Field theory. Galois groups and Galois extensions. The Correspondence of Galois. Some applications of the Galois Correspondence: Constructions with ruler and compass, Solubility of polynomial equations.

S. Gabelli, Teoria delle Equazioni e Teoria di Galois. Springer Italia, (2008).

C. Procesi, Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).
I. Stewart, Galois Theory. Chapman and Hall, (1989).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. 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.

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

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst-case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. The transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms concerning alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

1. Phenomenology, models. Examples.

2. Some exactly solvable ODE (Ordinary Differential Equations) classes (equations with separable variables, homogeneous equations, Bernoulli equation, Clairaut equation, exact differential equations, equations with constant coefficients, Hamiltonian equations with one degree of freedom)

3. General theory:
- Existence and uniqueness theorems (Gronwall's lemmas; Picard's theorem, Peano's theorem).
- Existence intervals and maximal solutions.
- Dependence on initial data and parameters.

4. Qualitative analysis of some simple EDO classes. Phase space.

5. Linear systems with constant coefficients. Exponential of matrices and Jordan's normal form theorem. Laplace transform.

6. Linear systems with variable coefficients. Solution spaces. The Wronskian. The periodic cae (Floquet Theory).

7. Periodic solutions and Fourier series.

8. Power series.

9. Stability.

10. Boundary problems for second order equations.

[AA] Shair Ahmad and Antonio Ambrosetti, Differential Equations. A first course on ODE and a brief introduction to PDE
Series: De Gruyter Textbook De Gruyter | 2019 DOI: https://doi.org/10.1515/9783110652864

[S] Schaum's Outline of Differential Equations, 4th Edition 4th Edition 0071824855 · 9780071824859

Classi theory of the Dirichlet problem: mean principle, maximum principle, Green foirmulas, Perron method, potential theory.

Sobolev spaces in one and $n$ dimensions, with the immersion theorems.

Elliptic systems of equations; weak solutions and the Lax-Milgram theorem.

Caccioppoli estimates and $W^{2,2}$ regularity of the solutions of elliptic sustems.

Maximal function of Hardy and Littlewood; a proof of the Lebesgue differentiation theorem; weak $L^p$ spaces and the Marcinkievitz interpolation theorem.

Protter-Weinberger, Maximum principles in Differential Equations.

L. V. Ahlfors, Compes Analysis.

H. Brezis, Analisi Funzionale.

M. Giaquinta, L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmoinic functions and minimal graphs.

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

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

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. 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 - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

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

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269

Taylor, J. - Introduzione all'analisi degli errori : lo studio delle incertezze nelle misure fisiche - Zanichelli Disponibile nella biblioteca Scientifica di Roma Tre

Dubi, A. - Monte Carlo applications in systems engineering - Wiley Disponibile nella biblioteca Scientifica di Roma Tre

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availability

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269
Taylor, J. - An introduction to error analysis - University Science Books Sausalito, California
Dubi, A. - Monte Carlo applications in systems engineering - Wiley

1. Machine learning. Types of learning. Loss functions. Empirical risk minimization. Generalization and overfitting.
2. Mathematical optimization problems. Convex functions. Gradient descent. Stochastic gradient descent.
3. Regression. Linear regression. Basis functions. Feature selection. Regularization.
4. Classification. Generative models. Nearest neighbor. Logistic regression. Support vector machines. Neural networks.
5. Combining models. Decision trees. Boosting.
6. Unsupervised learning. K-means clustering. Hierarchical clustering. Principal component analysis.
7. Application of the methods using the Python language. Examples using the NumPy, Pandas, SciKit-Learn, Keras, and TensorFlow libraries.

J. Watt, R. Borhani, A. Katsaggelos. Machine Learning Refined. Cambridge University Press, 2nd edition, 2020.

1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.

R. Trudeau: "La rivoluzione non Euclidea" Bollati Boringhieri
V, Nikulin, I. Shafarevich "Geometries and groups" Springer ed.

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

Introduction to Systems Biology: What Computational Biology is about; the role of mathematical modelling and bioinformatics; what are the aims and the main problems; what are the theoretical tools used.

Introduction to Molecular and Cell Biology; basic knowledge of genetics, proteomics and cellular processes; ecology and evolution; outline of the availability of open access biological data.

Recall of probability theory: discrete and continuous random variables; distributions; entropy; inference; sampling; estimation of probability; the EM algorithm (Expectation Maximisation).

Recall of information theory: Shannon entropy and other measures of entropy; measures of diversity (Shannon-Wiener index, Simpson index).

Introduction to Stochastic Processes: random walks, Markov chains.

Introduction to Machine Learning and Pattern Recognition: supervised and unsupervised learning; Bayesian methods; non-parametric techniques; Neural Networks; stochastic methods; clustering; k-means.

Sequence analysis: alignment algorithms (e.g., BLAST); Hidden Markov Models (HMM); internet services available.

Recall of Graph Theory: notation; types of graphs; representation; algorithms on graphs; statistical properties of the networks; centrality measures; motifs; network clustering.
Biological Networks: networks of signal transduction; gene regulatory networks; protein-protein interaction networks; metabolic networks; internet services available.

Bio-mathematical models; predictions by theoretical models; continuous models; recall of numerical methods for solving systems of ordinary differential equations; discrete models; spin models (Ising models); Cellular Automata; Boolean networks; Agent-based models; data fitting and parameter estimation; available software.

• E.S. Allman, J.A. Rhodes. Mathematical Models in Biology: An Introduction (2004) Cambridge University Press.

• W.J. Ewens, G.R. Grant. Statistical Methods in Bioinformatics, An Introduction (2005) Springer Verlag.

• R. Durbin, S. Eddy, A. Krogh, G. Mitchison. Biological sequence analysis - Probabilistic models of proteins and nucleic acids (1998) Cambridge University Press.

• B.H. Junker, F. Schreiber (eds). Analysis of biological networks (2008) John Wiley & Sons.

• R.O. Duda, P.E. Hart, D.G. Stork. Pattern Classification (2001) John Wiley & Sons.

