Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).

http://ricerca.mat.uniroma3.it/users/valerio/ge310_1819.html

http://ricerca.mat.uniroma3.it/users/valerio/GE310/prog18.pdf

- Classical theory of algebraic varieties in affine spaces over algebraically closed fields.
- Local geometry, normalization.
- Divisors, linear systems and morphisms of projective varieties.

Algebraic Geometry texts:
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1994.
J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1992.


Algebra texts:
* M. Artin, Algebra, Bollati Boringhieri 1997.
* M.F. Atiyah, I.G. Mac Donald, Introduzione all'algebra commutativa, Feltrinelli 1991.

Basic Definitions. Connected graphs. Eulerian graphs
Trees. Rooted trees. Spanning trees.
Cycle space. Cut space. Cyclomatic number.
Bipartite graphs. Matchings. Marriage theorem.
Existence of 1-factors and k-factors.
Connectivity. Structure of 2-connected and 3-connected graphs.
Hamiltonian graphs
Planar and Plane Graphs. Euler formula. Triangulations
Colourings.

R. Diestel. GRAPH THEORY. Springer GTM

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
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. 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.

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

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, 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. 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: 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. Transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms with respect to 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., COMPLEXITE' ET DECIDABILITE'. 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).

Part 1: Some preliminary notions.
Order relations and trees, inductive definitions, proofs by induction, axiom of choice and Kőnig's lemma.

Part 2: Provability and satisyability.
First order formal language: alphabet, terms, formulas, sequents. Structures for first order languages: structures, terms and formulas with parameters in a structure, value of terms, formulas and sequents. The calculus of sequents for first order logic: Gentzen's LK. Derivable sequents and derivations. Correctness of the rules of LK. Canonical analysis and fundamental theorem: construction of the canonical analysis (with and without cuts) and proof of the fundamental theorem of the canonical analysis. Consequences of the fundamental theorem: completeness theorem, compactness theorem, eliminability of cuts, L"owenheim-Skolem's theorem.

Part 3: Towards proof-theory: the cut-elimination theorem.
The cut-elimination procedure. Definition of the elementary steps of cut-elimination. First proof strategy (big reduction steps). Second proof strategy (reversion of derivations). The complexity of the cut-elimination procedure (sketch). Some immediate consequences of the cut-elimination theorem.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).

Towards model theory: some consequences of the compactness theorem.
Proof of the compactness theorem for languages of any cardinality. Languages with equality. The compactness theorem for languages with equality. Correctness and completeness for languages with equality. L"owenheim-Skolem's theorem for (denumerable) languages with equality. The limits of the expressive power of first order languages. Elementary equivalence, substructures, elementary substructures. Isomorphsims and elementary equivalence. The notion of substructure. Elementary substructures and diagrams. The preservation theorems. Generalisations of the L"owenheim-Skolem's theorem. Completeness of a theory.

Logic and Arithmetic: incompleteness

Decidability and fundamental results of recursion theory: primitive recursive functions and elementary functions, Ackermann's function and the (partial) recursive functions, arithmetical hierarchy and representation (in N) of recursive functions, arithmetization of syntax, fundamental theorems of recursion theory, decidability, semi-decidability, undecidability.

Peano arithmetic: Peano's axioms, the models of (first order) Peano arithmetic, the representable functions in (first order) Peano arithmetic, incompleteness and undecidability.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

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

Introductory example: the branching process.

Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.

Integration. Expected value.
Monotone convergence and the dominated convergence theorem.

Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.

Product measures. Fubini's theorem. Joint laws.

Conditional expectation with respect to a sub sigma-algebra.

Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.

Convergence in distribution and the central limit theorem.

D. Williams, Probability with martingales. Cambridge University Press, (1991).

R. Durrett, Probability: Theory and Examples. Thomson, (2000).

Introduction to statistics (1) (2): data collection and descriptive statistics, statistical inference and probabilistic models, population and sample, short history of statistics, sample and census survey, sample survey, sampling techniques, simple random sampling, problems; Descriptive statistics (1) (3): organization and description of data, tables and graphs of absolute and relative frequencies, grouping of data, histograms, olives, steam and leaf diagrams, the quantities that summarize the data, median average and sample fashion, sample variance and standard deviation, sample percentiles and box plots, Chebyshev inequality, normal samples, bivariate data set and sample correlation coefficient, problems; Parametric estimation (1): maximum likelihood estimators, evaluation of the efficiency of point estimators, confidence intervals for the average of a normal distribution with known variance, confidence intervals for the average of a normal distribution with unknown variance confidence intervals for the variance of a normal distribution, confidence intervals for the difference between the means of two normal distributions, approximate confidence intervals for the average of a Bernoulli distribution, problems; Hypothesis testing (1): significance levels, the verification of hypothesis on the average of a normal population, the case in which the variance is known, the case of unknown variance and the t test, checks whether two normal populations have the same average, the case of unknown but equal variances, the testing of hypotheses on a Bernoulli population, confidence intervals for the mean and bilateral tests, independence tests and contingency tables, proble me; Regression (4): estimation of regression parameters with the least squares method, a statistical solution for BLUE estimators, assumptions on the distribution of the model, estimate and hypothesis test for the regression parameters, coefficient of determination, estimate and forecast for a specific value of the explanatory variable, least squares lost (1), problems; Multiple regression (1), (5): estimate of the regression parameters with the least squares method, assumptions on the distribution of the model, estimate and hypothesis test for the regression parameters coefficient of multiple determination, prediction of future answers, problems ; Applications with R (6): examples for the sciences in R code.

(1) Probabilità e statistica, S. M. Ross, Apogeo - Maggioli Editore, 2015 (2) Lezioni di Statistica descrittiva, L. Pieraccini, A. Naccarato, Giappichelli Editore, 2003 (3) Statistica aziendale, B. Bracalente, M. Cossignani, A. Mulas, McGraw-Hill Editore, 2009 (4) Statistical Inference, G. Casella, R. Berger, Duxbury Advanced Series, 2002 (5) Econometrica, J. Johnston , Franco Angeli, 2001 (6) Introductory Statistics with R, P. Dalgaard, Springer, 2008

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) Notes_Convection_Diffusion.pdf
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)

Brief introduction to Julia language. Introduction to parallel architectures, Principle of parallel and distributed programming with Julia. Primitives of communication on synchronization: MPI paradigm. Languages based on directives: OpenMP. Performance metrics of parallel programs. Matrix operations and dense linear systems: introduction to BLAS, LAPACK, scaLAPACK. Sparse linear systems: introduction to CombBLAS, GraphBLAS. Collaborative development of course projects: heart-quake simulations; parallel LAR.

Blaise N. Barney, HPC Training Materials, per gentile concessione del Lawrence Livermore National Laboratory's Computational Training Center

[-] Abstract Machines. Interpreters and Compilers.
[-] Language constructs.
[-] Object Oriented Programming
[-] Functional Programming

GABBRIELLI, M., MARTINI, S., PROGRAMMING LANGUAGES: PRINCIPLES AND PARADIGMS. MCGRAW-HILL, Seconda Edizione

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.

