti: proprietà ed esempi (completo, alberi, ciclo, griglie). Proprietà combinatorie degli alberi. Grafi bipartiti e loro proprietà . Circuiti euleriani. Componenti connesse di un grafo e concetto di condensation graph. Periodo di un nodo. Caratterizzazione dell'aperiodicità e relative proprietà . - Matrice dei pesi di un grafo e sue potenze. Matrici di grafi aperiodici. Matrice stocastica e Laplaciano associati ad un grafo. Il teorema di Perron-Frobenius. Proprietà spettrale delle matrici stocastiche. Centralità (degree, eigenvector, Bonacich, Pagerank, Katz). - Matrici stocastiche reversibili. Forma di Dirichlet. Formulazione variazionale per il calcolo del secondo autovalore di una matrice reversibile. Conduttanza e stima di Cheeger. Esempi. - Connettività e flussi su rete: teorema di Menger e teorema del massimo flusso / minimo taglio. 2. Dinamiche averaging e di flusso lineari - Modello di averaging distribuito e sistemi compartimentali per reti chiuse e con ingressi esogeni. - Teoremi di convergenza, ruolo della centralità , matrici bi-stocastiche. - Applicazioni: stima distribuita su reti, algoritmo Metropolis, dinamiche di opinioni (modello di De Groot), saggezza delle folle, modelli di influenza in reti sociali, dinamiche lineari di flusso su rete, algoritmi distribuiti per il calcolo della centralità . - Stima dei tempi di convergenza per matrici stocastiche reversibili: ruolo del secondo autovalore. Spettro di matrici circolanti e prodotto di Kronecker di matrici. Calcolo di spettri di matrici associate a grafi notevoli: grafo completo, anelli, griglie, ipercubi, stelle. - Matrici sub-stocastiche e loro proprietà . Velocità di convergenza per averaging distribuito e sistemi compartimentali con ingressi. - Sistemi dinamici positivi e monotoni: definizione e proprietà di base. 4. Catene di Markov - Richiami sui processi Markoviani a tempo discreto: stati assorbenti, tempi e probabilità di assorbimento, tempi di ritorno, teorema di Kac, catene ergodiche, misure invarianti. - Esempi notevoli: catene reversibili e catene di nascita e morte, cammini casuali su grafi. Processi di ramificazione (Galton-Watson): probabilità di estinzione, soglia di criticità , processi supercritici e subcritici. - Processi markoviani a tempo continuo e loro rappresentazioni, processi di nascita e morte a tempo continuo, equazioni di Kolmogorov, distribuzioni invarianti ed equazioni di bilancio. - Coupling, distanza variazionale, mixing time. 5. Ottimizzazione di flussi su reti - Ottimizzazione di flussi su reti con costi convessi separabili: condizioni necessarie e sufficienti per ottimalità , dualità , interpretazione dei moltiplicatori di Lagrange come costi marginali - Reti elettriche: interpretazione delle leggi di Ohm e Kirchoff, nozione di resistenza effettiva, principio variazionale di Rayleigh. Applicazioni alle catene di Markov reversibili e ai modelli di influenza su reti sociali indirette. - Ottimizzazione del traffico su rete: ottimo di sistema, equilibrio di Wardrop, prezzo di anarchia, uso di incentivi e disincentivi. Applicazioni alle reti di trasporto. 6. Teoria evolutiva dei giochi - Introduzione alla teoria dei giochi non cooperativi. Giochi in forma strategica, nozione di best response, strategie dominanti, equilibri di Nash. Esempi classici. - Giochi potenziali, giochi di congestione. - Giochi su reti. Esempi: majority, minority, colorazione di grafi. - Analisi delle dinamiche best response e noisy best response. 7. Modelli epidemici e di interazione a coppie - Modelli di interazione di tipo pairwise. - Algoritmi gossip di averaging. - Modelli epidemici (SI, SIS, SIR), voter model. - Calcolo di probabilità e tempi di assorbimento su topologie semplici (completo, linea, stella). - Modelli ergodici (noisy voter model, SIS con mutazioni spontanee). 8. Limiti di campo medio - Modelli di interazione su grafi completi con matrice di transizione di tipo anonimo. Esempi. Processo delle frequenze empiriche. - Limite idrodinamico: Teorema di Kurtz e descrizione deterministica tramite ODE. - Limite idrodinamico di modelli binari: voter, SIS, majority, k-majority. - Modelli ergodici: la relazione tra misura invariante e punti di equilibrio della ODE. - Esempi multidimensionali: SIR, SIRS. Modelli eterogenei. 9. Grafi aleatori - Grafo aleatorio di Erdos-Renyi: definizione, proprietà elementari, distribuzione dei gradi, approssimazione con il processo di ramificazione, soglia di connettività , soglia per l’esistenza di una componente gigante. - Modello di grafo casuale con distribuzione dei gradi assegnata (configuration model): proprietà elementari, approssimazione con il processo di ramificazione, soglie per connessione ed esistenza di una componente gigante. - Modelli avanzati di grafi aleatori: preferential attachment, small world. - Limite idrodinamico per dinamiche senza memoria su grafi aleatori con distribuzione dei gradi fissata.Processi stocastici/Dinamiche su network (Processi stocastici)- Definizione di processi stocastici e loro principali caratterizzazioni e proprietà . - Processi di conteggio e renewal. - Processi di Poisson: definizioni equivalenti, risultati notevoli, processi di Poisson non omogenei. - Catene di Markov a tempo discreto e loro proprietà , matrici di transizione, equazioni di Chapman-Kolmogorov, classificazione degli stati, convergenza in distribuzione e teorema ergodico per catene irriducibili, esistenza e unicità di distribuzioni stazionarie, connessioni fra i tempi medi di ritorno e la distribuzione stazionaria. - Catene di Markov a tempo continuo e loro rappresentazioni, equazioni di Kolmogorov, distribuzioni stazionarie ed equazioni di bilancio. - Esempi notevoli: catene reversibili e catene di nascita e morte, elementi di teoria delle code, reti di Jackson. - Martingale e le loro proprietà , teorema di decomposizione di Doob, teoremi di arresto di Doob, teoremi di convergenza di Doob, disuguaglianza di Azuma-Hoeffding, esempi di applicazioni. - Moto Browniano (o processo di Wiener): definizione e risultati base.

