The objective of Linguaggi di Programmazione course is to introduce main formal language theory concepts and results as well as their application for programming language classification.
Most relevant approaches for syntactic analysis of programming languages are introduced. Learning how to recognize the structure of a programming language and the implementation techniques for the abrstract machine.
Understanding the Object Oriented paradigm together with other non imperative approaches.

[1] Maurizio Gabbrielli, Simone Martini,Programming Languages - Principles and paradigms, 2/ed. McGraw-Hill, (2011).
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, 2 edizione. O’Reilly Media, (2014).
[3] David Parsons, Foundational Java Key Elements and Practical Programming. Springer- Verlag, (2012).
Course Slides provided by the lecturer.

The crisis of classical physics. Waves and particles. State vectors and operators. Measurements, observables and uncertainty relation. The position operator. Translations and momentum. Time evolution and the Schrödinger equation. One-dimensional problems. Parity. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.

Lecture notes available on the course website

J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Zanichelli
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics - Addison-Wesley

§I.Relativistic Field Theory

The Poincaré Group. Symmetries: global vs local. Noether's first and second theorems and conservation laws. The canonical stress-energy and angular momentum tensors. Improvements. Belinfante's argument and symmetric energy-momentum tensor. Local symmetries and conserved quantities.

§II.Gravity as a relativistic field theory

Particles and fields in Special Relativity. Irreps of the Poincaré group: Wigner's induced representation method. Massless particles: ISO(D-2) little group and gauge invariance. From relativistic massless spin-2 particles to full GR. Fierz-Pauli quadratic Lagrangian. Nöther method and non-linear completions. Nöther's construction of Yang-Mills Lagrangian. The transverse-traceless gravitational cubic vertex. Weinberg's Equivalence Principle from relativistic invariance of the S matrix. Spin and the sign of static forces.

§III.Elements of differential geometry

Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Levi-Civita connection. Torsion. Differential forms: definition, wedge product, interior and exterior derivatives, Hodge dual. Lie derivative of forms and Cartan's formula. Yang-Mills theory in the language of forms. Weyl tensor. Riemann and Weyl tensors in various dimensions: counting components for irreps of GL(D). Conformal transformations of the metric tensor. Conformally flat spaces. Conformally coupled scalar fields.

§IV. The Cartan-Weyl formulation of GR and Fermionic couplings

Local inertial frames. The frame field and its relation to the metric field. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Gravity as a gauge theory of the Poincaré algebra. Connection one-forms on the Poincar\'e algebra. Local Poincar\'e transformations. Torsion and curvature over the Poincar\'e algebra. First-order formulation and Cartan-Weyl's action. Relation between gauge transformations and diffeomorphisms. Spinors on curved manifolds. Minimally coupled Fermionic matter. Dirac Lagrangian.

§V. Maximally symmetric spaces

Homogeneous and isotropic spaces. Characterisation of maximally symmetric spaces: curvature constant and signature. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients.

§VI. The Schwarzschild black hole

Spherically symmetric spaces. The Schwarzschild solution. Birkhoff's theorem. Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. Free-fall towards the horizon. The tortoise coordinate. Extension of a space-time. Eddington-Finkelstein coordinates. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal's diagram and eternal black holes. (A)dS-Schwarzschild space-time.

§VII. More general black holes

Conformal diagrams. Event horizons. Reissner-Nordström and Kerr black holes. Black hole thermodynamics.

§VII. Gravitational energy

Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Einstein-Hilbert equations for asymptotically flat metrics. Candidate for gravitational energy-momentum tensor. The superpotential. ADM energy and momentum. Example: ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Generic background with Killing vectors. Quadrupole radiation.

§VIII. Asymptotic symmetries

General notion of asymptotic symmetry group. The example of Maxwell's theory in flat space. Covariant phase space formalism. Asymptotically flat spacetime and Bondi-van der Burg-Metzner-Sachs supertranslations. Applications: soft theorems and memory effects.