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

D.M. Burton, Elementary Number theory, McGraw-Hill, (sesta edizione 2007)
G.H. Hardy e E.M. Wright, An introduction to the theory of numbers, Oxford University Press, (1979)
M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire's Theorem, Banach-Steinhaus Theorem, open map and closed graph theorem, applications.
Weak topologies, closed and convex sets, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)

1. Theory of Elliptic Curves
Weierstrass Equation, The structure of the group on rational points, formulas for the addition and duplication. Generalities on the intersections between lines and curves in P2(K)
Preparatory results for the proof of the associativity of points on elliptic curves. Proof of the associativity of the sum for the points of an elliptic curve. Other equations for elliptic curves, Legendre's equation, Cubic equations, Quartic equations, intersections of two cubic surfaces. The j-invariant, elliptic curve in characteristic 2, Endomorphisms, singular curves, elliptic curves module n.

2. Torsion points
Torsion points, Division polynomials. Weil's pairing

3. Elliptic curve on finite fields
Frobenius endomorphism. The problem of determining the order of the group. Curves on subfields, Legendre's symbols, Point orders, Shanks's "Baby Step, Giant Step" algorithm. Particular families of elliptic curves. Schoof's algorithm.

4. Cryptosystems on Elliptical Curves.
The Discrete Logarithm Problem. Algorithms for calculating the discrete logarithm: Baby-Step Giant-Step and Polig-Hellman. MOV attack. Attack on anomalous curves. Diffie-Hellman Key Exchange. Cryptosystems by Massey Omura and ElGamal. El Gamal Signature Scheme. Cryptosystems on elliptic curves analysis on the factorization problem. A cryptosystem based on Weil coupling. Factorization of internal numbers using elliptic curves. Using Pari.

Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003.

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Functional analysis, H. Bre'zis

- Recap of Topolgical notions: fundamental group of a topological space, covering spaces;
- Introduction to Category Theory and basics of Homological Algebra;
- Simplicial and Singular Homology;
- Cohomology;
- Persistent Homology and applications to Topological Data Analysis.

1. Notes of the lecturer;
2. A. Hatcher: Algebraic Topology (available online);

Introduction to algebraic number theory:
Rings of integers in number fields and unique factorisation of ideals.
Absolute values in number fields.

The Weil Height and the Mahler measure:
Definitions and properties.
The product Formula
Northcott’s Theorem.
Kroneker’s Theorem.

Thue equations:
Thue’s Theorem on diophantine approximation.
Siegel’s Lemma.
Thue equations have a finite number of integer solutions.

Arithmetic dynamics:
(Pre)periodic points.
The canonical height.
Rational functions.

Diophantine equations in roots of unity:
Revision about roots of unity and cyclotomic polynomials.
The Theorem of Ihare-Serre-Tate.

Equidistribution:
Definitions and examples.
Bilu’s Theorem.
Bogomolov’s Conjecture.

Zannier - Lecture notes on Diophantine Analysis
Hindry, Silverman - Diophantine Geometry

Highlights of Linear Algebra
Computations with Large Matrices
Low Rank and Compressed Sensing
Special Matrices
Probability and Statistics
Optimization
Learning from Data

G. Strang,
Linear Algebra and Learning from Data,
Wellesley-Cambridge Press

Complex numbers; some complex maps; the Riemann sphere; fractional-linear transformations; they preserve the cross ratio and the set of circles and straight lines. The apollonius circles. Integral of a complex function along a curve; length of a curve; properties of the integral; indicator function; formal sum of curves. Cauchy's theorem on rectangles and on all curves; Caucgy's formula. Liouville's principle. The identity principle for holomorphic functions; removable singularities; almost uniform convergence and its properties. Morera's lemma; mean value theorem and maximum principle; harmonic functions locally are the real part of holomorphic functions. Hadamard's three circles theorem. Euler's product for the sine. Harmonic functions and the electric potential; mean value theorem for harmonic functions, maximum principle and uniqueness for the Dirichlet problem. Dirichlet's kernel; the functions with the mean value property are harmonic. Schwarz's reflection principle. Laurent series; residues and residue calculus. The argument principle and Rouche's theorem. Holomorphic maps are open; a variant of the argument principle and Lagrange's inversion formula; the almost uniform limit of univalent functions is either univalent or constant. Schwarz's lemma; automorphisms of the disc; hyperbolic metric on the disc and its geodesics; the automorphisms of the disc preserve the hyperbolic metric. Analytic extension and monodromy theorem; some examples of Riemann surfaces. The Riemann mapping theorem; Picard's small theorem.

W. Rudin, Analisi reale e complessa.

J. B. Conway, Functions of one complex variable.

[1]Raoul Bott, Loring W. Tu,Differential forms in algebraic topology.Springer, (1986).
[2]Marco Abate, Francesca Tovena,Geometria Differenziale.Springer, (2011).

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

Additional material provided by the teacher.

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

Additional material provided by the teacher.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Convergence of cluster expansion for the high and low temperature expansions for the pressure of the d-dimensional Ising model
- The exact solution of the 2D Ising model with h=0

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Part II-
Statistical Mechanics models - Stochastic dynamics and applications

Mathematical model of different problems are presented as spread of epidemics,
sample problems, optimisation problems, physical problems, with numerical simulations.
Laboratory exercises are an essential part of the course.
Statistical Mechanics models, as the Ising model, and probability tools, as Markov Chain,
are applied, with references to relative theory.

S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction. In rete

O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52

Dynamical system and Reaction Diffusion problems; morphogenesis.

A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952

Numerical methods for simulating the behavior of statistical systems.
Use of the Julia programming language for parallel simulations.
Structures and algorithms.
Simulation of the Ising model.
Markov chains.
Analysis of the dynamics of Glauber, Metropolis, PCA, spin-flip.
Simulation of the dynamic evolution of a spin system towards the thermodynamic equilibrium.
Probabilistic Cellular Automata (PCA).
Coupling methods.
The Propp-Wilson algorithm.
Neuronal networks.
Simulation of the Hopfield model.