Processi stocastici/Dinamiche su network (Dinamiche su Network)Modalita di esame: Prova orale obbligatoria; Elaborato progettuale individuale; Processi stocastici/Dinamiche su network (Processi stocastici)Modalita di esame: Prova orale obbligatoria; Elaborato scritto individuale; Elaborato progettuale in gruppo;
Processi stocastici/Dinamiche su network (Dinamiche su Network)Exam: Compulsory oral exam; Individual project; Processi stocastici/Dinamiche su network (Processi stocastici)Exam: Compulsory oral exam; Individual essay; Group project;
... Processi stocastici/Dinamiche su network (Dinamiche su Network)La verifica dell’apprendimento consiste nelle seguenti parti: 1. Tre homework (HW1, HW2 e HW3) che vengono assegnati, rispettivamente, alla fine dei mesi di Ottobre, Novembre e Dicembre (prima delle vacanze natalizie). I primi due consistono di esercizi teorici ed applicativi ed hanno lo scopo di verificare l’assimilazione da parte dello studente delle teorie sviluppate nel modulo e a valutarne lo sviluppo delle capacità di modellizzazione matematica. Il terzo homework, oltre ad una parte strutturalmente simile ai primi due, prevede un progetto di tipo simulativo/numerico ed ha lo scopo di valutare le capacità progettuali dello studente. I tempi di consegna sono di due settimane per HW1 e HW2 e di tre per HW3. 2. Una prova orale finale che consiste di due parti: 2a) Discussione degli homework HW1, HW2, HW3 al fine di valutare la capacità critica dello studente rispetto al lavoro svolto. 2b) Presentazione da parte del candidato di un argomento teorico trattato nel corso e di eventuali relativi esempi di applicazione. L' argomento in questione viene scelto dalla commissione e comunicato al candidato, a cui vengono lasciati 30 minuti per la preparazione della risposta, e durante i quali possono essere consultati libri ed appunti. Scopo di questa seconda parte è la valutazione delle conoscenze teoriche di base acquisite e la capacità dello studente di presentarle in modo autonomo ed efficace. Punteggi: I punteggi massimi per gli homework, a seguito della discussione di cui a 2a), sono 4 per HW1 e HW2 ciascuno e di 6 per HW3. Nel caso in cui le consegne siano avvenute in ritardo i massimi scendono a 3 e 5 rispettivamente. Il punteggio massimo della prova 2b) è pari a 18. Al termine delle due prove il punteggio viene sommato: se esso è non superiore a 30, esso costituisce il voto finale, se è superiore il voto finale è 30 e lode. Processi stocastici/Dinamiche su network (Processi stocastici)- È prevista una prova orale obbligatoria. Durante la prova, il candidato dovrà discutere un argomento teorico trattato nel corso, con relative applicazioni. Verranno valutati i seguenti aspetti: . la conoscenza di definizioni e teoremi, incluse le relative dimostrazioni; . la comprensione della teoria; . la capacità di applicare la teoria in esercizi pratici o teorici; . la capacità espositiva e di presentazione. L'argomento principale dell'esame sarà scelto dalla commissione e comunicato al candidato, il quale avrà a disposizione 30 minuti per prepararsi. Durante questo tempo sarà possibile consultare libri e appunti. Al termine della prova orale verrà assegnato un punteggio massimo di 31 (equivalente a 30 e lode). - Oltre alla prova orale obbligatoria, lo studente può partecipare, in maniera facoltativa, alla seguente attività : . Ogni settimana, lo studente risponderà in autonomia a dei quiz pubblicati sulla pagina Moodle del corso. I quiz saranno preparati, alternativamente, dal docente o dagli studenti stessi (tramite un lavoro di gruppo svolto in classe, vedi sotto). . Ogni due settimane, gli studenti lavoreranno in gruppo per un'ora e mezza per risolvere esercizi o preparare i quiz basati sui contenuti del corso. Gli studenti che risponderanno ad almeno il 90% dei quiz settimanali e parteciperanno ad almeno il 70% delle attività di gruppo riceveranno un punto bonus, che verrà aggiunto al punteggio ottenuto nella prova orale.

Processi stocastici/Dinamiche su network (Dinamiche su Network)Parte degli argomenti del corso saranno coperti da dispense che verranno rese disponibili agli studenti tramite il Portale della Didattica. Processi stocastici/Dinamiche su network (Processi stocastici)Si consiglia di studiare il materiale del corso tramite slide con note rilasciate dal docente, e videoregistrazioni delle lezioni. Come ulteriore spunto per lo studio della materia si consigliano i seguenti libri, che pero' non verranno seguiti fedelmente dal docente e la cui lettura non è obbligatoria: - R. Durrett, "Essentials of Stochastic Processes", Springer, 2021 (online available). - D. A. Levin, Y. Peres, E. L. Wilmer, "Markov chains and mixing times", AMS, 2008 (online available). - S.M. Ross, "Introduction to Probability Models", Elsevier, 2010. - J.M. Steele "Stochastic Calculus and Financial Applications", Springer, 2000.