Note: some topics may be assigned as homework problems, as an alternative to the oral exam

-Carroll S, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972)
-Wald R, General Relativity (The University of Chicago Press, 1984)

The aim of the course is to provide the student with the general cognitive elements underlying the acquisition, control and monitoring systems of Nuclear and Subnuclear Physics experiments. The course is divided into the following topics:
-Introduction to DAQ
-Parallelism and Pipelining systems
-Derandomization
-DAQ and Trigger
-Data Transmission
-Front End Electronics
-Trigger
-Architecture Computing Systems
-Real Time Systems
-Real Time Operating Systems
-C Language
- VHDL Language
-TCP / IP Network Protocols
-DAQ Architecture
- Event Building
-VME Bus
-Run Control
-Farming
-Data Archiving

Lecture notes prepared by the teacher on the basis of the slides presented and available on the Moodle server:
https://matematicafisica.el.uniroma3.it

MATLAB desktop, command window, workspace, current folder, command history, MATLAB help, windows and preferences. Workspace management, loading/saving variables from/on file. Array Editor, manual editing of variables. Script Editor, basic commands for opening/saving/modifying script files.
Mathematical expressions, numbers and format, variables, display format, variable assignment, mathematical functions as operands, arithmetic operators, mathematical functions as operators, ordering modifiers, conversion functions.
Vectors and bidimensional matrices, building vectors and matrices, loading vectors and matrices, functions for vector/matrix generation (zeros, ones, rand, randn, eye etc.), concatenation, transposition, vector length, matrix dimension, matrix arithmetical operations, element-by-element operations, matrix functions, element-by-element functions, accessing/changing/deleting entries or blocks of matrices.
Norm of vectors and matrices, operator “:”, aggregate functions, indexing of vectors and matrices, single/double index, vectorial index. Boolean variables, relational operators, logical operators, logical expressions on scalars, vectors and matrices, logical indexing.
Multidimensional numerical arrays, characters and strings, function “char”. Cell array, cell array indexing, cell access, access to the cell content, function “cell”. Structure, function “struct”, structure indexing, access to the structure fields.
Polynomials, evaluation of polynomials, sum/difference/product/division of polynomials, polynomial derivation, polynomial roots, polynomials from the roots. Complex numbers, imaginary unit, building complex numbers, Cartesian and polar representation of complex numbers. Numerical sequences and series.
Graphical objects, types and hierarchy, handles. Reading/writing object properties, finding property values, copying/deleting objects. “Figure” objects, “Axes” objects, “Line” objects.
Colours, RGB representation. 2D graphics: function "plot" and "subplot", drawing points and lines on axes, plotting mathematical functions, plotting complex numbers, drawing multiple lines with matrices, plotting 2D parametric curves, “hystogram” function, other useful functions for 2D plots.
Line style, colours, markers, figure saving. 3D graphics: functions “plot3”, “surf” and “mesh”, bidimensional grid generation with “meshgrid”, plotting 3D parametric curves. Examples of 2D and 3D graphics.
MATLAB programming, M-files, script and functions, input/output commands, flux control, loops. Types of functions, primary functions, auxiliary functions, nested functions, anonymous functions, function handles. Global variables, script/function interruption, program debugging and comments.
Functions of functions for solving mathematical problems: graphs of functions, searching for the zeros of a mathematical function, solution of non-linear algebraic systems, definite integral computation, scalar function minimization, multidimensional non-linear constrained/non-constrained optimization, integration of first order Cauchy problems.

Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's construction.

Splitting fields and normal extensions.

Separable, inseparable and purely inseparable extensions. Primitive element theorem.

Galois extensions. Galois group and Galois correspondence for finite extensions.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

1. Introduction to information theory.
Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.

2. Source coding and data compression
Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. Ziv-Lempel compression.

3. Channel coding
Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.

4. Further codes and applications
Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.

David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2004.

1) Theory of Relativity: Special Relativity. Space-time. Four-vectors: relativistic velocity, momentum and energy. General relativity.
2) Quantum mechanics: Crisis of classical physics. The principles of quantum mechanics. Schrödinger equation and quantum systems. New phenomena, developments and interpretations.
3) Particles and Fields: Quantum Field Theory. The elementary constituents of matter. Theory of forces. The Standard Model. Physics beyond the Standard Model.
4) Quantum gravity.