1) The Julia Language
https://docs.julialang.org/en/v1/

2) O. Häggström and O.H.G. M.Finite Markov Chains and Algorithmic Appli-cations.
London Mathematical Society Student Texts. Cambridge UniversityPress, 2002.isbn: 9780521890014.
url:https://books.google.it/books?id=hpLxIJ9LwRgC

3) James Gary Propp and David Bruce Wilson.
“Exact Sampling with Cou-pled Markov Chains and Applications to Statistical Mechanics”.
In:RandomStruct. Algorithms9.1-2 (Aug. 1996),
url:http://dx.doi.org/10.1002/(SICI)1098-2418(199608/09)9:1/2%3C223::AID-RSA14%3E3.0.CO;2-O.

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

Additional material provided by the teacher.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Convergence of cluster expansion for the high and low temperature expansions for the pressure of the d-dimensional Ising model
- The exact solution of the 2D Ising model with h=0

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Part II-
Statistical Mechanics models - Stochastic dynamics and applications

Mathematical model of different problems are presented as spread of epidemics,
sample problems, optimisation problems, physical problems, with numerical simulations.
Laboratory exercises are an essential part of the course.
Statistical Mechanics models, as the Ising model, and probability tools, as Markov Chain,
are applied, with references to relative theory.

S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction. In rete

O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52

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

the course touches on basic algebraic topics:
divisibility, Euclid's algorithm,
continuous fractions, Fibonacci numbers,
combinatorics, straightedge and compass construction, cardinality, lattices and Boole Algebras.

testi consigliati (non da acquistare)
Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
Papick, Algebra Connections
Materiale e dispense online

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.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.

Further reading:
2) K. Huang, Meccanica Statistica, Zanichelli, 1997.
3) L. Peliti, Appunti di Meccanica Statistica, Bollati Boringhieri, 2003.
4) Joel L. Lebowitz, Statistical mechanics: A selective review of two central issues, Reviews of Modern Physics, 71, S346 (1999).

1. Classic Cryptography

- Basic cryptosystems: encryption by substitution, by translation, by permutation, affine cryptosystem, by Vigenère, by Hill. Stream encryption (synchronous and asynchronous), Linear feedback shift registers (LFSR) on finite fields, Autokey cypher. Product cyphers. Basic cryptanalysis: classification of attacks; cryptoanalysis for affine cyphers, for substitution cypher (frequency analysis), for Vigenere cypher: Kasiski test, coincidence index; cryptoanalysis of Hill's cypher and LFSR: algebraic attacks, cube attack.

2. Application of Shannon theory to cryptography

- Security of cyphers: computational security, provable security, unconditional security. Basics of probability: discrete random variables, joint probability, conditional probability, independent random variables, Bayes' theorem. Random variables associated with cryptosystems. Perfect secrecy for encryption systems. Vernam cryptosystem. Entropy. Huffman codes. Spurious Keys and Unicity distance.

3. Block cyphers

- iterative encryption schemes; Substitution-Permutation Networks (SPN); Linear cryptanalysis for SPN: Piling-Up Lemma, linear approximation of S-boxes, linear attacks on S-boxes; Differential cryptanalysis for SPN; Feistel cyphers; DES: description and analysis; AES: description; Notes on finite fields: operations on finite fields, Euclid's generalized algorithm for the computation of the GCD and inverse; Operating modes for block cyphers.

4. Hash functions and codes for message authentication

- Hash functions and data integrity. Safe hash functions: resistance to the pre-image, resistance to the second pre-image, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second pre-image, collision on the problem of the pre-image. Iterated hash functions; the construction of Merkle-Damgard. Safe Hash Algorithm (SHA-1). Authentication Codes (MAC): nested authentication codes (HMAC).

[1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press.
[2] Douglas Stinson, Cryptography: Theory and Practice, 3rd edition, (2006) Chapman and Hall/CRC.
[3] Delfs H., Knebl H., Introduction to Cryptography, (2007) Springer Verlag.

Kerberos e Diameter
IP Security
Web security
Authentication
Intrusion detection
Honey pot
Android security
Cloud security
Big data security
Forensics

Stallings’ Cryptography and Network Security, Seventh Edition

TEACHING MATHEMATICS WITH THE HELP OF A COMPUTER: GEOGEBRA AND MATHEMATICA SOFTWARES.
COMMANDS FOR NUMERICAL AND SYMBOLIC CALCULUS, GRAPHICS VISUALIZATION, PARAMETRIC SURFACES AND CURVES WITH ANIMATIONS IN CHANGING PARAMETERS.
SOLVING PROBLEMS: TRIANGLE'S PROPERTIES IN EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY WITH EXAMPLES, APPROXIMATION OF PI AND OTHER IRRATIONAL NUMBERS, SOLUTIONS OF EQUATIONS AND INEQUALITIES,SYSTEMS OF EQUATIONS, DEFINING AND VISUALIZING GEOMETRICAL LOCI, FUNCTION INTEGRAL AND DERIVATIVES, APPROXIMATION OF SURFACE AREA.

LIST OF PROBLEMS GIVEN IN CLASS WITH VISUALIZATION AND SOLUTIONS WITH THE HELP OF SOFTWARE MATHEMATICA OR GEOGEBRA.

NETWORKS AND GRAPHS
- Graphs, trees and networks
- Centrality measures and degree
- Random graphs, the Erdős and Rényi model

SMALL WORLDS NETWORKS
- Definition of Small World
- Clustering Coefficient
- The Watts-Strogatz model

GENERALISED RANDOM GRAPHS
- Statistical description of networks
- Degree Distributions of real networks
- Generalization of the Erdős–Rényi model
- Radom graphs with power-law degree distributions

GROWING GRAPHS
- Dynamical evolution of random graphs
- The Barabási–Albert model

DEGREE CORRELATIONS
- Correlated networks
- Assortative and Disassortative Networks, "Rich Club" behavior

WEIGHTED NETWORKS
- Beyond purely topological networks: tuning the interactions in a complex system
- The Barrat-Barthélemy-Vespignani model

INTRODUCTION TO DYNAMICAL PROCESSES: THEORY AND SIMULATION

Main text-book:
V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Cambridge University press (2017)

The course also follows selected parts of the book:
A. Barrat, M. Barthelemy, A. Vespignani, "Dynamical processes on complex networks", Cambridge University Press (2008)

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

Field theory. Galois groups and Galois extensions. The Correspondence of Galois. Some applications of the Galois Correspondence: Constructions with ruler and compass, Solubility of polynomial equations.