At the end of the course, students are expected to: * Know and understand the fundamental concepts of equilibrium statistical mechanics, including the Boltzmann distribution, entropy, variational principles, spontaneous symmetry breaking, linear response theory, and phase transitions. * Know and understand the theoretical description of magnetic systems and disordered systems, with particular emphasis on the Ising model, mean-field approximations, effective Hamiltonians, and critical phenomena in finite dimensions. * Know and understand the physical properties of spin glasses, including quenched disorder, frustration, ergodicity breaking, overlap order parameters, memory and chaos effects, and the phenomenology of experimental spin-glass systems. * Know and understand the main analytical methods used in the study of disordered systems, including replica theory, cavity methods, TAP equations, replica-symmetric and replica-symmetry-breaking solutions, and the Random Energy Model. * Know and understand stochastic growth processes and nonequilibrium statistical physics models, including the Edwards–Wilkinson and Kardar–Parisi–Zhang equations, scaling properties, and the relation between KPZ dynamics, Burgers equation, and directed polymers in random media. * Know and understand the statistical physics approach to optimization and constraint satisfaction problems on sparse random graphs, including Erdős–Rényi random graphs, graph coloring, random XOR-SAT problems, and message-passing algorithms such as warning propagation, belief propagation, and survey propagation. * Apply the acquired theoretical knowledge to solve exercises involving mean-field models, phase transitions, spin-glass order parameters, stochastic processes, and scaling arguments. * Apply statistical mechanics methods and probabilistic reasoning to the analysis of complex systems characterized by disorder, frustration, and randomness. * Apply the acquired knowledge to the formulation and interpretation of algorithms for optimization problems on random graphs and constraint satisfaction problems. * Develop the ability to connect theoretical models with physical phenomena and computational applications, recognizing the interdisciplinary role of disordered systems methods in physics, mathematics, computer science, and complex systems research.

The study of disordered systems plays a central role in modern statistical physics and complex systems science, with applications ranging from condensed matter physics and materials science to optimization, machine learning, information theory, and biological systems. Randomness, frustration, and collective phenomena generate rich emergent behaviors that cannot be understood within the framework of ordered equilibrium systems alone. The understanding of these phenomena requires both advanced theoretical tools and the ability to connect abstract models with experimentally observable properties and computational problems. Within this context, the course on Disordered Systems aims to provide the fundamental theoretical methods and conceptual tools necessary for the description of complex systems characterized by quenched disorder, frustration, and nontrivial collective dynamics. Particular emphasis is placed on the interplay between statistical mechanics, phase transitions, symmetry breaking, and probabilistic methods. The course introduces the theoretical framework of spin glasses and random systems, discussing both phenomenological and microscopic descriptions, together with the mathematical methods required for their analysis, including mean-field approximations, replica methods, cavity approaches, and effective Hamiltonian techniques. The course also addresses stochastic growth processes and nonequilibrium phenomena, with special attention to surface growth models and the Kardar-Parisi-Zhang universality class, highlighting the connections between statistical physics, stochastic differential equations, and random media. In addition, the course introduces modern approaches to optimization problems on sparse random graphs, discussing message-passing algorithms such as belief propagation, warning propagation, and survey propagation, as well as their relation to replica symmetry breaking and constraint satisfaction problems. Alongside the theoretical developments, illustrative examples and applications are discussed throughout the course in order to connect the formalism with physical phenomena and computational problems of current scientific interest. Particular emphasis is placed on the emergence of collective behavior from disorder, on the role of scaling and universality, and on the interdisciplinary relevance of disordered systems methods across physics, computer science, and complex systems research.

Part I – Introduction to Statistical Mechanics and Phase Transitions (6 hours) Fundamental concepts of equilibrium statistical mechanics, including the Boltzmann distribution, entropy, variational principles, and linear response theory. Magnetic systems, spontaneous symmetry breaking, the Ising model, and mean-field approaches are introduced as paradigmatic examples for the study of phase transitions and critical phenomena. Part II – Spin Glasses and Disordered Systems (20 hours) Phenomenology and theoretical description of spin glasses and systems with quenched disorder. Experimental properties, ergodicity breaking, overlap order parameters, replica methods, cavity approaches, and replica symmetry breaking are discussed within the framework of mean-field spin-glass models such as the Sherrington–Kirkpatrick model and the Random Energy Model. Part III – Stochastic Growth and KPZ Universality (10 hours) Introduction to stochastic surface growth models and nonequilibrium statistical physics. The Edwards–Wilkinson and Kardar–Parisi–Zhang equations are presented together with their scaling properties and their connections to Burgers dynamics and directed polymers in random media. Part IV – Optimization Problems on Sparse Random Graphs (24 hours) Statistical physics methods for combinatorial optimization and constraint satisfaction problems on sparse random graphs. The course covers Erdős–Rényi graphs, graph coloring, random XOR-SAT problems, and message-passing algorithms such as belief propagation and survey propagation, together with the limitations of replica-symmetric approximations.

Modalita di esame: Prova orale obbligatoria; Elaborato progettuale individuale;
Exam: Compulsory oral exam; Individual project;
... The final examination is aimed at assessing the acquisition of the expected knowledge and skills through an oral examination and the presentation of a project, both taking place on the same day. In order to evaluate the achievement of the learning objectives, including both the understanding of the theoretical concepts and the ability to apply them, the examination is divided into two parts. The first part consists of an oral examination focused on the theoretical topics covered during the course. Students are expected to discuss and critically analyze the fundamental concepts and methods of the statistical physics of disordered systems, including phase transitions, spin glasses, stochastic growth processes, and optimization problems on random graphs. Particular attention is devoted to the ability to formulate rigorous arguments, connect different theoretical frameworks, and discuss the physical interpretation of the models presented during the course. The second part consists of the presentation and discussion of a project developed by the student on a topic related to the course. The project may involve theoretical calculations, numerical simulations, analytical studies, or the discussion of relevant scientific literature. This part of the examination is aimed at evaluating the student’s ability to independently apply the methods introduced during the course, critically analyze results, and present scientific material in a clear and coherent way. The final grade is determined by considering both parts of the examination, with equal weight assigned to the oral discussion and to the project presentation. Honors (“cum laude�) may be awarded when the student demonstrates an outstanding level of understanding, methodological rigor, critical thinking, and clarity of presentation in both parts of the examination.