Lecture notes available on the course website

Physical quantities. Intensive and extensive physical quantities. Direct and indirect measurements.
Basic and derived quantities.
Units of measurement. Units of measurement systems. Change of units. Dimensions,
physical principle of homogeneity and dimensional analysis.
Measurement tools. Analogical and Digital Instruments.
Characteristics of the instruments: Range, Threshold, Resolution, Linearity and Sensitivity. Accuracy and Precision.
Uncertainty in measurements. Definition of measurement error. Random errors and
systematic errors. Concept of measurement uncertainty. Causes of uncertainties. Uncertainties of Type
A and Type B.
Graphical analysis of data. Usage of tables and graphs for representation and preliminary analysis
of data without the use of statistical tools. Linear, semi-logarithmic graphs,
Double-logarithmic. Histograms.
Propagation of uncertainties. Uncertainty in indirect measurements. Propagation
of uncertainties for independent quantities. Correlated random variables. Definition of
correlation coefficient. Propagation of uncertainties for correlated quantities.


Laboratory program
- Measurements of fundamental quantities: mass, length, time
- Determination of measurement uncertainty: sensitivity of the instrument,
- Standard deviation in repeated measurements, propagation of uncertainties
- Uncertainty on the average in repeated measurements and dependence on sample size
- Study of the pendulum: verification of the independence of the period from the mass, study of the dependence of the period on the length, measurement of g
- Study of the motion of a cart on the inclined plane, effect of friction, measurement of g
- Static and dynamic study of the elastic constant of a spring
- Measurement of resistances with voltamperometric method, study of a resistive voltage divider
- Study of diffraction
- verification of Snell's law

notes distributed during the classes

CONTENTS OF THE LECTURES: the numbers in round brackets refer to the chapter and section of the textbook adopted.
Kinetic theory of gases. Boltzmann equation and H theorem. (1, Par.2.1,2.2,2.3,2.4)
Maxwell-Boltzmann distribution. (1, Par. 2.5)
Phase space and Liouville theorem. (1, Par. 3.1,3.2)
Gibbs ensembles. Micro canonical ensemble. Definition of entropy. (1, Par. 3.3,3.4)
The ideal gas in the micro canonical ensemble. (1, Par. 3.6)
The equipartition theorem. (1, Par. 3.5)
The canonical ensemble. (1, Par.4.1).
The partition function and the free energy. Fluctuations of energy in the canonical ensemble. (1 Par. 4.4)
The grand canonical ensemble. The grand potential. The ideal gas in the grand canonical ensemble. (1 Par. 4.3).
Fluctuations of the particle number. (1 Par. 4.4)
Classical theory of the linear response and fluctuation-dissipation theorem. (1, Par. 8.4).
Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2).
Johnson-Nyqvist theory of thermal noise. (1 Par. 11.3).
Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4)
Fermi-Dirac and Bose-einstein quantum statistics. ( 1, Par. 7.1)
The Fermi gas. The Sommerfeld expansion and the electron specific heat. (1, Par. 7.2)
The Bose gas. The Bose-Einstein condensation. (1, Par. 7.3)
Quantum theory of black-body radiation. (1, Par. 7.5)

e-Learning Moodle Platform with Supplementary Material

Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.

Topics Part A

• Coordinates and Telescopes
• Elements of Spectroscopy
• Stars and Stellar Evolution • Galaxies
• Active Galactic Nuclei


Program Part A

• Overview

• Celestial coordinates (1.3)

• Telescopes and resolving power (6.1)

• Parallax distance (3.1)

• Flux, brightness, apparent and absolute magnitudes, colors (3.2, 3.3, 3.6)

• The black body (3.4, 3.5)

• Hertzsprung-Russel diagram (8.2)

• Open and globular clusters: position, stellar populations and HR diagram (13.3)

• White dwarfs, Novae and SuperNovae (notes and partly in 15 and 16)

• The classification of galaxies (24.1)

• The rotation curve of galaxies and dark matter (25.3)

• The center of the Galaxy and its Black Hole (25.4)

• Hubble's law and expansion of the Universe (27.2)

• Probability of collision between stars and galaxies (handouts)

• Black Holes: outline of General Relativity (outline 17)

• Active Galactic Nuclei (28.1, 28.2, 28.3)


Topics Part B

• Structure and stellar evolution
• Elements of Spectroscopy
• Distances and expansion of the Universe • Galaxies
• GRB and gravitational waves

Program Part B

• Acretion disks and X-ray emission in Active Galactic Nuclei (28.2)

• Stars of Neutrons and Pulsars (16.6, 16.7)

• Gamma Ray Bursts (handouts)

• Gravitational Waves (lecture notes)

• Spectroscopy: eq. Boltzmann-excitation and Saha-ionization (8.1)

• Spectroscopy: speed, temperature and density measurements (handouts)