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. Euclidean Geometry
Rudiments of Greek mathematics history. Ruler and compass constructions. Classical problems. The Elements. Axioms, definitions and postulates of Book I. Theorems I-XXVIII with proofs. Theorems XXIX, XXX, XXXI, XXXII: the role of V Postulate.
2. The question of V Postulate
The attempt by Posidonio. Equivalent propositions: Playfair, Wallis, transitivity of parallelism. Saccheri's quadrilateral. Quadrilateri di Saccheri. Saccheri-Lagrange theorem and the exclusion of the obtuse angle hypothesis. The non-euclidean geometries of Bolyai and Lobachevski.
3. Isometries of the plane
Even and odd isometries. Characterisation of an isometry by the image of three points not on a line. Chasles' Theorem. Products of reflections. Discrete groups of isometries. Finite groups, friezes, crystals. The theorem of addition of the angle. Leonardo's Theorem and the characterisation of finite groups.
Sketch of proof of the theorem of classification of frieze groups. Crystallographic restriction Theorem and the classification of wallpaper groups.
4. The geometry after Gauss
The geometry on the Sphere. Locally euclidean geometries. Uniformly discontinuous groups of isometries. The Torus, Moebius strip and the Klein bottle. Classification of uniformly discontinuous groups Sketch of the proof of the Theorem of Classification of locally euclidean geometries.
5. Geometries on the Torus and the Hyperbolic geometry
Similar geometries. Similar geometries on the Torus. The modular figure. Poincaré Half plane model. Lines and distance. What is repugnant for Saccheri, but not for Aristotle

R. Trudeau: La Rivoluzione non euclidea. Bollati Boringhieri ed, 1991

V. Nikulin, I. Shafarevich: Geometries and groups. Springer ed, 1987

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

Introductory example: the branching process.

Measure theory. Existence and uniqueness theorems for probability measures.
Borel-Cantelli lemma 1. Random variables. Independence.
Borel-Cantelli lemma 2. Kolmogorv's 0/1 law.

Integration. Expected value.
Monotone convergence and the dominated convergence theorem.

Inequalities: Markov, Jensen, Hoelder, Cauchy-Schwarz.
Laws of large numbers.

Product measures. Fubini's theorem. Joint laws.

Conditional expectation with respect to a sub sigma-algebra.

Martingales. Stopping times. Optional stopping and applications.
Hotting times. Convergence theorem for martingales bounded in L^1 and L^2.
Examples, Kolmogorov's strong law of large numbers.

Convergence in distribution and the central limit theorem.

D. Williams, Probability with martingales. Cambridge University Press, (1991).

R. Durrett, Probability: Theory and Examples. Thomson, (2000).

Introduction to statistics (1) (2): data collection and descriptive statistics, statistical inference and probabilistic models, population and sample, short history of statistics, sample and census survey, sample survey, sampling techniques, simple random sampling, problems; Descriptive statistics (1) (3): organization and description of data, tables and graphs of absolute and relative frequencies, grouping of data, histograms, olives, steam and leaf diagrams, the quantities that summarize the data, median average and sample fashion, sample variance and standard deviation, sample percentiles and box plots, Chebyshev inequality, normal samples, bivariate data set and sample correlation coefficient, problems; Parametric estimation (1): maximum likelihood estimators, evaluation of the efficiency of point estimators, confidence intervals for the average of a normal distribution with known variance, confidence intervals for the average of a normal distribution with unknown variance confidence intervals for the variance of a normal distribution, confidence intervals for the difference between the means of two normal distributions, approximate confidence intervals for the average of a Bernoulli distribution, problems; Hypothesis testing (1): significance levels, the verification of hypothesis on the average of a normal population, the case in which the variance is known, the case of unknown variance and the t test, checks whether two normal populations have the same average, the case of unknown but equal variances, the testing of hypotheses on a Bernoulli population, confidence intervals for the mean and bilateral tests, independence tests and contingency tables, proble me; Regression (4): estimation of regression parameters with the least squares method, a statistical solution for BLUE estimators, assumptions on the distribution of the model, estimate and hypothesis test for the regression parameters, coefficient of determination, estimate and forecast for a specific value of the explanatory variable, least squares lost (1), problems; Multiple regression (1), (5): estimate of the regression parameters with the least squares method, assumptions on the distribution of the model, estimate and hypothesis test for the regression parameters coefficient of multiple determination, prediction of future answers, problems ; Applications with R (6): examples for the sciences in R code.

(1) Probabilità e statistica, S. M. Ross, Apogeo - Maggioli Editore, 2015 (2) Lezioni di Statistica descrittiva, L. Pieraccini, A. Naccarato, Giappichelli Editore, 2003 (3) Statistica aziendale, B. Bracalente, M. Cossignani, A. Mulas, McGraw-Hill Editore, 2009 (4) Statistical Inference, G. Casella, R. Berger, Duxbury Advanced Series, 2002 (5) Econometrica, J. Johnston , Franco Angeli, 2001 (6) Introductory Statistics with R, P. Dalgaard, Springer, 2008

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) Notes_Convection_Diffusion.pdf
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)

The programme has the following topics:
- Introduction to DAQ systems
- Parallelisation and Pipelining
- De-randomisation
- DAQ and Trigger
- Data Transmission
- Front End Electronics
- Trigger
- Computing Systems Architectures
- Real Time Systems
- Real Time Operating Systems
- C Language
- TCP/IP Network Protocol
- DAQ Architectures
- Event Building
- VME Bus
- Run Control
- Farming
- Data Archiving

The course of Algorithms in cryptography is devoted to the study of encryption systems and their properties. In particular, we will study methods and algorithms developed to verify security level of cryptosystems, both from the point of view of formal verification (in the context of protocols) and from the point of view of cryptanalysis. Required as prerequisites are a basic level of computer knowledge of a Unix-like operating system (eg Linux) and programming in C or Java.

[1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press, in inglese;
[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.

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

To be selected according the thesis subject

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

The program will be selected upon agreement between the student, the internal supervisor and the subject offering the stage.

Textbooks will be selected according to the specific topic

A selected topic among computer sciences and/or scientific computing.

Textbooks will be chosen according the selected topic

Professional skills among computer sciences and/or scientific computing.

Textbooks will be chosen according the selected topic

1. Modules
Modules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.

2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.

3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.

4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings

5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.

R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972

1. Introduction. Reviews on Number Fields. Traces, Norms and Discriminant. Rings of integers.
2. Commutative Algebra. Noetherian rings and Dedekind rings. The ζ function of Dedekind.
3. Algebra. Finitely generated groups and reviews of Theory of Galois. Lattices.
4. Discriminant and Ramification. The Minkowsi Theorem. Dirichlet's Theorem. The Group of classes and the finiteness of the class group.
5. The class number formula.

Schoof, R., Algebraic Number Theory. dispense Università di Roma Tor Vergata, http://www.mat.uniroma2.it/ ̃eal/moonen.pdf, (2003).
Milne, J., Algebraic Number Theory. Lecture Notes, http://www.jmilne.org/math/CourseNotes/ANT.pdf, (2017).
Marcus, D, Number fields, 3rd Ed. Springer-Verlag, (1977)

Broad program:

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytical and geometric form, consequences.
First and second category spaces, Baire theorem, Banach-Steinhaus theorem, open map and closed graph, applications.
Weak, closed and convex topologies, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in one dimension, 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)

The complex field. Holomorphic functions; Cauchy-Riemann equations.
Series and Abel's theorem. Exponential and logarithms.
Elementary conformal mappings. Complex integration; Cauchy's theorem; Cauchy's formula.
Local properties of holomorphic functions (singularities, zeroes and poles; local mapping theorem and
maximum principle).
Residues.
Harmonic functions.
Series expansions (Weierstrass' theorem, Taylor's series).
Partial fractions and infinite products.
Supplementary arguments (depending on time): entire functions and Hadamard's theorem. Riemann zeta function.
Riemann mapping theorem.

Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1

Definitions and first properties of elliptic curves. Rational points. Weil pairing. Elliptic curve cryptography. Primality test.

Lawrence C. Washington, Elliptic curves: Number Theory and Criptography, Chapman & Hall (CRC)

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.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

A) STRUCTURAL RULES EXPRESSED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC, LUDICS
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

On line lecture notes

Basics of functional analysis (normed spaces, Hilbert spaces, Banach spaces, linear and limited hearts).
Lp spaces (completeness, duality. Hilbert's space L2).
Regularization and approximation through smooth functions: convolution, approximate delta.
Weak derivatives (test functions, distributions, weak derivatives in Lp).
The spaces of Sobolev Wk, p:
The space of Sobolev W1, p.
The space W01, p.
Some examples of limit problems.
Maximum principle.
Density theorems.
Immersion theorems.
Potential estimates.
Compactness.
Extensions and interpolations.
Sobolev spaces and variational formulation of problems at the limits in dimension N:
Definition and elementary properties of Sobolev spaces W^1, p (D)
Extension operators.
Sobolev inequalities.
The space W^01, p
Variational formulation of some elliptic boundary problems.
Existence of weak solutions.
Regularity of weak solutions.
Maximum principle.