S. Gabelli, Teoria delle Equazioni e Teoria di Galois. Springer Italia, (2008).

C. Procesi, Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).
I. Stewart, Galois Theory. Chapman and Hall, (1989).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. 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.

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

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst-case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. The transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms concerning alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

- Recap of Topolgical notions: fundamental group of a topological space, covering spaces;
- Introduction to Category Theory and basics of Homological Algebra;
- Simplicial and Singular Homology;
- Cohomology;
- Persistent Homology and applications to Topological Data Analysis.

1. Notes of the lecturer;
2. A. Hatcher: Algebraic Topology (available online);

1. Phenomenology, models. Examples.

2. Some exactly solvable ODE (Ordinary Differential Equations) classes (equations with separable variables, homogeneous equations, Bernoulli equation, Clairaut equation, exact differential equations, equations with constant coefficients, Hamiltonian equations with one degree of freedom)

3. General theory:
- Existence and uniqueness theorems (Gronwall's lemmas; Picard's theorem, Peano's theorem).
- Existence intervals and maximal solutions.
- Dependence on initial data and parameters.

4. Qualitative analysis of some simple EDO classes. Phase space.

5. Linear systems with constant coefficients. Exponential of matrices and Jordan's normal form theorem. Laplace transform.

6. Linear systems with variable coefficients. Solution spaces. The Wronskian. The periodic cae (Floquet Theory).

7. Periodic solutions and Fourier series.

8. Power series.

9. Stability.

10. Boundary problems for second order equations.

[AA] Shair Ahmad and Antonio Ambrosetti, Differential Equations. A first course on ODE and a brief introduction to PDE
Series: De Gruyter Textbook De Gruyter | 2019 DOI: https://doi.org/10.1515/9783110652864

[S] Schaum's Outline of Differential Equations, 4th Edition 4th Edition 0071824855 · 9780071824859

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

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

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. 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 - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

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

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269

Taylor, J. - Introduzione all'analisi degli errori : lo studio delle incertezze nelle misure fisiche - Zanichelli Disponibile nella biblioteca Scientifica di Roma Tre

Dubi, A. - Monte Carlo applications in systems engineering - Wiley Disponibile nella biblioteca Scientifica di Roma Tre

1. Machine learning. Types of learning. Loss functions. Empirical risk minimization. Generalization and overfitting.
2. Mathematical optimization problems. Convex functions. Gradient descent. Stochastic gradient descent.
3. Regression. Linear regression. Basis functions. Feature selection. Regularization.
4. Classification. Generative models. Nearest neighbor. Logistic regression. Support vector machines. Neural networks.
5. Combining models. Decision trees. Boosting.
6. Unsupervised learning. K-means clustering. Hierarchical clustering. Principal component analysis.
7. Application of the methods using the Python language. Examples using the NumPy, Pandas, SciKit-Learn, Keras, and TensorFlow libraries.

J. Watt, R. Borhani, A. Katsaggelos. Machine Learning Refined. Cambridge University Press, 2nd edition, 2020.

1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.

R. Trudeau: "La rivoluzione non Euclidea" Bollati Boringhieri
V, Nikulin, I. Shafarevich "Geometries and groups" Springer ed.

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

Complex numbers; some complex maps; the Riemann sphere; fractional-linear transformations; they preserve the cross ratio and the set of circles and straight lines. The apollonius circles. Integral of a complex function along a curve; length of a curve; properties of the integral; indicator function; formal sum of curves. Cauchy's theorem on rectangles and on all curves; Caucgy's formula. Liouville's principle. The identity principle for holomorphic functions; removable singularities; almost uniform convergence and its properties. Morera's lemma; mean value theorem and maximum principle; harmonic functions locally are the real part of holomorphic functions. Hadamard's three circles theorem. Euler's product for the sine. Harmonic functions and the electric potential; mean value theorem for harmonic functions, maximum principle and uniqueness for the Dirichlet problem. Dirichlet's kernel; the functions with the mean value property are harmonic. Schwarz's reflection principle. Laurent series; residues and residue calculus. The argument principle and Rouche's theorem. Holomorphic maps are open; a variant of the argument principle and Lagrange's inversion formula; the almost uniform limit of univalent functions is either univalent or constant. Schwarz's lemma; automorphisms of the disc; hyperbolic metric on the disc and its geodesics; the automorphisms of the disc preserve the hyperbolic metric. Analytic extension and monodromy theorem; some examples of Riemann surfaces. The Riemann mapping theorem; Picard's small theorem.

W. Rudin, Analisi reale e complessa.

J. B. Conway, Functions of one complex variable.

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire's Theorem, Banach-Steinhaus Theorem, open map and closed graph theorem, applications.
Weak topologies, closed and convex sets, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)

1. Theory of Elliptic Curves
Weierstrass Equation, The structure of the group on rational points, formulas for the addition and duplication. Generalities on the intersections between lines and curves in P2(K)
Preparatory results for the proof of the associativity of points on elliptic curves. Proof of the associativity of the sum for the points of an elliptic curve. Other equations for elliptic curves, Legendre's equation, Cubic equations, Quartic equations, intersections of two cubic surfaces. The j-invariant, elliptic curve in characteristic 2, Endomorphisms, singular curves, elliptic curves module n.

2. Torsion points
Torsion points, Division polynomials. Weil's pairing

3. Elliptic curve on finite fields
Frobenius endomorphism. The problem of determining the order of the group. Curves on subfields, Legendre's symbols, Point orders, Shanks's "Baby Step, Giant Step" algorithm. Particular families of elliptic curves. Schoof's algorithm.

4. Cryptosystems on Elliptical Curves.
The Discrete Logarithm Problem. Algorithms for calculating the discrete logarithm: Baby-Step Giant-Step and Polig-Hellman. MOV attack. Attack on anomalous curves. Diffie-Hellman Key Exchange. Cryptosystems by Massey Omura and ElGamal. El Gamal Signature Scheme. Cryptosystems on elliptic curves analysis on the factorization problem. A cryptosystem based on Weil coupling. Factorization of internal numbers using elliptic curves. Using Pari.

Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003.