General Statistical Mechanics * Mehran Kardar, Statistical Physics of Particles and Statistical Physics of Fields, Cambridge University Press. Useful for the introductory part on statistical mechanics, phase transitions, scaling, and field-theory concepts. * My lecture notes for the course on Complex Systems that wil be available on the course portal. Disordered Systems and Spin Glasses * Marc Mézard, Giorgio Parisi, and Miguel Virasoro, Spin Glass Theory and Beyond, World Scientific. A classic reference on replica methods, spin glasses, and replica symmetry breaking. * Fischer, Konrad H., and John A. Hertz. Spin glasses. No. 1. Cambridge University Press, 1993. Another classic in the field. KPZ and Nonequilibrium Growth * Timothy Halpin-Healy and Yi-Cheng Zhang, “Kinetic Roughening Phenomena, Stochastic Growth, Directed Polymers and All That,� Physics Reports (1995). A standard review on KPZ universality and stochastic growth. * Ivan Corwin, “The Kardar–Parisi–Zhang Equation and Universality Class,� Random Matrices: Theory and Applications (2012). A modern introduction to KPZ theory. Optimization and Sparse Random Graphs * Marc Mézard and Andrea Montanari, Information, Physics, and Computation, Oxford University Press. A key reference for belief propagation, cavity methods, random graphs, satisfiability problems, and optimization. * F. Krzakala et al. (eds.), Statistical Physics, Optimization, Inference, and Message-Passing Algorithms, Oxford University Press (2015). Lecture notes and reviews on message-passing algorithms and constraint satisfaction problems. Additional / Advanced References * Patrick Charbonneau et al., Spin Glass Theory and Far Beyond, World Scientific World Scientific (2023). A modern and broad overview of spin glasses, optimization, machine learning, and complex systems applications. * Francesco Zamponi, Mean Field Theory of Spin Glasses (lecture notes). Useful as supplementary material for replica and cavity methods. Available at https://arxiv.org/abs/1008.4844

With this course, students will learn some fundamental notions of the physics of critical phenomena in many-body systems, such as scale invariance, the renormalization group and the universality of a physical theory. By means of the exercises proposed and solved in the class, they will become familiar with mathematical tools of daily use in the most advanced areas of research in theoretical and statistical physics.

Mandatory course for the Master in Physics of Complex Systems (national track), 2nd year, 1st term. The course aims to provide an introduction to statistical field theory and its application to the study of phase transitions and critical phenomena in complex systems. The formulation of interacting many-body systems in terms of continuous fields makes possible to go beyond mean-field approximations (Landau theory) to evaluate the effects of spatial fluctuations and correlations and their influence on the collective behaviour of systems near a phase transition. To this purpose, the course will provide an introduction to the mathematical tools (functional integrals, perturbative expansions, Feynman diagrams) and physical concepts (scale invariance, renormalization) that are at the basis of modern statistical physics. The revolutionary idea of ​Wilson’s renormalization group will be introduced and discussed by means of examples. The same methods will be also applied to the study of collective non-equilibrium phenomena.

1. Landau theory (~10 h): - phenomenological description of phase transitions, saddle-point approximation and mean-field theory, critical exponents; - continuous and discrete symmetry breaking (Goldstone modes and domain walls), lower-critical dimensions; - Gaussian integrals, role of fluctuations, Ginzburg criterion, upper critical dimensions. 2. From Scaling invariance to Renormalization (~10h): - scaling theory and invariance, coarse-graining process, the effective Hamiltonian, functional derivatives, Green functions; - Gaussian model, Wilson’s momentum-shell renormalization procedure, fixed-points and critical exponents for the Gaussian model. 3. Perturbative Renormalization Group (~20h): - perturbation theory in coordinate and momentum space, Wick’s theorem, Feynman diagrams, vertex functions and Legendre transform, loop expansion in the phi4 theory, 1/N expansion in the O(N) theory; - power counting and divergences, renormalization of mass, field and couplings, dimensional regularization, minimal subtraction scheme, RG flow equations, fixed points for the phi4 and O(n) theories, critical exponents; - basics of functional (non-perturbative) renormalization group methods. 4. Field Theory and Non-equilibrium Statistical Physics (~20h). Selected topics among: - equilibrium dynamics of a field, linear response and fluctuation-dissipation theorem; - Martin-Siggia-Rose formalism, non-equilibrium critical dynamics, interface growth, Edwards-Wilkinson and Kardar-Parisi-Zhang universality classes; - Peliti-Doi formalism, reaction-diffusion systems and absorbing phase transitions; - slow relaxation in the dynamics of the p-spin spherical model, mode-coupling equations, ergodicity breaking.

E' possibile sostenere l?esame in anticipo rispetto all?acquisizione della frequenza

• Mehran Kardar, Statistical Physics of Fields, Cambridge University Press, 2007 - John Cardy, Scaling and Renormalisation in Statistical Physics, Cambridge University Press, 1996 - Nigel Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, CRC Press, 2018 - Giorgio Parisi, Statistical Field Theory, Addison-Wesley, 1988. - additional material provided by the teacher

The internship leads the student to apply the knowledge and methodologies acquired in the first three terms and to expand her/his knowledge at a very deep level within a well-defined research problem in physics of complex systems, chosen in agreement with the supervisor. The training, guided by the supervisor, is typically based on the study of recent scientific publications. The work, under the supervisor's guidance, helps the student to develop autonomy and, together with the Spring College, to orient her/his future choices. The Spring College completes the programme, introducing the student to a selection of current research topics in the physics of complex systems. The activity is open to master and PhD students from any country. The topics covered, as well as the teachers, vary from year to year. The Spring College leads the student to apply the knowldge acquired in the courses of the first three terms and, together with the internships, is particularly important in helping the student o orient her/his future choices, which could be towards the job market or towards a PhD program.

The final project work is composed of 2 parts: an internship in a research group or laboratory at one of the partner institutions, corresponding to 18 credits; an intensive school (Spring College on the Physics of Complex Systems) at ICTP, Trieste, corrisponding to 12 credits. .