[GT] D. Gilbarg, N.S. Trudinger Elliptic partial differential equations of second order

Ordinary Differential Equations
Finite difference methods for Ordinary Differential Equations: Euler method. Consistency, stability, absolute stability. Second-order Runge-Kutta methods. Implicit one-step methods: backward Euler, Crank-Nicolson. Convergence of one-step methods.
Multistep methods: general structure, complexity, absolute stability. Stability and consistency for multistep methods. Adams methods. BDF methods. Predictor-Corrector methods. (reference: chapter 7 of the notes "Appunti del corso di Analisi Numerica")

Difference schemes for Partial Differential Equations
General concepts about finite difference approximations. Semi-discrete approximations and their convergence. Lax-Richtmeyer theorem.
The advection equation: method of characteristics. Semi-dicrete and fully discrete upwind method, consistency and stability. The heat equation: Fourier approximation. Centred finite difference approximation, consistency and stability. Poisson equation: Fourier and centred difference approximations, convergence study. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected material from chapters 1, 2, 3, 12, 13)

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

Additional notes provided by the teacher

1. Brownian motion I. Gaussian multivariate distribution. Processes with incremental and independent increments. Definition and continuity properties of Brownian motion. Non-differentiability of the trajectories. Property of Markov. Strong Markov property and reflection principle. 2. Brownian motorcycle II. Brownian motorcycle in multiple dimensions. Harmonic functions and Dirichlet problem. Solution of the Dirichlet problem through Brownian motion for regular domains. Poisson's problem and its solution for regular domains. Law of the iterated log arithm. Skorohod embedding. Donsker invariance principle. Applications: arcosine laws and the law of the maximum of random walks. 3. Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Itˆo formula and applications. Local weather and Tanaka formula. Ito formula in more dimensions and for general stochastic differential. 4. Stochastic differential equations. Stochastic differential linear equations: examples of solutions. Existence and uniqueness theorem for stochastic differential equations. Diffusion process within zero noise limit. Infinitesimal generator of a diffusion and partial differential equations. Feynman-Kac formula and applications.

[1]P. M ̈orters and Y. Peres,Brownian Motion.Cambridge University Press, (2010).
[2]L.C. Evans,An Introduction to Stochastic Differential Equations.AMS bookstore, (2013).
[3]T.M.Liggett,Continuous time Markov processes.AMS, (2010).

1. Random walks and Markov chains Successions of random variables. Random walks. Discrete and continuous time Markov chains. Invariant measurement, time-reversal and reversibility a2. Examples and classic models. Random walks on graphs. Birth and death processes. Exclusion processes. Monte Carlo method: Glauber's dynamic Metropolise type algorithms for Ising modeling, coloring of a graph and other interacting systems. 3. Convergence to equilibrium I. Distance in variation, mixing times. Teoremiergodici. Coupling techniques. Strong stationary times. Applications to the coupon collector problem and to the shuffling of a deck of cards.4. Convergence to equilibrium II. Convergence in norm L2. Spectral gap and estimated relaxation times. Cheeger inequality, conductance and cam method. "Comparison" method. Spectral gap for the d-dimensional sultore exclusion process. Convergence to equilibrium in terms of logarithmic entropy and inequality. Esempi.5. Other topics chosen. Glauber's dynamics for the Ising model: dynamic phase transition for the medium field model and for the suZ2 model. The "cut-off" phenomenon. Logarithmic Sobolev inequalities and equilibrium convergence. Algorithm for the "perfect simulation".

[1]D. Levine, Y. Peres, E. Wilmer,Markov chains and mixing times..AMS bookstore, (2009).
[2]O. Haggstrom,Finite Markov chains and algorithmic applications..Cambridge Univ. Press,(2002).
[3]J. Norris,Markov chains.Cambridge Univ. Press, (2008).
[4]L. Saloffe-Coste,Lectures on finite Markov chains..Springer Lecture Notes in Math.1665, (1997).

Principles of Object Oriented Design
Abstraction, Polimorphism, Inheritance, Aggregation
Object Oriented Programming Models and UML
UML Use Case, Sequence, Class e Object, Deployment diagrams
Software Analysis and Developmenmt for Java Virtual Machine: I/O, Stream, Networking
(Scientific, Real-time,...) Efficient Distributed Computing, Multithreading and Concurrency in Cloud e Mobile

MANUALE DI JAVA 9 - DE SIO CESARI CLAUDIO Hoepli

1. Introduction to graph theory: graph, directed graph, tree, rooted tree, connection, strong connection, acyclic graph; graphs isomorphisms, planarity, Kuratowski's theorem, Euler's formula; graph coloring; Eulerian paths, Hamiltonian circuits.
2. Theory of algorithms and optimization: recalls on algorithms and structured programming; computational complexity of an algorithm, complexity classes for problems, classes P, NP, NP-complete, NP-hard; decision-making, research, enumeration and optimization problems; non-linear programming problems, convex programming, linear programming and integer linear programming; combinatorial optimization problems.
Recalls on the elements of combinatorial calculation, algorithms for the generation of the set of parts of a finite set, permutations and combinations of the elements of a set, calculation of the binomial coefficient; the problem of the "Latin squares" and the game of Sudoku; a recursive algorithm for the solution of the game.
3. Optimization problems on graphs and network flows: graph visit, verification of fundamental properties of a graph: connection, strong connection, presence of cycles. Topological sorting of an directed acyclic graph.
The problem of the minimum spanning tree, Kruskal's algorithm, Prim's algorithm, formulation of the problem in terms of Integer Linear Programming.
Search for minimum cost paths on a weighed graph; minimum cost path with single source, Dijkstra's algorithm, Bellman-Ford's algorithm; minimum cost path between all pairs of vertices of the graph, dynamic programming algorithms, Floyd-Warshall's algorithm, transitive closure of a graph.
Network flows and the maximum flow on a network, Theorem of maximum flow and minimum cut, Ford-Fulkerson's algorithm, Edmonds-Karp's algorithm, preflow algorithm, "push-relabel" algorithm.
Problems of partitioning of graphs, trees and paths, problems for the optimal partitioning of trees and paths into p connected components; objective functions, algorithmic techniques for the solution of this class of problems (dynamic programming, linear programming, shifting).
Stable marriage problem, problem definition and matching stability criterion, applications, Gale and Shapley algorithm.
4. Programming laboratory for developing Python language programs for optimizations problems, also with Wolfram Mathematica software.

Cormen, Leiserson, Rivest, Stein, "Introduction to algorithms", third edition, The MIT Press, 2009

Brief introduction to Julia language for scientific computing. Intro to geometric modeling and scientific visualization. Simplicial, cellular and chain complexes. Boundary and coboundary operators. Algebraic operators of incidence and adjacency. Duality. Extraction of geometric models from 3D imaging. Delaunay triangulations and Voronoi complexes. Morse functions and Reeb graphs. Elements of topological structures on big data. Persistent homology. Matrix operations and dense linear systems: outline of BLAS, LAPACK, scaLAPACK. Sparse linear systems. CombBLAS, GraphBLAS. Development of a collaborative project.

Herbert Edelsbrunner and John Harer, Computational Topology. An Introduction, AMS, 2011.