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Functional analysis, H. Bre'zis

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

Additional material provided by the teacher.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Convergence of cluster expansion for the high and low temperature expansions for the pressure of the d-dimensional Ising model
- The exact solution of the 2D Ising model with h=0

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Part II-
Statistical Mechanics models - Stochastic dynamics and applications

Mathematical model of different problems are presented as spread of epidemics,
sample problems, optimisation problems, physical problems, with numerical simulations.
Laboratory exercises are an essential part of the course.
Statistical Mechanics models, as the Ising model, and probability tools, as Markov Chain,
are applied, with references to relative theory.

S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction. In rete

O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

Additional material provided by the teacher.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Convergence of cluster expansion for the high and low temperature expansions for the pressure of the d-dimensional Ising model
- The exact solution of the 2D Ising model with h=0

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Part II-
Statistical Mechanics models - Stochastic dynamics and applications

Mathematical model of different problems are presented as spread of epidemics,
sample problems, optimisation problems, physical problems, with numerical simulations.
Laboratory exercises are an essential part of the course.
Statistical Mechanics models, as the Ising model, and probability tools, as Markov Chain,
are applied, with references to relative theory.

S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction. In rete

O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52

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

the course touches on basic algebraic topics:
divisibility, Euclid's algorithm,
continuous fractions, Fibonacci numbers,
combinatorics, straightedge and compass construction, cardinality, lattices and Boole Algebras.

testi consigliati (non da acquistare)
Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
Papick, Algebra Connections
Materiale e dispense online

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.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.

Further reading:
2) K. Huang, Meccanica Statistica, Zanichelli, 1997.
3) L. Peliti, Appunti di Meccanica Statistica, Bollati Boringhieri, 2003.
4) Joel L. Lebowitz, Statistical mechanics: A selective review of two central issues, Reviews of Modern Physics, 71, S346 (1999).

1. Classic Cryptography

- Basic cryptosystems: encryption by substitution, by translation, by permutation, affine cryptosystem, by Vigenère, by Hill. Stream encryption (synchronous and asynchronous), Linear feedback shift registers (LFSR) on finite fields, Autokey cypher. Product cyphers. Basic cryptanalysis: classification of attacks; cryptoanalysis for affine cyphers, for substitution cypher (frequency analysis), for Vigenere cypher: Kasiski test, coincidence index; cryptoanalysis of Hill's cypher and LFSR: algebraic attacks, cube attack.

2. Application of Shannon theory to cryptography

- Security of cyphers: computational security, provable security, unconditional security. Basics of probability: discrete random variables, joint probability, conditional probability, independent random variables, Bayes' theorem. Random variables associated with cryptosystems. Perfect secrecy for encryption systems. Vernam cryptosystem. Entropy. Huffman codes. Spurious Keys and Unicity distance.

3. Block cyphers

- iterative encryption schemes; Substitution-Permutation Networks (SPN); Linear cryptanalysis for SPN: Piling-Up Lemma, linear approximation of S-boxes, linear attacks on S-boxes; Differential cryptanalysis for SPN; Feistel cyphers; DES: description and analysis; AES: description; Notes on finite fields: operations on finite fields, Euclid's generalized algorithm for the computation of the GCD and inverse; Operating modes for block cyphers.

4. Hash functions and codes for message authentication

- Hash functions and data integrity. Safe hash functions: resistance to the pre-image, resistance to the second pre-image, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second pre-image, collision on the problem of the pre-image. Iterated hash functions; the construction of Merkle-Damgard. Safe Hash Algorithm (SHA-1). Authentication Codes (MAC): nested authentication codes (HMAC).

[1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press.
[2] Douglas Stinson, Cryptography: Theory and Practice, 3rd edition, (2006) Chapman and Hall/CRC.
[3] Delfs H., Knebl H., Introduction to Cryptography, (2007) Springer Verlag.

NETWORKS AND GRAPHS
- Graphs, trees and networks
- Centrality measures and degree
- Random graphs, the Erdős and Rényi model

SMALL WORLDS NETWORKS
- Definition of Small World
- Clustering Coefficient
- The Watts-Strogatz model

GENERALISED RANDOM GRAPHS
- Statistical description of networks
- Degree Distributions of real networks
- Generalization of the Erdős–Rényi model
- Radom graphs with power-law degree distributions

GROWING GRAPHS
- Dynamical evolution of random graphs
- The Barabási–Albert model

DEGREE CORRELATIONS
- Correlated networks
- Assortative and Disassortative Networks, "Rich Club" behavior

WEIGHTED NETWORKS
- Beyond purely topological networks: tuning the interactions in a complex system
- The Barrat-Barthélemy-Vespignani model

INTRODUCTION TO DYNAMICAL PROCESSES: THEORY AND SIMULATION

Main text-book:
V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Cambridge University press (2017)

The course also follows selected parts of the book:
A. Barrat, M. Barthelemy, A. Vespignani, "Dynamical processes on complex networks", Cambridge University Press (2008)

Kerberos e Diameter
IP Security
Web security
Authentication
Intrusion detection
Honey pot
Android security
Cloud security
Big data security
Forensics

Stallings’ Cryptography and Network Security, Seventh Edition

TEACHING MATHEMATICS WITH THE HELP OF A COMPUTER: GEOGEBRA AND MATHEMATICA SOFTWARES.
COMMANDS FOR NUMERICAL AND SYMBOLIC CALCULUS, GRAPHICS VISUALIZATION, PARAMETRIC SURFACES AND CURVES WITH ANIMATIONS IN CHANGING PARAMETERS.
SOLVING PROBLEMS: TRIANGLE'S PROPERTIES IN EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY WITH EXAMPLES, APPROXIMATION OF PI AND OTHER IRRATIONAL NUMBERS, SOLUTIONS OF EQUATIONS AND INEQUALITIES,SYSTEMS OF EQUATIONS, DEFINING AND VISUALIZING GEOMETRICAL LOCI, FUNCTION INTEGRAL AND DERIVATIVES, APPROXIMATION OF SURFACE AREA.

LIST OF PROBLEMS GIVEN IN CLASS WITH VISUALIZATION AND SOLUTIONS WITH THE HELP OF SOFTWARE MATHEMATICA OR GEOGEBRA.

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

Field theory. Galois groups and Galois extensions. The Correspondence of Galois. Some applications of the Galois Correspondence: Constructions with ruler and compass, Solubility of polynomial equations.