• Eq. of star structure, time and Kelvin-Helmholtz instability (11.1-4)

• Nuclear reactions of hydrogen (11.3)

• Jeans mass of gravitational collapse, free-fall time and Initial Mass Function (12.2, 12.3)

• The Milky Way and the local group (25.1, 25.2)

• Metallicity (25.2)

• Transit of Venus and measurement of the Earth-Sun distance (handouts)

• Distance scale (27.1)

• Hubble's law and expansion of the Universe (27.2)

• Local Group, Clusters of Galaxies, large scale structure of the Universe (27.3)

• The Big Bang and the background radiation (29.2 brief notes and lecture notes)

A copy of the lecture notes can be downloaded from the course website.

In brackets, the paragraphs from “An Introduction to Modern Astrophysics, II ed. - B.W. Carrol, D.A. Ostlie - Ed. Pearson, Addison Wesley ”(copies available in the library). The discussion in the course has been simplified compared to what is reported in the text. Alternative text in Italian: Attilio Ferrari, Stars, Galaxies, Universe - Fundamentals of Astrophysics - Ed. Springer

1. The themes of logic
2. Classical logic: propositions, proofs
3. Classical logic: connectives
4. Classical logic: types, variables, quantifiers
5. First-order Classical logic
6. Classes and sets
7. Codes, digits, Boolean algebra
8. Turing Machine
9. Axiomatization and formalization of first-order classical logic
10. Logic and other disciplines

V. Michele Abrusci, LOGICA - Lezioni di primo livello, Quarta edizione, Wolters Kluwer, 2018

Lebesgue integration in Rn, Fourier transform, theory of differential equations

Luigi Chierchia, Analisi matematica due
Reed-Simon

The objective of Linguaggi di Programmazione course is to introduce main formal language theory concepts and results as well as their application for programming language classification.
Most relevant approaches for syntactic analysis of programming languages are introduced. Learning how to recognize the structure of a programming language and the implementation techniques for the abrstract machine.
Understanding the Object Oriented paradigm together with other non imperative approaches.

[1] Maurizio Gabbrielli, Simone Martini,Programming Languages - Principles and paradigms, 2/ed. McGraw-Hill, (2011).
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, 2 edizione. O’Reilly Media, (2014).
[3] David Parsons, Foundational Java Key Elements and Practical Programming. Springer- Verlag, (2012).
Course Slides provided by the lecturer.

The crisis of classical physics. Waves and particles. State vectors and operators. Measurements, observables and uncertainty relation. The position operator. Translations and momentum. Time evolution and the Schrödinger equation. One-dimensional problems. Parity. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.

Lecture notes available on the course website

J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Zanichelli
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics - Addison-Wesley

§I.Relativistic Field Theory

The Poincaré Group. Symmetries: global vs local. Noether's first and second theorems and conservation laws. The canonical stress-energy and angular momentum tensors. Improvements. Belinfante's argument and symmetric energy-momentum tensor. Local symmetries and conserved quantities.

§II.Gravity as a relativistic field theory

Particles and fields in Special Relativity. Irreps of the Poincaré group: Wigner's induced representation method. Massless particles: ISO(D-2) little group and gauge invariance. From relativistic massless spin-2 particles to full GR. Fierz-Pauli quadratic Lagrangian. Nöther method and non-linear completions. Nöther's construction of Yang-Mills Lagrangian. The transverse-traceless gravitational cubic vertex. Weinberg's Equivalence Principle from relativistic invariance of the S matrix. Spin and the sign of static forces.

§III.Elements of differential geometry

Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Levi-Civita connection. Torsion. Differential forms: definition, wedge product, interior and exterior derivatives, Hodge dual. Lie derivative of forms and Cartan's formula. Yang-Mills theory in the language of forms. Weyl tensor. Riemann and Weyl tensors in various dimensions: counting components for irreps of GL(D). Conformal transformations of the metric tensor. Conformally flat spaces. Conformally coupled scalar fields.

§IV. The Cartan-Weyl formulation of GR and Fermionic couplings

Local inertial frames. The frame field and its relation to the metric field. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Gravity as a gauge theory of the Poincaré algebra. Connection one-forms on the Poincar\'e algebra. Local Poincar\'e transformations. Torsion and curvature over the Poincar\'e algebra. First-order formulation and Cartan-Weyl's action. Relation between gauge transformations and diffeomorphisms. Spinors on curved manifolds. Minimally coupled Fermionic matter. Dirac Lagrangian.