Modalita di esame: Accertamento (esame senza voto); Exam: Check; ... The graduation grade for the Master's degree will be calculated starting from the average of the exams, to which a maximum of 8 additional points can be added. Points available to the Committee (maximum 6): - maximum 4 points for the Thesis; - maximum 2 points for the Thesis presentation. Points for the academic career (maximum 2): - max 1 point based on the number of honors (0.5 points for every 5 credits of exams with honors) - max 1 point based on the duration of the studies: * 1 point for graduating within the November/December session of the 2nd year of enrollment; * 0.5 points for graduating within the March/April session of the 3rd year of enrollment; * 0 points for graduating from the July session of the 3rd year. [Students with a path that requires an additional semester: 3rd year March/April and July - 1 point 3rd year October and November/December - 0.5 points Students with initial enrollment in March without advances ("anticipo crediti"): 3rd year March/April and July - 1 point 3rd year October and November/December - 0.5 points] If the final grade is equal to or greater than 112.51, the student will be awarded the maximum grade of 110 with honors (110 cum laude).

The internship leads the student to apply the knowledge and methodologies acquired in the first three terms and to expand her/his knowledge at a very deep level within a well-defined research problem in physics of complex systems, chosen in agreement with the supervisor. The training, guided by the supervisor, is typically based on the study of recent scientific publications. The work, under the supervisor's guidance, helps the student to develop autonomy and, together with the Spring College, to orient her/his future choices. The Spring College completes the programme, introducing the student to a selection of current research topics in the physics of complex systems. The activity is open to master and PhD students from any country. The topics covered, as well as the teachers, vary from year to year. The Spring College leads the student to apply the knowldge acquired in the courses of the first three terms and, together with the internships, is particularly important in helping the student o orient her/his future choices, which could be towards the job market or towards a PhD program.

The final project work is composed of 2 parts: an internship in a research group or laboratory at one of the partner institutions, corresponding to 18 credits; an intensive school (Spring College on the Physics of Complex Systems) at ICTP, Trieste, corrisponding to 12 credits. .

Modalita di esame: Accertamento (esame senza voto);
Exam: Check;
... The graduation grade for the Master's degree will be calculated starting from the average of the exams, to which a maximum of 8 additional points can be added. Points available to the Committee (maximum 6): - maximum 4 points for the Thesis; - maximum 2 points for the Thesis presentation. Points for the academic career (maximum 2): - max 1 point based on the number of honors (0.5 points for every 5 credits of exams with honors) - max 1 point based on the duration of the studies: * 1 point for graduating within the November/December session of the 2nd year of enrollment; * 0.5 points for graduating within the March/April session of the 3rd year of enrollment; * 0 points for graduating from the July session of the 3rd year. [Students with a path that requires an additional semester: 3rd year March/April and July - 1 point 3rd year October and November/December - 0.5 points Students with initial enrollment in March without advances ("anticipo crediti"): 3rd year March/April and July - 1 point 3rd year October and November/December - 0.5 points] If the final grade is equal to or greater than 112.51, the student will be awarded the maximum grade of 110 with honors (110 cum laude).

.

The internship leads the student to apply the knowledge and methodologies acquired in the first three terms and to expand her/his knowledge at a very deep level within a well-defined research problem in physics of complex systems, chosen in agreement with the supervisor. The training, guided by the supervisor, is typically based on the study of recent scientific publications. The work, under the supervisor's guidance, helps the student to develop autonomy and, together with the Spring College, to orient her/his future choices. The Spring College completes the programme, introducing the student to a selection of current research topics in the physics of complex systems. The activity is open to master and PhD students from any country. The topics covered, as well as the teachers, vary from year to year. The Spring College leads the student to apply the knowldge acquired in the courses of the first three terms and, together with the internships, is particularly important in helping the student o orient her/his future choices, which could be towards the job market or towards a PhD program.

The final project work is composed of 2 parts: an internship in a research group or laboratory at one of the partner institutions, corresponding to 18 credits; an intensive school (Spring College on the Physics of Complex Systems) at ICTP, Trieste, corrisponding to 12 credits. .

Modalita di esame: Accertamento (esame senza voto);
Exam: Check;
... The graduation grade for the Master's degree will be calculated starting from the average of the exams, to which a maximum of 8 additional points can be added. Points available to the Committee (maximum 6): - maximum 4 points for the Thesis; - maximum 2 points for the Thesis presentation. Points for the academic career (maximum 2): - max 1 point based on the number of honors (0.5 points for every 5 credits of exams with honors) - max 1 point based on the duration of the studies: * 1 point for graduating within the November/December session of the 2nd year of enrollment; * 0.5 points for graduating within the March/April session of the 3rd year of enrollment; * 0 points for graduating from the July session of the 3rd year. [Students with a path that requires an additional semester: 3rd year March/April and July - 1 point 3rd year October and November/December - 0.5 points Students with initial enrollment in March without advances ("anticipo crediti"): 3rd year March/April and July - 1 point 3rd year October and November/December - 0.5 points] If the final grade is equal to or greater than 112.51, the student will be awarded the maximum grade of 110 with honors (110 cum laude).

• understanding basics notions of molecular biology; • understanding basic approaches to sequence alignment; • being familiar with structural inference and maximum entropy techniques; • being able to code simple sequence-alignment algorithms; • being able to apply basic machine-learning methods to given genetic and biological problems.

The course provides an introduction to molecular biology and to quantitative methods that can be used to extract information from complex biological systems, including the analysis of DNA, RNA and protein sequences, the reconstruction of phylogenetic trees, and the use of machine-learning techniques to analyze the structure of gene and protein networks.