S. Gabelli, Teoria delle Equazioni e Teoria di Galois. Springer Italia, (2008).

C. Procesi, Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).
I. Stewart, Galois Theory. Chapman and Hall, (1989).

1. Abstract integration theory
Riemann integration theory. The concept of measurability. Step functions. Elementary properties of measures. Arithmetic in [0,∞]. Integration of positive functions. Integration of complex functions. Importance of sets
with null measure.
2. Positive Borel measures
Vector spaces. Topological preliminaries. Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions.
3. L^p spaces
Inequalities and convex functions. L^p spaces. Approximation through continuous functions.
4. Basic theory of Hilbert spaces
Inner products and linear functionals. Dual space of L^2
5. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
6. Complex measures
Total variation. Absolute continuity. Radon-Nykodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem.

"Analisi reale e complessa”, W. Rudin. Bollati Boringhieri.

1. Topological classification of curves and compact surfaces. Triangulations, Euler carachetristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc lenght. 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.

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

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.

Introduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst-case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. The transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms concerning alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

1. Phenomenology, models. Examples.

2. Some exactly solvable ODE (Ordinary Differential Equations) classes (equations with separable variables, homogeneous equations, Bernoulli equation, Clairaut equation, exact differential equations, equations with constant coefficients, Hamiltonian equations with one degree of freedom)

3. General theory:
- Existence and uniqueness theorems (Gronwall's lemmas; Picard's theorem, Peano's theorem).
- Existence intervals and maximal solutions.
- Dependence on initial data and parameters.

4. Qualitative analysis of some simple EDO classes. Phase space.

5. Linear systems with constant coefficients. Exponential of matrices and Jordan's normal form theorem. Laplace transform.

6. Linear systems with variable coefficients. Solution spaces. The Wronskian. The periodic cae (Floquet Theory).

7. Periodic solutions and Fourier series.

8. Power series.

9. Stability.

10. Boundary problems for second order equations.

[AA] Shair Ahmad and Antonio Ambrosetti, Differential Equations. A first course on ODE and a brief introduction to PDE
Series: De Gruyter Textbook De Gruyter | 2019 DOI: https://doi.org/10.1515/9783110652864

[S] Schaum's Outline of Differential Equations, 4th Edition 4th Edition 0071824855 · 9780071824859

Classi theory of the Dirichlet problem: mean principle, maximum principle, Green foirmulas, Perron method, potential theory.

Sobolev spaces in one and $n$ dimensions, with the immersion theorems.

Elliptic systems of equations; weak solutions and the Lax-Milgram theorem.

Caccioppoli estimates and $W^{2,2}$ regularity of the solutions of elliptic sustems.

Maximal function of Hardy and Littlewood; a proof of the Lebesgue differentiation theorem; weak $L^p$ spaces and the Marcinkievitz interpolation theorem.

Protter-Weinberger, Maximum principles in Differential Equations.

L. V. Ahlfors, Compes Analysis.

H. Brezis, Analisi Funzionale.

M. Giaquinta, L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmoinic functions and minimal graphs.

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

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

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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

1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility.
2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems.
3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling.
4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality.
5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.

D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak formo of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. 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 - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

1. Classic Cryptography

- Basic cryptosystems: encryption by substitution, by translation, by permutation, affine cryptosystem, by Vigenère, by Hill. Stream encryption (synchronous and asynchronous), Linear feedback shift registers (LFSR) on finite fields, Autokey cypher. Product cyphers. Basic cryptanalysis: classification of attacks; cryptoanalysis for affine cyphers, for substitution cypher (frequency analysis), for Vigenere cypher: Kasiski test, coincidence index; cryptoanalysis of Hill's cypher and LFSR: algebraic attacks, cube attack.

2. Application of Shannon theory to cryptography

- Security of cyphers: computational security, provable security, unconditional security. Basics of probability: discrete random variables, joint probability, conditional probability, independent random variables, Bayes' theorem. Random variables associated with cryptosystems. Perfect secrecy for encryption systems. Vernam cryptosystem. Entropy. Huffman codes. Spurious Keys and Unicity distance.

3. Block cyphers

- iterative encryption schemes; Substitution-Permutation Networks (SPN); Linear cryptanalysis for SPN: Piling-Up Lemma, linear approximation of S-boxes, linear attacks on S-boxes; Differential cryptanalysis for SPN; Feistel cyphers; DES: description and analysis; AES: description; Notes on finite fields: operations on finite fields, Euclid's generalized algorithm for the computation of the GCD and inverse; Operating modes for block cyphers.

4. Hash functions and codes for message authentication

- Hash functions and data integrity. Safe hash functions: resistance to the pre-image, resistance to the second pre-image, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second pre-image, collision on the problem of the pre-image. Iterated hash functions; the construction of Merkle-Damgard. Safe Hash Algorithm (SHA-1). Authentication Codes (MAC): nested authentication codes (HMAC).

[1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press.
[2] Douglas Stinson, Cryptography: Theory and Practice, 3rd edition, (2006) Chapman and Hall/CRC.
[3] Delfs H., Knebl H., Introduction to Cryptography, (2007) Springer Verlag.

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

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269

Taylor, J. - Introduzione all'analisi degli errori : lo studio delle incertezze nelle misure fisiche - Zanichelli Disponibile nella biblioteca Scientifica di Roma Tre

Dubi, A. - Monte Carlo applications in systems engineering - Wiley Disponibile nella biblioteca Scientifica di Roma Tre

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availability

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269
Taylor, J. - An introduction to error analysis - University Science Books Sausalito, California
Dubi, A. - Monte Carlo applications in systems engineering - Wiley

1. Machine learning. Types of learning. Loss functions. Empirical risk minimization. Generalization and overfitting.
2. Mathematical optimization problems. Convex functions. Gradient descent. Stochastic gradient descent.
3. Regression. Linear regression. Basis functions. Feature selection. Regularization.
4. Classification. Generative models. Nearest neighbor. Logistic regression. Support vector machines. Neural networks.
5. Combining models. Decision trees. Boosting.
6. Unsupervised learning. K-means clustering. Hierarchical clustering. Principal component analysis.
7. Application of the methods using the Python language. Examples using the NumPy, Pandas, SciKit-Learn, Keras, and TensorFlow libraries.