§V. Maximally symmetric spaces

Homogeneous and isotropic spaces. Characterisation of maximally symmetric spaces: curvature constant and signature. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients.

§VI. The Schwarzschild black hole

Spherically symmetric spaces. The Schwarzschild solution. Birkhoff's theorem. Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. Free-fall towards the horizon. The tortoise coordinate. Extension of a space-time. Eddington-Finkelstein coordinates. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal's diagram and eternal black holes. (A)dS-Schwarzschild space-time.

§VII. More general black holes

Conformal diagrams. Event horizons. Reissner-Nordström and Kerr black holes. Black hole thermodynamics.

§VII. Gravitational energy

Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Einstein-Hilbert equations for asymptotically flat metrics. Candidate for gravitational energy-momentum tensor. The superpotential. ADM energy and momentum. Example: ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Generic background with Killing vectors. Quadrupole radiation.

§VIII. Asymptotic symmetries

General notion of asymptotic symmetry group. The example of Maxwell's theory in flat space. Covariant phase space formalism. Asymptotically flat spacetime and Bondi-van der Burg-Metzner-Sachs supertranslations. Applications: soft theorems and memory effects.

Note: some topics may be assigned as homework problems, as an alternative to the oral exam

-Carroll S, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972)
-Wald R, General Relativity (The University of Chicago Press, 1984)

The aim of the course is to provide the student with the general cognitive elements underlying the acquisition, control and monitoring systems of Nuclear and Subnuclear Physics experiments. The course is divided into the following topics:
-Introduction to DAQ
-Parallelism and Pipelining systems
-Derandomization
-DAQ and Trigger
-Data Transmission
-Front End Electronics
-Trigger
-Architecture Computing Systems
-Real Time Systems
-Real Time Operating Systems
-C Language
- VHDL Language
-TCP / IP Network Protocols
-DAQ Architecture
- Event Building
-VME Bus
-Run Control
-Farming
-Data Archiving

Lecture notes prepared by the teacher on the basis of the slides presented and available on the Moodle server:
https://matematicafisica.el.uniroma3.it

MATLAB desktop, command window, workspace, current folder, command history, MATLAB help, windows and preferences. Workspace management, loading/saving variables from/on file. Array Editor, manual editing of variables. Script Editor, basic commands for opening/saving/modifying script files.
Mathematical expressions, numbers and format, variables, display format, variable assignment, mathematical functions as operands, arithmetic operators, mathematical functions as operators, ordering modifiers, conversion functions.
Vectors and bidimensional matrices, building vectors and matrices, loading vectors and matrices, functions for vector/matrix generation (zeros, ones, rand, randn, eye etc.), concatenation, transposition, vector length, matrix dimension, matrix arithmetical operations, element-by-element operations, matrix functions, element-by-element functions, accessing/changing/deleting entries or blocks of matrices.
Norm of vectors and matrices, operator “:”, aggregate functions, indexing of vectors and matrices, single/double index, vectorial index. Boolean variables, relational operators, logical operators, logical expressions on scalars, vectors and matrices, logical indexing.
Multidimensional numerical arrays, characters and strings, function “char”. Cell array, cell array indexing, cell access, access to the cell content, function “cell”. Structure, function “struct”, structure indexing, access to the structure fields.
Polynomials, evaluation of polynomials, sum/difference/product/division of polynomials, polynomial derivation, polynomial roots, polynomials from the roots. Complex numbers, imaginary unit, building complex numbers, Cartesian and polar representation of complex numbers. Numerical sequences and series.
Graphical objects, types and hierarchy, handles. Reading/writing object properties, finding property values, copying/deleting objects. “Figure” objects, “Axes” objects, “Line” objects.
Colours, RGB representation. 2D graphics: function "plot" and "subplot", drawing points and lines on axes, plotting mathematical functions, plotting complex numbers, drawing multiple lines with matrices, plotting 2D parametric curves, “hystogram” function, other useful functions for 2D plots.
Line style, colours, markers, figure saving. 3D graphics: functions “plot3”, “surf” and “mesh”, bidimensional grid generation with “meshgrid”, plotting 3D parametric curves. Examples of 2D and 3D graphics.
MATLAB programming, M-files, script and functions, input/output commands, flux control, loops. Types of functions, primary functions, auxiliary functions, nested functions, anonymous functions, function handles. Global variables, script/function interruption, program debugging and comments.
Functions of functions for solving mathematical problems: graphs of functions, searching for the zeros of a mathematical function, solution of non-linear algebraic systems, definite integral computation, scalar function minimization, multidimensional non-linear constrained/non-constrained optimization, integration of first order Cauchy problems.

Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's construction.

Splitting fields and normal extensions.

Separable, inseparable and purely inseparable extensions. Primitive element theorem.

Galois extensions. Galois group and Galois correspondence for finite extensions.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available from the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available from the course page

Slides of the lessons, available from the course page

Branching processes, introduction to Sigma-algebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pi-systems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. Borel-Cantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 0-1 law.
Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws.
Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers.
Doob's inequalities for martingales and sub-martingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.

D. Williams, Probability with martingales
R. Durrett, Probability: Theory and examples

Evolution equations of Mathematical Physics: transport, wave and heat equations. Introduction to quantum mechanics. Fourier transform

[B] P. Butta', Note del corso di Fisica Matematica
[Cr] W. Craig, A course on Partial Differential Equations
[L1] V. Lubicz, Apppunti di Meccanica Quantistica

Introduction to the study of partial derivative equations in mathematical physics. Wave equation: transport; wavefront; conservation laws; Duhamel's principle. Heat equation: heat core; maximum principle; entropy. Introduction to quantum mechanics: historical background, free Schroedinger equation, Stern-Garlach experiment, postulates of quantum mechanics, properties of Hilbert spaces, harmonic oscillator, hydrogen atom.

1. Introduction to information theory.
Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.

2. Source coding and data compression
Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. Ziv-Lempel compression.

3. Channel coding
Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.

4. Further codes and applications
Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.

David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2004.

1) Theory of Relativity: Special Relativity. Space-time. Four-vectors: relativistic velocity, momentum and energy. General relativity.
2) Quantum mechanics: Crisis of classical physics. The principles of quantum mechanics. Schrödinger equation and quantum systems. New phenomena, developments and interpretations.
3) Particles and Fields: Quantum Field Theory. The elementary constituents of matter. Theory of forces. The Standard Model. Physics beyond the Standard Model.
4) Quantum gravity.

Lecture notes available on the course website

Physical quantities. Intensive and extensive physical quantities. Direct and indirect measurements.
Basic and derived quantities.
Units of measurement. Units of measurement systems. Change of units. Dimensions,
physical principle of homogeneity and dimensional analysis.
Measurement tools. Analogical and Digital Instruments.
Characteristics of the instruments: Range, Threshold, Resolution, Linearity and Sensitivity. Accuracy and Precision.
Uncertainty in measurements. Definition of measurement error. Random errors and
systematic errors. Concept of measurement uncertainty. Causes of uncertainties. Uncertainties of Type
A and Type B.
Graphical analysis of data. Usage of tables and graphs for representation and preliminary analysis
of data without the use of statistical tools. Linear, semi-logarithmic graphs,
Double-logarithmic. Histograms.
Propagation of uncertainties. Uncertainty in indirect measurements. Propagation
of uncertainties for independent quantities. Correlated random variables. Definition of
correlation coefficient. Propagation of uncertainties for correlated quantities.


Laboratory program
- Measurements of fundamental quantities: mass, length, time
- Determination of measurement uncertainty: sensitivity of the instrument,
- Standard deviation in repeated measurements, propagation of uncertainties
- Uncertainty on the average in repeated measurements and dependence on sample size
- Study of the pendulum: verification of the independence of the period from the mass, study of the dependence of the period on the length, measurement of g
- Study of the motion of a cart on the inclined plane, effect of friction, measurement of g
- Static and dynamic study of the elastic constant of a spring
- Measurement of resistances with voltamperometric method, study of a resistive voltage divider
- Study of diffraction
- verification of Snell's law