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

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 Availabilty

Further applications

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

• Euclidean Geometry: triangles centers , circle inversion.
• Ordered Geometry and Sylvester problem.
• Projective Geometry: axioms, Desargues, collineations and
correlations
• Platonic Solids Euler formula. Politops, 4-dimensional space politops.
• Topology of surfaces and graphs 4 colors problem and 6 colors theorem, Heawood theorem.
• Delaunay triangulations.
• Plane curves, local study of singularities, Newton polygons .

1) H.S.M. Coxeter Introduction to geometry, Wiley 1970;
2) G. Fisher Plane algebraic curves, AMS Students Mathematical Library V.
15, AMS 2001.
moreover
3) M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 1998;
4) S. Rebay, Tecniche di Generazione di Griglia per il Calcolo Scientifico-Triangolazione di Delaunay, slides Univ. Studi di Brescia;
5) B. Sturmfels, Polynomial equations and convex polytopes, American Mathematical Monthly 105 (1998) 907-922.
6) Shuhong Gao, Absolute Irreducibility of Polynomials via Newton Polytopes, J. of Algebra

MODULE 1
1 A rapid introduction to MATLAB
1.1 MATLAB basics: Preliminary elements; Variable assignment; Workspace; Arithmetic operations; Vectors and matrices; Standard operations of linear algebra; Element-by-element multiplication and division; Colon (:) operator; Predefined function; inline Function; Anonymous Function.
1.2 M-file: Script and Function
1.3 Programming fundamentals: if, else, and elseif scheme; for loops; while loops
1.4 Matlab graphics
1.5 Preliminary exercises on programming
1.6 Exercises on the financial evaluation basics

MODULE 2
2 Preliminary elements on Probability Theory and Statistics
2.1 Random variables
2.2 Probability distributions
2.3 Continuous random variable
2.4 Higher-order moments and synthetic indices of a distribution
2.5 Some probability distributions: Uniform, Normal, Log-normal, Chi-square, Student-t
3 Linear and Non-linear Programming
3.1 Some Matlab built-in functions for optimization problems
3.2 Multi-objective optimization: Determining the efficient frontier
4 Portfolio Optimization
4.1 Portfolio of equities: Prices and returns
4.2 Risk-return analysis: Mean-Variance; Effects of the diversification in an Equally Weighted portfolio; Mean-MAD; Mean-MinMax; VaR; Mean-CVaR; Mean-Gini portfolios
4.3 Bond portfolio immunization

MODULE 3
5 Further elements on Probability Theory and Statistics
5.1 Introduction to the Monte Carlo simulation
5.2 Stochastic processes: Brownian motion; Ito’s Lemma; Geometrical Brownian motion
6 Pricing of derivatives with an underlying security
6.1 Binomial model (CRR): A replicating portfolio of stocks and bonds; Calibration of the model; Multi-period case
6.2 Black-Scholes model: Assumptions of the model; Pricing of a European call; Pricing equation for a call; Implied Volatility
6.3 Option Pricing with Monte Carlo Method: Solution in integral form; Path Dependent Derivatives

F. Cesarone, Computational Finance: a MATLAB oriented modeling, draft

Ordinary Differential Equations
Finite difference methods for Ordinary Differential Equations: Euler method. Consistency, stability, absolute stability. Second-order Runge-Kutta methods. Implicit one-step methods: backward Euler, Crank-Nicolson. Convergence of one-step methods.
Multistep methods: general structure, complexity, absolute stability. Stability and consistency for multistep methods. Adams methods. BDF methods. Predictor-Corrector methods. (reference: chapter 7 of the notes "Appunti del corso di Analisi Numerica")

Difference schemes for Partial Differential Equations
General concepts about finite difference approximations. Semi-discrete approximations and their convergence. Lax-Richtmeyer theorem.
The advection equation: method of characteristics. Semi-dicrete and fully discrete upwind method, consistency and stability. The heat equation: Fourier approximation. Centred finite difference approximation, consistency and stability. Poisson equation: Fourier and centred difference approximations, convergence study. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected material from chapters 1, 2, 3, 12, 13)

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

Additional notes provided by the teacher

1. Random walks and Markov chains Successions of random variables. Random walks. Discrete and continuous time Markov chains. Invariant measurement, time-reversal and reversibility a2. Examples and classic models. Random walks on graphs. Birth and death processes. Exclusion processes. Monte Carlo method: Glauber's dynamic Metropolise type algorithms for Ising modeling, coloring of a graph and other interacting systems. 3. Convergence to equilibrium I. Distance in variation, mixing times. Teoremiergodici. Coupling techniques. Strong stationary times. Applications to the coupon collector problem and to the shuffling of a deck of cards.4. Convergence to equilibrium II. Convergence in norm L2. Spectral gap and estimated relaxation times. Cheeger inequality, conductance and cam method. "Comparison" method. Spectral gap for the d-dimensional sultore exclusion process. Convergence to equilibrium in terms of logarithmic entropy and inequality. Esempi.5. Other topics chosen. Glauber's dynamics for the Ising model: dynamic phase transition for the medium field model and for the suZ2 model. The "cut-off" phenomenon. Logarithmic Sobolev inequalities and equilibrium convergence. Algorithm for the "perfect simulation".

[1]D. Levine, Y. Peres, E. Wilmer,Markov chains and mixing times..AMS bookstore, (2009).
[2]O. Haggstrom,Finite Markov chains and algorithmic applications..Cambridge Univ. Press,(2002).
[3]J. Norris,Markov chains.Cambridge Univ. Press, (2008).
[4]L. Saloffe-Coste,Lectures on finite Markov chains..Springer Lecture Notes in Math.1665, (1997).

1. Brownian motion I. Gaussian multivariate distribution. Processes with incremental and independent increments. Definition and continuity properties of Brownian motion. Non-differentiability of the trajectories. Property of Markov. Strong Markov property and reflection principle. 2. Brownian motorcycle II. Brownian motorcycle in multiple dimensions. Harmonic functions and Dirichlet problem. Solution of the Dirichlet problem through Brownian motion for regular domains. Poisson's problem and its solution for regular domains. Law of the iterated log arithm. Skorohod embedding. Donsker invariance principle. Applications: arcosine laws and the law of the maximum of random walks. 3. Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Itˆo formula and applications. Local weather and Tanaka formula. Ito formula in more dimensions and for general stochastic differential. 4. Stochastic differential equations. Stochastic differential linear equations: examples of solutions. Existence and uniqueness theorem for stochastic differential equations. Diffusion process within zero noise limit. Infinitesimal generator of a diffusion and partial differential equations. Feynman-Kac formula and applications.

[1]P. M ̈orters and Y. Peres,Brownian Motion.Cambridge University Press, (2010).
[2]L.C. Evans,An Introduction to Stochastic Differential Equations.AMS bookstore, (2013).
[3]T.M.Liggett,Continuous time Markov processes.AMS, (2010).

Introductory notions
Recap of Special Relativity. Lorentz transformations in Minkowski’s space. Vectors in Minkowski’s space. Basis of the tangent space.
Cotangents space and dual vectors in Minkowski’s space. Basis of cotangent space. Lorentz transformations of vectors and dual vectors. Tensors in Minkowski’s space. Properties of vectors, dual vectors and tensors in Minkowski’s space. Definition of symmetric and antisymmetric tensor. Symmetrization and antisymmetrization of a generic tensor. Metric in Minkowski’s space: definition and properties. Operations related to the metric: scalar products, rising and lowering indices of a tensor, contractions and trace of a tensor. Equivalence between inertial and gravitational mass. Weak Equivalence Principle (WEP), Einstein’s equivalence Principle (EEP).