J. Watt, R. Borhani, A. Katsaggelos. Machine Learning Refined. Cambridge University Press, 2nd edition, 2020.

1.Euclidean Geometry; rudiments of history of Greek Mathematics. Ruler and compass constructions. The classical problems. Euclid's Elements.
2. The V Postulate's problem: Posidonius attempt. Equivalent Postulates: Playfair, Wallis, transitivity of parallelism. Saccheri's work. Saccheri's Quadrilaterals. The three hyposes. Saccheri-Lagrange's Theorem and the exclusion of obtuse angle hypothesis. The birth of Non-Euclidean Geometry in Bolyai and Lobatchewaki.
3. Symmetries of the plane: Isometries of the plane and their type. Characterization of an isometry by the image of three points not on a line, Chasles' Theorem. Discrete groups of isometries. Rosettes, Friezes and Wallpaper groups. The Theorem of addiction of the angles. Leonardo's Theorem and the classification of finite groups of isometries. Sketch of the classification of the Frieze groups. Crystallographic restriction's Theorem and the classification of wallpaper groups.
4. Gauss' Geometry; The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Mobius band, Klein bottle. Classification of uniforly discontinuous groups of isometries. Sketch of the proof of the Theorem of classification of locally euclidean geometries
5. Moduli of geometries on the torus and hyperbolic geometry: Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré upper plane model.Lines and distance. What was repugnant to Saccheri and that was not for Aristotle.

R. Trudeau: "La rivoluzione non Euclidea" Bollati Boringhieri
V, Nikulin, I. Shafarevich "Geometries and groups" Springer ed.

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

Introduction to Systems Biology: What Computational Biology is about; the role of mathematical modelling and bioinformatics; what are the aims and the main problems; what are the theoretical tools used.

Introduction to Molecular and Cell Biology; basic knowledge of genetics, proteomics and cellular processes; ecology and evolution; outline of the availability of open access biological data.

Recall of probability theory: discrete and continuous random variables; distributions; entropy; inference; sampling; estimation of probability; the EM algorithm (Expectation Maximisation).

Recall of information theory: Shannon entropy and other measures of entropy; measures of diversity (Shannon-Wiener index, Simpson index).

Introduction to Stochastic Processes: random walks, Markov chains.

Introduction to Machine Learning and Pattern Recognition: supervised and unsupervised learning; Bayesian methods; non-parametric techniques; Neural Networks; stochastic methods; clustering; k-means.

Sequence analysis: alignment algorithms (e.g., BLAST); Hidden Markov Models (HMM); internet services available.

Recall of Graph Theory: notation; types of graphs; representation; algorithms on graphs; statistical properties of the networks; centrality measures; motifs; network clustering.
Biological Networks: networks of signal transduction; gene regulatory networks; protein-protein interaction networks; metabolic networks; internet services available.

Bio-mathematical models; predictions by theoretical models; continuous models; recall of numerical methods for solving systems of ordinary differential equations; discrete models; spin models (Ising models); Cellular Automata; Boolean networks; Agent-based models; data fitting and parameter estimation; available software.

• E.S. Allman, J.A. Rhodes. Mathematical Models in Biology: An Introduction (2004) Cambridge University Press.

• W.J. Ewens, G.R. Grant. Statistical Methods in Bioinformatics, An Introduction (2005) Springer Verlag.

• R. Durbin, S. Eddy, A. Krogh, G. Mitchison. Biological sequence analysis - Probabilistic models of proteins and nucleic acids (1998) Cambridge University Press.

• B.H. Junker, F. Schreiber (eds). Analysis of biological networks (2008) John Wiley & Sons.

• R.O. Duda, P.E. Hart, D.G. Stork. Pattern Classification (2001) John Wiley & Sons.

Selected topics useful for the preparation of the final thesis; this work has to be approved by the thesis supervisor.

To be selected according to the thesis subject

CONGRUENCES AND POLYNOMIALS. LINEAR DIOPHANTINE EQUATIONS. SYSTEMS OF LINEAR CONGRUENCES. POLYNOMIAL CONGRUENCES. POLYNOMIAL CONGRUENCES MOD p: LAGRANGES’S THEOREM. p-ADIC APPROXIMATION. THE EXISTENCE OF PRIMITIVE ROOTS MOD p. QUADRATIC CONGRUENCES. QUADRATIC RESIDUES. THE LEGENDRE SYMBOL. GAUSS’S LEMMA AND THE LAW OF QUADRATIC RECIPROCITY.THE JACOBI SYMBOL. SUMS OF TWO, THREE, FOUR SQUARES. MULTIPLICATIVE FUNCTIONS. THE MÖBIUS INVERSION FORMULA. SOME NONLINEAR DIOPHANTINE EQUATIONS. CONTINUED FRACTIONS.

D.M. Burton, Elementary Number theory, McGraw-Hill, (sesta edizione 2007)
G.H. Hardy e E.M. Wright, An introduction to the theory of numbers, Oxford University Press, (1979)
M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire's Theorem, Banach-Steinhaus Theorem, open map and closed graph theorem, applications.
Weak topologies, closed and convex sets, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986)
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010)
W. Rudin - Functional Analysis - McGraw-Hill (1991)

Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''.
Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length.
Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semi-Eulerian graphs.
Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac.
the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph.
Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem.
Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations.
Euler's formula for planar graphs. The dual of a planar graph.
Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties.
Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces.
Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4-color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem.
Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability.
The Max-Flow Min-Cut theorem and Menger's theorem.
Complexity of algorithms and applications to the theory of graphs.
Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability.
Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids.
Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids.
Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix.
Basis for the space of the cycles and for the space of the cuts of a graph. The Riemann-Roch theorem for graphs.
Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph.
Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications.
Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank.
Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected.
Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.