• Introduction to Molecular Biology: central dogma, DNA, RNA, proteins; gene regulation; metabolism; (20 h, C. Bosia) • Hidden Markov models: from pairwise to multiple sequence alignments; inference in protein families; phylogeny reconstruction; RNA folding; (20 h, A. Gamba) • Machine learning techniques: introduction to widely used methodologies (e.g. neural networks, random forests, convolution neural networks, Bayesian neural networks, recurrent neural networks, LSTM); application to genetic and biological studies; (20 h, E. Ficarra).

E' possibile sostenere l?esame in anticipo rispetto all?acquisizione della frequenza

• Course handouts • B. Alberts et al., Molecular Biology of the Cell, Garland Science, 2015 • R. Durbin et al., Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids, Cambridge Un. Press, 2002 • H.C. Nguyen, R. Zecchina and J. Berg, Inverse statistical problems: from the inverse Ising problem to data science, Adv. Phys., 66 (2017) 197-261. • S. Cocco et al., Inverse statistical physics of protein sequences: a key issues review, Rep. Progr. Phys. 81 (2018) 032601. • J. Felsenstein, Inferring phylogenies, Sinauer Associates, 2004 • C.M. Bishop, Pattern Recognition and Machine Learning, Springer, 2011 • Suggested scientific publications

Lo studente imparerà a • Sviluppare e valutare i modelli matematici relativi a diversi fenomeni biologici e medici; • Utilizzare i metodi di integrazione analitica e numerica di equazioni differenziali ordinarie e di sistemi di equazioni differenziali ordinarie; • Identificare le configurazioni di equilibrio e le proprietà di stabilità degli stessi, riportandole in opportuni diagrammi di biforcazione; • Riconoscere le caratteristiche qualitative delle soluzioni delle equazioni alle derivate parziali; • Utilizzare gli strumenti della statistica descrittiva per rappresentare ed interpretare i dati sperimentali; • Utilizzare gli strumenti della statistica inferenziale per estrapolare informazioni relative a popolazioni e processi biologici partendo dai dati sperimentali; • Utilizzare il software scientifico Matlab per la soluzione di sistemi di equazioni differenziali e per l’esecuzione di test statistici; • Utilizzare il software scientifico Comsol Multiphysics per la soluzione di modelli biologici complessi; • Presentare in un elaborato scientifico i risultati numerici e/o analitici relativi ad un particolare problema biologico (a scelta dello studente)

Scopo del corso è fornire conoscenze di matematica applicata e di metodologie di costruzione e valutazione di modelli matematici applicati ai processi biologi (sia fisiologici, sia patologici) e alla medicina. Una parte importante è dedicata allo studio dell'intero ciclo di modellizzazione di fenomeni in ambito biomedico, partendo dalla formulazione dei modelli per arrivare, attraverso la corretta formulazione di problemi differenziali, alla soluzione quantitativa ottenuta tramite opportuni metodi matematici e programmi di calcolo numerico (Matlab e Comsol Multiphysics). Inoltre, verranno introdotti alcuni concetti fondamentali per l’interpretazione di dati sperimentali e la deduzione di conclusioni con valenza statistica. Dal punto di vista didattico si intende quindi rivoltare l'abituale sequenza teoria-applicazioni, facendo emergere la necessità dell'utilizzo di metodi matematici, numerici e statistici dall'esigenza del problema di interesse specifico. Coerentemente con questa attitudine, il rapporto tra lezioni teoriche, esercitazioni e laboratori informatici sarà sostanzialmente paritetico (ossia 36 ore di teoria, 12 ore di esercitazioni e 12 ore in laboratorio).

• Il ciclo della modellizzazione matematica. Cosa vuol dire dedurre un modello matematico in biomedicina. Modelli di crescita. Dinamica delle popolazioni. Crescita tumorale. Soluzione di equazioni differenziali del prim'ordine lineari e non lineari. Soluzione di equazioni differenziali lineari del secondo ordine. Problemi ai valori iniziali. Concetto di equilibrio e stabilità . Programmi scientifici per l'integrazione di equazioni differenziali (metodi di Eulero esplicito ed implicito, metodo di Runge-Kutta). • Interpretazione di dati sperimentali e definizione di trial clinici. Statistica descrittiva. Introduzione alla statistica inferenziale (distribuzioni importanti, stima di parametri, intervalli di confidenza, test di ipotesi). Trattazione di problemi caratterizzati da incertezze nel dato iniziale o nei parametri. Analisi statistica dei dati. • Modelli epidemiologici. Modelli di farmacocinetica/farmacodinamica. Modelli di reazioni chimiche. Sistemi di equazioni differenziali. Analisi di stabilità di un sistema di equazioni differenziali. Diagrammi di biforcazione: biforcazione a forchetta, supercritica e subcritica, Turning points e cicli di isteresi. Programmi scientifici per l'integrazione di sistemi di equazioni differenziali e di problemi stiff. Fitting dei dati. Applicazione di metodi statistici. • Processi di diffusione lineare e non lineare in biologia. Omeostasi cellulare (elettrodiffusione). Equazioni paraboliche. Soluzioni fondamentali. Metodo di Fourier per la risoluzione di PDE a variabili separabili. Metodo delle differenze finite. Cenni ai metodi degli elementi finiti. Equazioni ellittiche. Problemi al contorno. Cenni a metodi di risoluzione analitica e numerica. • Modelli di Lotka-Volterra per descrivere la competizione tra specie cellulari e popolazioni. Modellizzazione della propagazione di segnali nervosi (modelli di Hodgkin-Huxley e FitzHugh-Nagumo). Cicli limite. Biforcazione di Hopf. Stabilità di cicli limite: generazione soft ed hard di cicli limite. • Diffusione di sostanze chimiche e popolazioni cellulari attraverso un’interfaccia. Sistema respiratorio. Modelli per dialisi. Migrazione di cellule attraverso membrane sottili. Interfacciamento di domini. Soluzioni discontinue. Level-set method. • Equazione del trasporto. Sedimentazione di particelle. Equazione di bilancio di massa e applicazioni ai modelli tumorali. Chemotassi ed equazione di Keller-Segel. Caratteristiche fisiche della soluzione di un'equazione del trasporto. Metodo delle caratteristiche.