Basic notions of differential geometry
Introduction to the notion of manifold. Definition and properties of maps. Injective and suriective maps (some examples included). Composition of charts. Invertible charts. Definition of diffeomorphism.
Definition of chart (or coordinate system). Definition of atlas. Definition of manifold. Product of manifolds.
Formal coordinate independent definition of vector. Demonstration that the dimension of the tangent space coincides with the one of the corresponding manifold. Basis (or coordinate system) of the tangent space. Coordinate transformations. Coordinate transformations of the components of a vector. Definition and properties of the tangent field. Definition of one parameter group of diffeomorphisms. Definition of integral curves. Commutator of two vectors. Coordinate independent definition of dual vector (one-form). Cotangent space and corresponding basis. Coordinate transformation of the components of a one-form. Coordinate independent definition of tensor. Demonstration that the partial derivative of a tensor is not a tensor.
Metric: signature and canonical form.
Tensor densities.
Differential forms. Wedge product. Exterior derivative. Closed and exact form. Poincarre Lemma (statement only). Hodge duality. Maxwell equations expressed in term of exterior derivative and hodge duality (only small reference).
Integration over a manifold: volume element in terms of the determinant of the metric.
Maps between manifold: pullback and pushforward. Pullback and pushforward associated to diffeomorphisms. Equivalence between diffeomorphisms and coordinate transformations. Vector field associated to diffeomorphisms. Lie Derivative: definition and general properties. Action of Lie’s derivative on scalars, vectors, one-forms and tensors. General Relativity as diffeomorphism invariant theory. Analogy between gauge transformations and diffeomorphisms.

Symmetries.
Notion of submanifold. Immersed and embedded submanifolds. Notion of hypersurface and boundary of a manifold.
Integration on manifolds again: differential form as generic volume element. Orientation and orientable manifold. Covering of the manifold through partition of unity. Integration of p-forms over submanifold. Demonstrations that the volume element can be expressed in terms of the determinant of the metric. Stokes theorem (no demonstration).


Connection, Covariant Derivative, Curvature
Lie’s Algebra and Lie’s group. Action from the right and from the left. Left- and right-invariant vectors. Structure constants. Examples of Lie groups. Maurer-Cartan forms. Maurer-Cartan’s equations. Action of Lie Groups on manifolds. Definition of free, effective and transitive action. Orbit and stabilizer.
Algebric definition of connection and covariant derivative. General properties of covariant derivatives. Action of coordinate transformations on the connection.
Demonstration that the difference of Christoffel coefficients associated to two different connections transforms as a tensor; torsion tensor, torsion-free and metric connection. Demonstrations that for any given metric exists a connection (metric connections) for which the covariant derivative of the metric is zero.
Formal construction of the covariant derivative from the notion of parallel transport (qualitative introduction).
Fiber bundle. Trivial and locally trivializable bundles. Local trivilizations. Maps between fiber bundles (notions).
Defintion of bundle atlas, G-atlas, G-structure. Fiber Bundle with structure group G. Definition of Principal Bundle. Definition of section of a bundle. Vector bundle and bundle of basis, definition and general properties. Relation between principle bundle, vectorbundle and bundle of frames (definition of associated vector bundle to a principal bundle.
Construction of the covariante derivative on a vectorbundle (only the knowledge of the fundamental logical steps is required for the exam).
Curvature tensor as 2-form on a fiber bundle. Geometrical interpretation of the curvature. Bianchi identity. Fiber metric. Ortogonal basis.
Connections and gauge theories: electromagnetism as simple example.
Soldering form. Choice of the gauge. Ortonormal and metric gauge.
Levi-Civita connection; Riemann’s tensor : definition and properties. Ricci’s tensor and scalar, Weyl’s tensor. Globally and locally inertial coordinates.

Einstein’s theory of gravity

Minimal coupling.
Particle in a gravitational field: affine parameter, self-parallel curves. Geodesic’s equations. Geodesic deviation.
Derivation of the Einstein’s equations from Newton’s limit.
Lagrangian derivations of Einstein’s equations.
General considerations on the structure of Einstein’s equations. Choice of the gauge. Energy conditions.
Symmetries and Killing vectors: version of Noether’s theorem from general relativity. Maximal number of linearly independent Killing vectors on a manifold. Homogenous and isotropic manifold. Spaces at constant curvature. Metric in spaces at constant curvature.

Notable solutions of Einstein’s equations

Static spherically symmetric spacetimes. Determination of Schwarzschild’s metric.
Cosmological solution. Spatially homogeneous and isotropic spacetime. Frieman’s Robertson-Walker metric. Friedman’s equations.
Coordinate singularities. Case of study: Schwarzschild radius. Rindler metric. Kruskal coordinates. Black hole solution.
Perturbation around a background metric. Case of study:perturbation of flat metric. Degrees of freedom. Linearized Einstein’s equations. Choice of the gauge. Linearized Einstein’s equations in vacuum: gravitational waves. Solutions in presence of the source (only few words).


Advanced concepts
Conformal transformations. Cotton’s tensor. Conformally flat metric. Demonstration of the theorem: a metric is conformally flat if and only if Weyl (Cotton) tensor is null. Conformal group. Conformal Killing vectors.
Alternative theories of gravity. Scalar-tensor theories. Jordan and Einstein’s frames.

1. S. Carrol Space time and Geometry: An Introduction to General Relativity (Addison Wesley, 2004);
2. R. Wald General Relativity (The Chicago Press, 1984);
3. B. Schutz A First Course in General Relativity (Cambridge Press)
4. people.sissa.it/~percacci/lectures/general/index.html
5. B. Schutz Geometrical Methods of Mathematical Physics (Cambridge Press)
6. S. Weinberg Gravitation and Cosmology-principles and application of the general theory of relativity (John Weiley & Sons, 1972);

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

To be selected according the thesis subject

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

The program will be selected upon agreement between the student, the internal supervisor and the subject offering the stage.

Textbooks will be selected according to the specific topic

A selected topic among computer sciences and/or scientific computing.

Textbooks will be chosen according the selected topic

Professional skills among computer sciences and/or scientific computing.

Textbooks will be chosen according the selected topic

Part 1: Some preliminary notions.
Order relations and trees, inductive definitions, proofs by induction, axiom of choice and Kőnig's lemma.

Part 2: Provability and satisyability.
First order formal language: alphabet, terms, formulas, sequents. Structures for first order languages: structures, terms and formulas with parameters in a structure, value of terms, formulas and sequents. The calculus of sequents for first order logic: Gentzen's LK. Derivable sequents and derivations. Correctness of the rules of LK. Canonical analysis and fundamental theorem: construction of the canonical analysis (with and without cuts) and proof of the fundamental theorem of the canonical analysis. Consequences of the fundamental theorem: completeness theorem, compactness theorem, eliminability of cuts, L"owenheim-Skolem's theorem.

Part 3: Towards proof-theory: the cut-elimination theorem.
The cut-elimination procedure. Definition of the elementary steps of cut-elimination. First proof strategy (big reduction steps). Second proof strategy (reversion of derivations). The complexity of the cut-elimination procedure (sketch). Some immediate consequences of the cut-elimination theorem.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).

Towards model theory: some consequences of the compactness theorem.
Proof of the compactness theorem for languages of any cardinality. Languages with equality. The compactness theorem for languages with equality. Correctness and completeness for languages with equality. L"owenheim-Skolem's theorem for (denumerable) languages with equality. The limits of the expressive power of first order languages. Elementary equivalence, substructures, elementary substructures. Isomorphsims and elementary equivalence. The notion of substructure. Elementary substructures and diagrams. The preservation theorems. Generalisations of the L"owenheim-Skolem's theorem. Completeness of a theory.

Logic and Arithmetic: incompleteness

Decidability and fundamental results of recursion theory: primitive recursive functions and elementary functions, Ackermann's function and the (partial) recursive functions, arithmetical hierarchy and representation (in N) of recursive functions, arithmetization of syntax, fundamental theorems of recursion theory, decidability, semi-decidability, undecidability.

Peano arithmetic: Peano's axioms, the models of (first order) Peano arithmetic, the representable functions in (first order) Peano arithmetic, incompleteness and undecidability.

V.M. Abrusci, L. Tortora de Falco, Logica Volume 1- Dimostrazioni e modelli al primo
ordine. Springer, (2014).