notes distributed during the classes

CONTENTS OF THE LECTURES: the numbers in round brackets refer to the chapter and section of the textbook adopted.
Kinetic theory of gases. Boltzmann equation and H theorem. (1, Par.2.1,2.2,2.3,2.4)
Maxwell-Boltzmann distribution. (1, Par. 2.5)
Phase space and Liouville theorem. (1, Par. 3.1,3.2)
Gibbs ensembles. Micro canonical ensemble. Definition of entropy. (1, Par. 3.3,3.4)
The ideal gas in the micro canonical ensemble. (1, Par. 3.6)
The equipartition theorem. (1, Par. 3.5)
The canonical ensemble. (1, Par.4.1).
The partition function and the free energy. Fluctuations of energy in the canonical ensemble. (1 Par. 4.4)
The grand canonical ensemble. The grand potential. The ideal gas in the grand canonical ensemble. (1 Par. 4.3).
Fluctuations of the particle number. (1 Par. 4.4)
Classical theory of the linear response and fluctuation-dissipation theorem. (1, Par. 8.4).
Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2).
Johnson-Nyqvist theory of thermal noise. (1 Par. 11.3).
Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4)
Fermi-Dirac and Bose-einstein quantum statistics. ( 1, Par. 7.1)
The Fermi gas. The Sommerfeld expansion and the electron specific heat. (1, Par. 7.2)
The Bose gas. The Bose-Einstein condensation. (1, Par. 7.3)
Quantum theory of black-body radiation. (1, Par. 7.5)

e-Learning Moodle Platform with Supplementary Material

Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.

Topics Part A

• Coordinates and Telescopes
• Elements of Spectroscopy
• Stars and Stellar Evolution • Galaxies
• Active Galactic Nuclei


Program Part A

• Overview

• Celestial coordinates (1.3)

• Telescopes and resolving power (6.1)

• Parallax distance (3.1)

• Flux, brightness, apparent and absolute magnitudes, colors (3.2, 3.3, 3.6)

• The black body (3.4, 3.5)

• Hertzsprung-Russel diagram (8.2)

• Open and globular clusters: position, stellar populations and HR diagram (13.3)

• White dwarfs, Novae and SuperNovae (notes and partly in 15 and 16)

• The classification of galaxies (24.1)

• The rotation curve of galaxies and dark matter (25.3)

• The center of the Galaxy and its Black Hole (25.4)

• Hubble's law and expansion of the Universe (27.2)

• Probability of collision between stars and galaxies (handouts)

• Black Holes: outline of General Relativity (outline 17)

• Active Galactic Nuclei (28.1, 28.2, 28.3)


Topics Part B

• Structure and stellar evolution
• Elements of Spectroscopy
• Distances and expansion of the Universe • Galaxies
• GRB and gravitational waves

Program Part B

• Acretion disks and X-ray emission in Active Galactic Nuclei (28.2)

• Stars of Neutrons and Pulsars (16.6, 16.7)

• Gamma Ray Bursts (handouts)

• Gravitational Waves (lecture notes)

• Spectroscopy: eq. Boltzmann-excitation and Saha-ionization (8.1)

• Spectroscopy: speed, temperature and density measurements (handouts)

• Eq. of star structure, time and Kelvin-Helmholtz instability (11.1-4)

• Nuclear reactions of hydrogen (11.3)

• Jeans mass of gravitational collapse, free-fall time and Initial Mass Function (12.2, 12.3)

• The Milky Way and the local group (25.1, 25.2)

• Metallicity (25.2)

• Transit of Venus and measurement of the Earth-Sun distance (handouts)

• Distance scale (27.1)

• Hubble's law and expansion of the Universe (27.2)

• Local Group, Clusters of Galaxies, large scale structure of the Universe (27.3)

• The Big Bang and the background radiation (29.2 brief notes and lecture notes)

A copy of the lecture notes can be downloaded from the course website.

In brackets, the paragraphs from “An Introduction to Modern Astrophysics, II ed. - B.W. Carrol, D.A. Ostlie - Ed. Pearson, Addison Wesley ”(copies available in the library). The discussion in the course has been simplified compared to what is reported in the text. Alternative text in Italian: Attilio Ferrari, Stars, Galaxies, Universe - Fundamentals of Astrophysics - Ed. Springer

1. The themes of logic
2. Classical logic: propositions, proofs
3. Classical logic: connectives
4. Classical logic: types, variables, quantifiers
5. First-order Classical logic
6. Classes and sets
7. Codes, digits, Boolean algebra
8. Turing Machine
9. Axiomatization and formalization of first-order classical logic
10. Logic and other disciplines

V. Michele Abrusci, LOGICA - Lezioni di primo livello, Quarta edizione, Wolters Kluwer, 2018

Lebesgue integration in Rn, Fourier transform, theory of differential equations

Luigi Chierchia, Analisi matematica due
Reed-Simon