Modalita di esame: Prova scritta (in aula); Prova orale obbligatoria; Elaborato progettuale in gruppo;
Exam: Written test; Compulsory oral exam; Group project;
... L'esame consiste in - una prova scritta con due domande aperte sulle caratteristiche generali dei modelli matematici presentati durante le lezioni al fine di valutare le conoscenze apprese durante il corso; - un elaborato di carattere applicativo che usi le metodologie esposte nelle lezioni per lo sviluppo e lo studio analitico e/o numerico di specifici modelli matematici su tematica scelta dallo studente tra quelle proposte. L'elaborato dovrà essere ben documentato e difeso tramite presentazione orale. La prova orale inerente la teoria è individuale, l’elaborato può essere svolto individualmente o in gruppi di due-tre persone. L’elaborato scritto unitamente ai programmi utilizzati devono essere consegnati un paio di giorni prima della presentazione e della discussione del lavoro svolto. La presentazione e la difesa dei contenuti del lavoro svolto non potranno durare più di 15 minuti per gruppo. Considerando sempre la varietà di applicazioni biomediche, di massima la struttura dell'elaborato dovrà essere composta come riportato qui di seguito. Parte I: Introduzione 1- Introduzione generale al problema (1 pagina) 2- Osservazione fenomenologica, esigenza biomedica e relativa domanda (1 pagina) 3- Modelli in letteratura (1 pagina) Parte II: Metodi 4- Deduzione del modello matematico (3 pagine) Parte III: Risultati 5- Simulazioni numeriche (5 pagine) 6- Quadro riassuntivo delle simulazioni numeriche (3 pagine) 7- Discussione dei risultati e risposta suggerita dal modello matematico (max 1 pagina) Il risultato finale verrà così identificato - 40% per la qualità dell’elaborato - 30% per la qualità della presentazione e la difesa dei contenuti del lavoro svolto - 30% per la prova scritta inerente gli argomenti del programma.

Testo principale L. Mesin, Mathematical Models for biomedicine, ilmiolibro self publishing 2017 Testi di approfondimento J. P. Keener and J. Sneyd, Mathematical Physiology, Springer, 1998. R. Barr and R.L. Plonsey; Bioelectricity: A Quantitative Approach. Plenum press, 1988. N. Bellomo, L. Preziosi, "Mathematical Modelling, Methods and Scientific Computation", CRC Press J.D. Murray, Mathematical Biology I, An Introduction, Springer, 2002. J.D. Murray, Mathematical Biology II, Spatial Models and Biomedical Applications, Springer, 2002 S.J. Farlow, “Partial differential equations for scientists and engineers�, Dover S.M. Ross, “Probabilità e statistica per l’ingegneria e le scienze�, Apogeo Verranno inoltre forniti articoli scientifici da cui prendere spunto per la preparazione delle tesine.

The student will acquire knowledge of the basic tools used in machine learning, and an introductory insight on the functioning of the optimization algorithms that form the inner computational “engine� of these tools. The student will gain some experience in visualizing and analyzing labeled and unlabeled high-dimensional data sets and in extracting useful information from them. Complementing a student’s background in statistics, optimization or data mining, this course will help forming the skills of a Junior Data Scientist.

Machine Learning (ML) encompasses a variety of methodologies and computational algorithms, mainly grounded in Bayesian statistics, for extracting information, clustering, detecting patterns, making decisions and predictions or, more generally, understanding phenomena from available data. Classical learning models created in the 1970s, such as Neural Networks, as well as later techniques such as Support Vector Machines (SVM), are witnessing a new wave of resurgence in both theory and applications in the present era of Big Data, where the deluge of unstructured information calls for automated and highly efficient methods of data analysis. Contemporary Machine Learning, in turn, constitutes an essential part of Data Science, an interdisciplinary field for which industry has a global excess demand of experts. In this course, we present the main tools for supervised learning (regression, regularization, classification) and unsupervised learning (clustering, dimensionality reduction), with a focus on the structure and features of the optimization algorithms that are needed to actually solve numerically the learning problems of interest. The course is structured into lectures in class, in which the context and methodologies are explained, and computer lab sessions, in which the students apply the methodologies to real-world data sets and problems coming from various fields, such as finance, business analytics, news, biology, medical diagnosis, etc.

Introduction to Machine Learning. Supervised and unsupervised learning. Parametric and nonparametric models. Classical examples in pattern analysis (e.g., hand-writing recognition). A brief historical perspective. Review of probability theory and statistics. Marginal and conditional distributions. Bayes theorem. Prior, likelihood, posterior. Bayesian inference. Regression problems. Over-fitting. Bias-variance tradeoff. Regularized regression. Linear regression with sparsity-inducing penalties. Ridge regression. The Lasso. The Elastic-Net. Applications (e.g., in image analysis and in computational finance). Logistic regression. Sparse logistic regression and applications (e.g., to text categorization). Algorithms for large-scale regularized regression: First-order methods. Proximal methods. The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). Coordinate descent and block-coordinate descent methods. Classifiers. Neural Networks. Training and the back-propagation algorithm. Maximum margin classifiers. Dual representation. Kernel methods and the Support Vector Machine (SVM). Clustering. K-means clustering. Gaussian mixtures and the Expectation-Minimization (EM) algorithm. Singular value decomposition and the Principal Component Analysis (PCA). Interpretability and the Sparse-PCA. Fast algorithms for Sparse-PCA.

Modalita di esame: Prova scritta (in aula);
Exam: Written test;
... The final exam consists in a written test, which will contain a mixture of methodological questions and numerical exercises (to be executed with pen and paper; use of a calculator is allowed). Depending on the numerosity of the course and exam participants, the written examination may take either the form of an open question test, or the form of a multiple-choice questionar.