V.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

Introduction: Cardano equations for the solubility of third-degree equations, rings and fields, the characteristics of a field, known facts about rings of polynomials, field extensions, construction of some field extensions, the sub-ring generated by a subset, the subfield generated by a subset, algebraic and transcendent elements, algebraically closed fields.

Splitting fields: Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness up to isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental Theorem of Galois Theory: Group of the automorphisms of a field, normal, separable and Galois extensions, characterizations of separable extensions, Fundamental Theorem of Galois Correspondence, examples, Galois group of a polynomial, Radical extensions, solvable groups and Galois's Theorem on solving equations , Theorem of the existence of primitive elements.

The computation of Galois group: Galois groups as subgroups of $ S_n $, transitive subgroups of $ S_n $, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in $ A_n $, Theory of discriminants, Galois groups of polynomials with degree up to $4$, examples of polynomials with Galois group $S_p$.

Cyclotomic fields: Definitions, Galois group, maximal real subfields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Theorem of the inverse Galois theory for abelian groups.

Finite fields: Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields. Construction of the algebraic closure of a finite field with $ p $ elements.

Constructions with ruler and compass: Definition of constructible points of the plane, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, trisection of angles, quadrature of the circle and Gauss's theorem for the construction of regular polygons with ruler and compass.

J. S. Milne.Fields and Galois Theory. Course Notes v4.22 (March 30, 2011).
S. Gabelli. Teoria delle Equazioni e Teoria di Galois. Springer UNITEXT (La Matematica per il 3+2) 2008, XVII, 410 pagg., ISBN: 978-88-470-0618-8
E. Artin.Galois Theory. NOTRE DAME MATHEMATICAL LECTURES Number 2. 1942.
C. Procesi.Elementi di Teoria di Galois. Decibel, Zanichelli, (Seconda ristampa, 1991).

1. Modules
Modules and submodules. Operations between submodules. Omomorphisms
and quotient modules. Generators and bases. Free modules. Invariance of rank.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem
of Caylay-Hamilton. The Nakayama Lemma.

2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals.
Zorn's lemma. Maximal and minimal ideals. Jacobson radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remeinder Theorem. Prime
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.

3. Rings and fraction modules
Multiplicative parts. Saturated multiplicative parts. Rings and fraction modules.
Extension and contraction of ideals. Prime and primary ideals in fraction rings. Local rings.
Local properties. Ring of formal series on a field.

4. Integral dependence
Integral dependence and integral closure. Properties of stability and transitivity
of integral dependence. Lying over, Inc and Going up. Krull dimension of the
integral closure. Notes on the noetherianity of integral closure. Valuation rings and their characterizations. Discrete valuation rings. The Theorem of
Krull on integralclosure. Dedekind rings

5. Noetherian and Artinian rings and modules.
Chain conditions and equivalent properties. Noetherian and Artininan rings.
Modules and algebras on noetherian rings. The Hilbert Base Theorem. The Cohen Theorem.
Primary decomposition of ideals. Uniqueness theorems. Prime associates
and zerodivisori. Rings and artinian modules. Characterization theorem for Artinian rings
The Principal Ideal Theorem.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.

R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972

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

http://ricerca.mat.uniroma3.it/users/valerio/ge310_1819.html

http://ricerca.mat.uniroma3.it/users/valerio/GE310/prog18.pdf

- Classical theory of algebraic varieties in affine spaces over algebraically closed fields.
- Local geometry, normalization.
- Divisors, linear systems and morphisms of projective varieties.

Algebraic Geometry texts:
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1994.
J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1992.


Algebra texts:
* M. Artin, Algebra, Bollati Boringhieri 1997.
* M.F. Atiyah, I.G. Mac Donald, Introduzione all'algebra commutativa, Feltrinelli 1991.

Basic Definitions. Connected graphs. Eulerian graphs
Trees. Rooted trees. Spanning trees.
Cycle space. Cut space. Cyclomatic number.
Bipartite graphs. Matchings. Marriage theorem.
Existence of 1-factors and k-factors.
Connectivity. Structure of 2-connected and 3-connected graphs.
Hamiltonian graphs
Planar and Plane Graphs. Euler formula. Triangulations
Colourings.

R. Diestel. GRAPH THEORY. Springer GTM

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
4. Integration on product spaces
Measurability on cartesian products. Product measure. Fubini theorem.
4. 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.

Linear dynamic systems. Planar systems. Gradient systems. Stability theorems. Limit cycles. Angles of Euler, Equations of Euler. Lagrange top.
Perturbation theory. Homological equation. Perturbative series. Birkhoff series. Perturbation theory to all orders for isochronous systems. KAM theorem.

G. Gentile, Introduzione ai sistemi dinamici. 1. Equazioni differenziali ordinarie, analisi qualitativa e alcune applicazioni, available online
G. Gentile, Introduzione ai sistemi dinamici. 2. Meccanica lagrangiana e hamiltoniana, available online

Linear dynamic systems. Planar systems. Gradient systems. Stability theorems. Limit cycles. Angles of Euler, Equations of Euler. Lagrange top.
Perturbation theory. Homological equation. Perturbative series. Birkhoff series. Perturbation theory to all orders for isochronous systems. KAM theorem.

G. Gentile, Introduzione ai sistemi dinamici. 1. Equazioni differenziali ordinarie, analisi qualitativa e alcune applicazioni, available online
G. Gentile, Introduzione ai sistemi dinamici. 2. Meccanica lagrangiana e hamiltoniana, available online

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) Notes_Convection_Diffusion.pdf
classroom notes

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)

4) Computational Science and Engineering
Gilbert Strang, Wellesley-Cambridge Press, 2007 Cap 3.1, page 236~241;
Cap. 7.2 Iterative methods, page 563~567

1. Introduction to graph theory: graph, directed graph, tree, rooted tree, connection, strong connection, acyclic graph; graphs isomorphisms, planarity, Kuratowski's theorem, Euler's formula; graph coloring; Eulerian paths, Hamiltonian circuits.
2. Theory of algorithms and optimization: recalls on algorithms and structured programming; computational complexity of an algorithm, complexity classes for problems, classes P, NP, NP-complete, NP-hard; decision-making, research, enumeration and optimization problems; non-linear programming problems, convex programming, linear programming and integer linear programming; combinatorial optimization problems.
Recalls on the elements of combinatorial calculation, algorithms for the generation of the set of parts of a finite set, permutations and combinations of the elements of a set, calculation of the binomial coefficient; the problem of the "Latin squares" and the game of Sudoku; a recursive algorithm for the solution of the game.
3. Optimization problems on graphs and network flows: graph visit, verification of fundamental properties of a graph: connection, strong connection, presence of cycles. Topological sorting of an directed acyclic graph.
The problem of the minimum spanning tree, Kruskal's algorithm, Prim's algorithm, formulation of the problem in terms of Integer Linear Programming.
Search for minimum cost paths on a weighed graph; minimum cost path with single source, Dijkstra's algorithm, Bellman-Ford's algorithm; minimum cost path between all pairs of vertices of the graph, dynamic programming algorithms, Floyd-Warshall's algorithm, transitive closure of a graph.
Network flows and the maximum flow on a network, Theorem of maximum flow and minimum cut, Ford-Fulkerson's algorithm, Edmonds-Karp's algorithm, preflow algorithm, "push-relabel" algorithm.
Problems of partitioning of graphs, trees and paths, problems for the optimal partitioning of trees and paths into p connected components; objective functions, algorithmic techniques for the solution of this class of problems (dynamic programming, linear programming, shifting).
Stable marriage problem, problem definition and matching stability criterion, applications, Gale and Shapley algorithm.
4. Programming laboratory for developing Python language programs for optimizations problems, also with Wolfram Mathematica software.