R. Diestel: Graph theory, Spriger GTM 173.
R. Wilson: Introduction to Graph theory, Prentice Hall.
B. Bollobas: Modern Graph theory, Springer GTM 184.
J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244.
N. Biggs: Algebraic graph theory, Cambridge University Press.
C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207.
J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

1. Theory of Elliptic Curves
Weierstrass Equation, The structure of the group on rational points, formulas for the addition and duplication. Generalities on the intersections between lines and curves in P2(K)
Preparatory results for the proof of the associativity of points on elliptic curves. Proof of the associativity of the sum for the points of an elliptic curve. Other equations for elliptic curves, Legendre's equation, Cubic equations, Quartic equations, intersections of two cubic surfaces. The j-invariant, elliptic curve in characteristic 2, Endomorphisms, singular curves, elliptic curves module n.

2. Torsion points
Torsion points, Division polynomials. Weil's pairing

3. Elliptic curve on finite fields
Frobenius endomorphism. The problem of determining the order of the group. Curves on subfields, Legendre's symbols, Point orders, Shanks's "Baby Step, Giant Step" algorithm. Particular families of elliptic curves. Schoof's algorithm.

4. Cryptosystems on Elliptical Curves.
The Discrete Logarithm Problem. Algorithms for calculating the discrete logarithm: Baby-Step Giant-Step and Polig-Hellman. MOV attack. Attack on anomalous curves. Diffie-Hellman Key Exchange. Cryptosystems by Massey Omura and ElGamal. El Gamal Signature Scheme. Cryptosystems on elliptic curves analysis on the factorization problem. A cryptosystem based on Weil coupling. Factorization of internal numbers using elliptic curves. Using Pari.

Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003.

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Functional analysis, H. Bre'zis

- Recap of Topolgical notions: fundamental group of a topological space, covering spaces;
- Introduction to Category Theory and basics of Homological Algebra;
- Simplicial and Singular Homology;
- Cohomology;
- Persistent Homology and applications to Topological Data Analysis.

1. Notes of the lecturer;
2. A. Hatcher: Algebraic Topology (available online);

Introduction to algebraic number theory:
Rings of integers in number fields and unique factorisation of ideals.
Absolute values in number fields.

The Weil Height and the Mahler measure:
Definitions and properties.
The product Formula
Northcott’s Theorem.
Kroneker’s Theorem.

Thue equations:
Thue’s Theorem on diophantine approximation.
Siegel’s Lemma.
Thue equations have a finite number of integer solutions.

Arithmetic dynamics:
(Pre)periodic points.
The canonical height.
Rational functions.

Diophantine equations in roots of unity:
Revision about roots of unity and cyclotomic polynomials.
The Theorem of Ihare-Serre-Tate.

Equidistribution:
Definitions and examples.
Bilu’s Theorem.
Bogomolov’s Conjecture.

Zannier - Lecture notes on Diophantine Analysis
Hindry, Silverman - Diophantine Geometry

Highlights of Linear Algebra
Computations with Large Matrices
Low Rank and Compressed Sensing
Special Matrices
Probability and Statistics
Optimization
Learning from Data

G. Strang,
Linear Algebra and Learning from Data,
Wellesley-Cambridge Press

Complex numbers; some complex maps; the Riemann sphere; fractional-linear transformations; they preserve the cross ratio and the set of circles and straight lines. The apollonius circles. Integral of a complex function along a curve; length of a curve; properties of the integral; indicator function; formal sum of curves. Cauchy's theorem on rectangles and on all curves; Caucgy's formula. Liouville's principle. The identity principle for holomorphic functions; removable singularities; almost uniform convergence and its properties. Morera's lemma; mean value theorem and maximum principle; harmonic functions locally are the real part of holomorphic functions. Hadamard's three circles theorem. Euler's product for the sine. Harmonic functions and the electric potential; mean value theorem for harmonic functions, maximum principle and uniqueness for the Dirichlet problem. Dirichlet's kernel; the functions with the mean value property are harmonic. Schwarz's reflection principle. Laurent series; residues and residue calculus. The argument principle and Rouche's theorem. Holomorphic maps are open; a variant of the argument principle and Lagrange's inversion formula; the almost uniform limit of univalent functions is either univalent or constant. Schwarz's lemma; automorphisms of the disc; hyperbolic metric on the disc and its geodesics; the automorphisms of the disc preserve the hyperbolic metric. Analytic extension and monodromy theorem; some examples of Riemann surfaces. The Riemann mapping theorem; Picard's small theorem.

W. Rudin, Analisi reale e complessa.

J. B. Conway, Functions of one complex variable.

[1]Raoul Bott, Loring W. Tu,Differential forms in algebraic topology.Springer, (1986).
[2]Marco Abate, Francesca Tovena,Geometria Differenziale.Springer, (2011).

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

Additional material provided by the teacher.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Convergence of cluster expansion for the high and low temperature expansions for the pressure of the d-dimensional Ising model
- The exact solution of the 2D Ising model with h=0

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Part II-
Statistical Mechanics models - Stochastic dynamics and applications

Mathematical model of different problems are presented as spread of epidemics,
sample problems, optimisation problems, physical problems, with numerical simulations.
Laboratory exercises are an essential part of the course.
Statistical Mechanics models, as the Ising model, and probability tools, as Markov Chain,
are applied, with references to relative theory.

S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction. In rete

O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52

Fundamentals of the approximation of Ordinary Differential Equations systems. One-step methods of forward/backward Euler type and their convergence.
Convergence of numerical methods for time-dependent Partial Differential Equations, Lax-Richtmyer theorem.
Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, Lax-Friedrichs.
Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, Rankine-Hougoniot condition. Fundamentals on convergence theory for finite-volume approximations. Monotone finite-volume methods: Godunov, Lax-Friedrichs, Rusanov.
Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems.
The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions.
The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization.
Modelling of incompressible fluids: the Navier-Stokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the Vorticity-Streamfunction formulation.

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press

Additional material provided by the teacher.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Convergence of cluster expansion for the high and low temperature expansions for the pressure of the d-dimensional Ising model
- The exact solution of the 2D Ising model with h=0

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Part II-
Statistical Mechanics models - Stochastic dynamics and applications

Mathematical model of different problems are presented as spread of epidemics,
sample problems, optimisation problems, physical problems, with numerical simulations.
Laboratory exercises are an essential part of the course.
Statistical Mechanics models, as the Ising model, and probability tools, as Markov Chain,
are applied, with references to relative theory.

S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems -
A concrete mathematical introduction. In rete

O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications,
London Mathematical Society-Student Texts 52

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