Course slides, handouts, and lab practice sheets will be made available to the students via the PoliTo Web portal. Useful reference textbooks are the following ones: C.M. Bishop, Pattern Recognition and Machine Learning, Springer, 2006. J. Friedman, T. Hastie and R. Tibshirani, The Elements of Statistical Learning, Springer, 2009. G.C. Calafiore and L. El Ghaoui, Optimization Models, Cambridge Univ. Press, 2014.

After this course the student is expected to learn some general theoretical methods and techniques to approach problems of out of equilibrium physics. Furthermore, the student will be able to apply these methods to some problems of condensed matter physics, and to explicitly compute quantities that are experimentally measurable.

Out of equilibrium physics is a fascinating and multi-disciplinary topic that finds applications in various contexts, where standard equilibrium approach cannot be applied. The course is meant to provide an introduction to some general theoretical methods and techniques to approach out of equilibrium problems. A particular focus will be devoted to quantum systems, with examples and a hands-on approach to solve problems in condensed matter physics and to discuss their experimental realization.

Part 1: The Classical Boltzmann Equation -Short review of motion of a classical particle under a stochastic force, the phenomenological Drude model for transport. -Construction of the Boltzmann equation, hypothesis, the collision integral, and properties. -Example: The Boltzmann Equation in a random potential. The diffusion limit and Brownian motion Part 2: Towards out of equilibrium physics of quantum systems -Quantum ingredients in the Semi-Classical Boltzmann Equation -The relaxation-time approximation. -Linear response theory, Fluctuation-dissipation theorem. Onsager relations and its applications -Applications to Condensed Matter: derivation of Ohm’s law, scattering processes of electrons in solids, charge and heat transport Part 3: Quantum approaches to out of Equilibrium Quantum Systems A) The coherent limit. Introduction to quantum mesoscopic physics. Scattering matrix formalism, the Landauer-Büttiker approach to compute the electron current in mesoscopic systems. The quantum of resistance. Examples of quantum interference phenomena in nanosystems (Aharonov-Bohm effect, Weak localization). Optional: The noise of electrons. Josephson effect and mesoscopic effects with superconductors B) Effects of dissipation and decoherence. -Phenomenological approach: the T1-T2 model, applications to a two-level system exposed to a time-dependent perturbation Depending on student interest: -Microscopic approach 1: The Density matrix formalism, applications to a two-level system coupled to a thermal bath. Markov limit, reduced density matrix, the Quantum Optical Master Equation. -Microscopic approach 2: The Keldysh-Schwinger formalism and non-equilibrium Green functions. Applications to quantum transport in nanodevices: the Meir-Wingreen formula for conductance, the Coulomb blockade effect.

Modalita di esame: Prova orale obbligatoria;
Exam: Compulsory oral exam;
... The exam is based on an oral test at the blackboard, with questions on 2-3 topics of the material discussed during the course. The first topic can be chosen by the student. The knowledge of the out of equilibrium physics methods is tested by asking the student to illustrate the key concepts, to derive proofs of the main results, and to apply the techniques to some examples and applications discussed during the course.

R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford Univ Press) L. E. Reichl, A modern course in Stitistical Physics (John Wiley & Sons) J. Rammer, Quantum Field Theory of nonequilibrium states (Cambridge Univ. Press) A. Kamenev, Field Theory of Nonequilibrium systems (Cambridge Univ. Press) H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial markets (World Scientific) Haug-Jauho, Quantum kinetics in Transport and optics of semiconductors (Springer)

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Statistical field theory (J-B Fournier. 6 ECTS) While statistical mechanics deals with averages and fluctuations of global quantities, statistical field theory copes with their spatial fluctuations and correlations. By the so-called coarse-graining procedure, one can define for any quantity of interest a field, and, based on symmetry arguments, an effective Hamiltonian that describes its fluctuations. Advanced tools, such as functional integrals, Feynman diagrams, Wick theorem, and renormalization group (RG), are used to perform calculations. Statistical field theory is particularly adapted to study critical phase transitions, in which scale-free fluctuations are responsible for universal behaviours. The course introduces percolation as a geometrical example of critical phenomena, and develops detailed statistical field theoretical calculations for magnetic systems described by the O(n) and φ4 coarse-grained models, which are found in many other areas of condensed and soft matter physics.

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Modalita di esame: Prova scritta (in aula);
Exam: Written test;
... .

Lecture on Phase Transitions and the Renormalization Group, N. Goldenfeld, Frontiers in Physics. Statistical Physics of Fields, M. Kardar, Cambridge University Press. Principles of Condensed Matter Physics, P. Chaikin and T. C. Lubensky, Cambridge University Press.

Stochastic phenomena, the description of which involves randomness, are ubiquitous in physics, chemistry, biology and beyond (information theory, computer science, economy etc.). The goal is to equip students with the essential tools for their modeling and understanding. The course starts with a probability bootstrap with emphasis on large deviations, establishing a bridge with statistical physics approaches. Markov processes, Langevin and Fokker-Planck equations are then presented, together with the stochastic differential equation framework and stochastic calculus (Stratonovich versus Ito-Döblin rules). The linear-response theory features, found along the way, will be put in a broader perspective. The second half of the course is devoted to fluctuation theorems, first passage properties. Follows an introduction to functionals of Brownian motion and the Feynman-Kac formula.

1. Few useful concepts of probability theory 2. Introduction to Langevin equation 3. Markov processes and Master equation 4. Stochastic differential equations 5. Fokker-Planck equation 6. Functionals of stochastic processes

Modalita di esame: Prova scritta (in aula);
Exam: Written test;

READING MATERIALS: • The Fokker-Planck Equation: Methods of Solutions and Applications, H. Risken, Springer. • Stochastic processes in physics and chemistry, N. G. van Kampen, Elsevier. • Handbook of stochastic methods, C. W. Gardiner, Springer. STUDY MATERIALS: can be found at https://physics-complex-systems.fr/cursus.html