Cormen, Leiserson, Rivest, Stein, "Introduction to algorithms", third edition, The MIT Press, 2009

Brief introduction to Julia language. Introduction to parallel architectures, Principle of parallel and distributed programming with Julia. Primitives of communication on synchronization: MPI paradigm. Languages based on directives: OpenMP. Performance metrics of parallel programs. Matrix operations and dense linear systems: introduction to BLAS, LAPACK, scaLAPACK. Sparse linear systems: introduction to CombBLAS, GraphBLAS. Collaborative development of course projects: heart-quake simulations; parallel LAR.

Blaise N. Barney, HPC Training Materials, per gentile concessione del Lawrence Livermore National Laboratory's Computational Training Center

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.

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]

A selected topic among computer sciences and/or scientific computing.

Textbooks will be chosen according the selected topic

Professional skills among computer sciences and/or scientific computing.

Textbooks will be chosen according the selected topic

The program will be selected upon agreement between the student, the internal supervisor and the subject offering the stage.

Textbooks will be selected according to the specific topic

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.M. Abrusci, L. Tortora de Falco, Logica Volume 2- Incompletezza, teoria assiomatica degli insiemi. Springer, (2018).

A) STRUCTURAL RULES EXPRESSED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC, LUDICS
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

On line lecture notes

Arithmetic functions and their properties:
-Definition and Dirichlet convolution.
-Number and sum of divisors function.
-Möbius function.
-Euler function.

Congruences:
-Sets of residues.
-Polynomial congruences.
-Primitive roots.

Quadratic residues:
-Legendre symbol.
-Quadratic reciprocity.
-Jacobi symbol.

Sums of squares:
-Sums of two squares.
-Number of representations.
-Sums of four squares.
-Sums of three squares.

Continued fractions and diophantine approximation:
-Simple continued fractions.
-Continued fractions and diophantine approximation.
-Infinite simple continued fractions.
-Periodic continued fractions.
-Pell's equation.
-Liouville's Theorem.

Script by W. Chen
http://www.williamchen-mathematics.info/lnentfolder/lnent.html

M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense

The complex field. Holomorphic functions; Cauchy-Riemann equations.
Series and Abel's theorem. Exponential and logarithms.
Elementary conformal mappings. Complex integration; Cauchy's theorem; Cauchy's formula.
Local properties of holomorphic functions (singularities, zeroes and poles; local mapping theorem and
maximum principle).
Residues.
Harmonic functions.
Series expansions (Weierstrass' theorem, Taylor's series).
Partial fractions and infinite products.
Supplementary arguments (depending on time): entire functions and Hadamard's theorem. Riemann zeta function.
Riemann mapping theorem.

Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1

Broad program:

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytical and geometric form, consequences.
First and second category spaces, Baire theorem, Banach-Steinhaus theorem, open map and closed graph, applications.
Weak, closed and convex topologies, Banach-Alaoglu theorem, separability and reflexivity.
Sobolev spaces in one dimension, 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)

Ordinary Differential Equations
Finite difference methods for Ordinary Differential Equations: Euler method. Consistency, stability, absolute stability. Second-order Runge-Kutta methods. Implicit one-step methods: backward Euler, Crank-Nicolson. Convergence of one-step methods.
Multistep methods: general structure, complexity, absolute stability. Stability and consistency for multistep methods. Adams methods. BDF methods. Predictor-Corrector methods. (reference: chapter 7 of the notes "Appunti del corso di Analisi Numerica")

Difference schemes for Partial Differential Equations
General concepts about finite difference approximations. Semi-discrete approximations and their convergence. Lax-Richtmeyer theorem.
The advection equation: method of characteristics. Semi-dicrete and fully discrete upwind method, consistency and stability. The heat equation: Fourier approximation. Centred finite difference approximation, consistency and stability. Poisson equation: Fourier and centred difference approximations, convergence study. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected material from chapters 1, 2, 3, 12, 13)

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

Additional notes provided by the teacher

1. Brownian motion I. Gaussian multivariate distribution. Processes with incremental and independent increments. Definition and continuity properties of Brownian motion. Non-differentiability of the trajectories. Property of Markov. Strong Markov property and reflection principle. 2. Brownian motorcycle II. Brownian motorcycle in multiple dimensions. Harmonic functions and Dirichlet problem. Solution of the Dirichlet problem through Brownian motion for regular domains. Poisson's problem and its solution for regular domains. Law of the iterated log arithm. Skorohod embedding. Donsker invariance principle. Applications: arcosine laws and the law of the maximum of random walks. 3. Stochastic integration. Paley-Wiener-Zygmund integral. Stochastic integral with respect to Brownian motion. Itˆo formula and applications. Local weather and Tanaka formula. Ito formula in more dimensions and for general stochastic differential. 4. Stochastic differential equations. Stochastic differential linear equations: examples of solutions. Existence and uniqueness theorem for stochastic differential equations. Diffusion process within zero noise limit. Infinitesimal generator of a diffusion and partial differential equations. Feynman-Kac formula and applications.

[1]P. M ̈orters and Y. Peres,Brownian Motion.Cambridge University Press, (2010).
[2]L.C. Evans,An Introduction to Stochastic Differential Equations.AMS bookstore, (2013).
[3]T.M.Liggett,Continuous time Markov processes.AMS, (2010).

Principles of Object Oriented Design
Abstraction, Polimorphism, Inheritance, Aggregation
Object Oriented Programming Models and UML
UML Use Case, Sequence, Class e Object, Deployment diagrams
Software Analysis and Developmenmt for Java Virtual Machine: I/O, Stream, Networking
(Scientific, Real-time,...) Efficient Distributed Computing, Multithreading and Concurrency in Cloud e Mobile

MANUALE DI JAVA 9 - DE SIO CESARI CLAUDIO Hoepli

[-] Abstract Machines. Interpreters and Compilers.
[-] Language constructs.
[-] Object Oriented Programming
[-] Functional Programming

GABBRIELLI, M., MARTINI, S., PROGRAMMING LANGUAGES: PRINCIPLES AND PARADIGMS. MCGRAW-HILL, Seconda Edizione

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

Introductory notions
Recap of Special Relativity. Lorentz transformations in Minkowski’s space. Vectors in Minkowski’s space. Basis of the tangent space.
Cotangents space and dual vectors in Minkowski’s space. Basis of cotangent space. Lorentz transformations of vectors and dual vectors. Tensors in Minkowski’s space. Properties of vectors, dual vectors and tensors in Minkowski’s space. Definition of symmetric and antisymmetric tensor. Symmetrization and antisymmetrization of a generic tensor. Metric in Minkowski’s space: definition and properties. Operations related to the metric: scalar products, rising and lowering indices of a tensor, contractions and trace of a tensor. Equivalence between inertial and gravitational mass. Weak Equivalence Principle (WEP), Einstein’s equivalence Principle (EEP).

Basic notions of differential geometry
Introduction to the notion of manifold. Definition and properties of maps. Injective and suriective maps (some examples included). Composition of charts. Invertible charts. Definition of diffeomorphism.
Definition of chart (or coordinate system). Definition of atlas. Definition of manifold. Product of manifolds.
Formal coordinate independent definition of vector. Demonstration that the dimension of the tangent space coincides with the one of the corresponding manifold. Basis (or coordinate system) of the tangent space. Coordinate transformations. Coordinate transformations of the components of a vector. Definition and properties of the tangent field. Definition of one parameter group of diffeomorphisms. Definition of integral curves. Commutator of two vectors. Coordinate independent definition of dual vector (one-form). Cotangent space and corresponding basis. Coordinate transformation of the components of a one-form. Coordinate independent definition of tensor. Demonstration that the partial derivative of a tensor is not a tensor.
Metric: signature and canonical form.
Tensor densities.
Differential forms. Wedge product. Exterior derivative. Closed and exact form. Poincarre Lemma (statement only). Hodge duality. Maxwell equations expressed in term of exterior derivative and hodge duality (only small reference).
Integration over a manifold: volume element in terms of the determinant of the metric.
Maps between manifold: pullback and pushforward. Pullback and pushforward associated to diffeomorphisms. Equivalence between diffeomorphisms and coordinate transformations. Vector field associated to diffeomorphisms. Lie Derivative: definition and general properties. Action of Lie’s derivative on scalars, vectors, one-forms and tensors. General Relativity as diffeomorphism invariant theory. Analogy between gauge transformations and diffeomorphisms.

Symmetries.
Notion of submanifold. Immersed and embedded submanifolds. Notion of hypersurface and boundary of a manifold.
Integration on manifolds again: differential form as generic volume element. Orientation and orientable manifold. Covering of the manifold through partition of unity. Integration of p-forms over submanifold. Demonstrations that the volume element can be expressed in terms of the determinant of the metric. Stokes theorem (no demonstration).


Connection, Covariant Derivative, Curvature
Lie’s Algebra and Lie’s group. Action from the right and from the left. Left- and right-invariant vectors. Structure constants. Examples of Lie groups. Maurer-Cartan forms. Maurer-Cartan’s equations. Action of Lie Groups on manifolds. Definition of free, effective and transitive action. Orbit and stabilizer.
Algebric definition of connection and covariant derivative. General properties of covariant derivatives. Action of coordinate transformations on the connection.
Demonstration that the difference of Christoffel coefficients associated to two different connections transforms as a tensor; torsion tensor, torsion-free and metric connection. Demonstrations that for any given metric exists a connection (metric connections) for which the covariant derivative of the metric is zero.
Formal construction of the covariant derivative from the notion of parallel transport (qualitative introduction).
Fiber bundle. Trivial and locally trivializable bundles. Local trivilizations. Maps between fiber bundles (notions).
Defintion of bundle atlas, G-atlas, G-structure. Fiber Bundle with structure group G. Definition of Principal Bundle. Definition of section of a bundle. Vector bundle and bundle of basis, definition and general properties. Relation between principle bundle, vectorbundle and bundle of frames (definition of associated vector bundle to a principal bundle.
Construction of the covariante derivative on a vectorbundle (only the knowledge of the fundamental logical steps is required for the exam).
Curvature tensor as 2-form on a fiber bundle. Geometrical interpretation of the curvature. Bianchi identity. Fiber metric. Ortogonal basis.
Connections and gauge theories: electromagnetism as simple example.
Soldering form. Choice of the gauge. Ortonormal and metric gauge.
Levi-Civita connection; Riemann’s tensor : definition and properties. Ricci’s tensor and scalar, Weyl’s tensor. Globally and locally inertial coordinates.

Einstein’s theory of gravity

Minimal coupling.
Particle in a gravitational field: affine parameter, self-parallel curves. Geodesic’s equations. Geodesic deviation.
Derivation of the Einstein’s equations from Newton’s limit.
Lagrangian derivations of Einstein’s equations.
General considerations on the structure of Einstein’s equations. Choice of the gauge. Energy conditions.
Symmetries and Killing vectors: version of Noether’s theorem from general relativity. Maximal number of linearly independent Killing vectors on a manifold. Homogenous and isotropic manifold. Spaces at constant curvature. Metric in spaces at constant curvature.

Notable solutions of Einstein’s equations

Static spherically symmetric spacetimes. Determination of Schwarzschild’s metric.
Cosmological solution. Spatially homogeneous and isotropic spacetime. Frieman’s Robertson-Walker metric. Friedman’s equations.
Coordinate singularities. Case of study: Schwarzschild radius. Rindler metric. Kruskal coordinates. Black hole solution.
Perturbation around a background metric. Case of study:perturbation of flat metric. Degrees of freedom. Linearized Einstein’s equations. Choice of the gauge. Linearized Einstein’s equations in vacuum: gravitational waves. Solutions in presence of the source (only few words).


Advanced concepts
Conformal transformations. Cotton’s tensor. Conformally flat metric. Demonstration of the theorem: a metric is conformally flat if and only if Weyl (Cotton) tensor is null. Conformal group. Conformal Killing vectors.
Alternative theories of gravity. Scalar-tensor theories. Jordan and Einstein’s frames.

1. S. Carrol Space time and Geometry: An Introduction to General Relativity (Addison Wesley, 2004);
2. R. Wald General Relativity (The Chicago Press, 1984);
3. B. Schutz A First Course in General Relativity (Cambridge Press)
4. people.sissa.it/~percacci/lectures/general/index.html
5. B. Schutz Geometrical Methods of Mathematical Physics (Cambridge Press)
6. S. Weinberg Gravitation and Cosmology-principles and application of the general theory of relativity (John Weiley & Sons, 1972);

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

MODULE 1
1 A rapid introduction to MATLAB
1.1 MATLAB basics: Preliminary elements; Variable assignment; Workspace; Arithmetic operations; Vectors and matrices; Standard operations of linear algebra; Element-by-element multiplication and division; Colon (:) operator; Predefined function; inline Function; Anonymous Function.
1.2 M-file: Script and Function
1.3 Programming fundamentals: if, else, and elseif scheme; for loops; while loops
1.4 Matlab graphics
1.5 Preliminary exercises on programming
1.6 Exercises on the financial evaluation basics

MODULE 2
2 Preliminary elements on Probability Theory and Statistics
2.1 Random variables
2.2 Probability distributions
2.3 Continuous random variable
2.4 Higher-order moments and synthetic indices of a distribution
2.5 Some probability distributions: Uniform, Normal, Log-normal, Chi-square, Student-t
3 Linear and Non-linear Programming
3.1 Some Matlab built-in functions for optimization problems
3.2 Multi-objective optimization: Determining the efficient frontier
4 Portfolio Optimization
4.1 Portfolio of equities: Prices and returns
4.2 Risk-return analysis: Mean-Variance; Effects of the diversification in an Equally Weighted portfolio; Mean-MAD; Mean-MinMax; VaR; Mean-CVaR; Mean-Gini portfolios
4.3 Bond portfolio immunization

MODULE 3
5 Further elements on Probability Theory and Statistics
5.1 Introduction to the Monte Carlo simulation
5.2 Stochastic processes: Brownian motion; Ito’s Lemma; Geometrical Brownian motion
6 Pricing of derivatives with an underlying security
6.1 Binomial model (CRR): A replicating portfolio of stocks and bonds; Calibration of the model; Multi-period case
6.2 Black-Scholes model: Assumptions of the model; Pricing of a European call; Pricing equation for a call; Implied Volatility
6.3 Option Pricing with Monte Carlo Method: Solution in integral form; Path Dependent Derivatives

F. Cesarone, Computational Finance: a MATLAB oriented modeling, draft

• Euclidean Geometry: triangles centers , circle inversion.
• Ordered Geometry and Sylvester problem.
• Projective Geometry: axioms, Desargues, collineations and
correlations
• Platonic Solids Euler formula. Politops, 4-dimensional space politops.
• Topology of surfaces and graphs 4 colors problem and 6 colors theorem, Heawood theorem.
• Delaunay triangulations.
• Plane curves, local study of singularities, Newton polygons .

1) H.S.M. Coxeter Introduction to geometry, Wiley 1970;
2) G. Fisher Plane algebraic curves, AMS Students Mathematical Library V.
15, AMS 2001.
moreover
3) M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 1998;
4) S. Rebay, Tecniche di Generazione di Griglia per il Calcolo Scientifico-Triangolazione di Delaunay, slides Univ. Studi di Brescia;
5) B. Sturmfels, Polynomial equations and convex polytopes, American Mathematical Monthly 105 (1998) 907-922.
6) Shuhong Gao, Absolute Irreducibility of Polynomials via Newton Polytopes, J. of Algebra

A selected topic among computer sciences and/or scientific computing.

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Professional skills among computer sciences and/or scientific computing.

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The program will be selected upon agreement between the student, the internal supervisor and the subject offering the stage.

Textbooks will be selected according to the specific topic