Course

Credits

Scientific Disciplinary Sector Code

Contact Hours

Exercise Hours

Laboratory Hours

Personal Study Hours

Type of Activity

Language

Optional group:
CURRICULUM MODELLISTICA FISICA E SIMULAZIONI NUMERICHE: scegliere 2 Insegnamenti (15 CFU) nei seguenti SSD MAT/01, MAT/02, MAT/03, MAT/05 tra le attività caratterizzanti (B), di cui almeno 1 Insegnamento (6 CFU) nel SSD MAT/01  (show)

15








20410408 
AL310  ELEMENTS OF ADVANCED ALGEBRA
(objectives)
Acquire a good knowledge of the concepts and methods of the theory of polynomial equations in one variable. Learn how to apply the techniques and methods of abstract algebra. Understand and apply the fundamental theorem of Galois correspondence to study the "complexity" of a polynomial.

Derived from
20410408 AL310  ISTITUZIONI DI ALGEBRA SUPERIORE in Matematica L35 PAPPALARDI FRANCESCO, TOLLI FILIPPO, CAPUANO LAURA
( syllabus)
Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions, the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.
Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for 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 automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.
The calculation of the 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, Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p, Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.
Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic subfields, Galois groups, cyclotomic polynomials and their properties, Galois inverse 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 plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.
( reference books)
J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

9

MAT/02

48

24





Core compulsory activities

ITA 
20410409 
AM310  ELEMENTS OF ADVANCED ANALYSIS
(objectives)
To acquire a good knowledge of the abstract integration theory and of the functional spaces L^p.

Derived from
20410409 AM310  ISTITUZIONI DI ANALISI SUPERIORE in Matematica LM40 BATTAGLIA LUCA
( syllabus)
Measure theory, outer measures, construction of Borel measures and the Lebesgue measure. Integration theory, limit theorems, convergence in mean and in measure, integration on product spaces, change of variables for the Lebesgue integral. Radon measures, regularity, positive linear functionals, Riesz representation theorem. Signed measures, decomposition theorems, differentiation, BV functions, fundamental theorem of calculus. Lp spaces, basic properties, dual spaces, density theorems.
( reference books)
G. Folland  "Real Analysis"  Wiley

9

MAT/05

48

24





Core compulsory activities

ITA 
20410411 
GE310  ELEMENTS OF ADVANCED GEOMETRY
(objectives)
Topology: topological classification of curves and surfaces. Differential geometry: study of the geometry of curves and surfaces in R^3 to provide concrete and easily calculable examples on the concept of curvature in geometry. The methods used place the geometry in relation to calculus of several variables, linear algebra and topology, providing the student with a broad view of some aspects of mathematics.

9

MAT/03

48

24





Core compulsory activities

ITA 
20410449 
GE410  ALGEBRAIC GEOMETRY 1
(objectives)
Introduce to the study of topology and geometry defined through algebraic tools. Refine the concepts in algebra through applications to the study of algebraic varieties in affine and projective spaces.

Derived from
20410449 GE410  GEOMETRIA ALGEBRICA 1 in Matematica LM40 LOPEZ ANGELO, VIVIANI FILIPPO
( syllabus)
Affine Spaces Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.
Varieties Projective spaces and their Zariski topology. Quasiprojective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasiprojective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.
Local geometry Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.
( reference books)
L. Caporaso Introduction to algebraic geometry Notes of the course
I. Shafarevich Basic Algebraic geometry SpringerVerlag, Berlin, 1994

9

MAT/03

48

24





Core compulsory activities

ITA 
20410417 
IN410Computability and Complexity
(objectives)
Improve the understanding of the mathematical aspects of the notion of computation, and study the relationships between different computational models and the computational complexity.

Derived from
20410417 IN410CALCOLABILITÀ E COMPLESSITÀ in Scienze Computazionali LM40 PEDICINI MARCO
( syllabus)
1) Computability, complexity and representability:
 Introduction to decision problems, algorithmic and nonalgorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semidecidability 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 semirings. Nondeterministic finite automata. Regular Languages. Equivalence between deterministic and nondeterministic automata.
 Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worstcase 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 multitape 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).
 Timeconstructible functions. The notion of Tclock. Examples of some time constructible function. Closure by composition.
 Nondeterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial nondeterministic functions. NPcomplete problems.
2) Lambda calculus and functional programming:
 Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. αequivalence. alphaequivalence passes to the context. The transitive closure of a relationship, owned by ChurchRosser. Listing of lambdaterms concerning alphaequivalence.
 Definition of betareduction and betaequivalence. ChurchRosser's theorem for betareduction. Normal forms for betareduction. Betareduction strategies. Normalizing strategy: left reduction (left mostouter 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 lambdacalculus: pairs, lists, trees, the solution of recursive equations on lambdaterms ([2] chapters 1, 2, 5).
( reference books)
[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGERVERLAG, (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).

9

MAT/01

48

24





Core compulsory activities

ITA 
20410451 
LM410 THEOREMS IN LOGIC 1
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems.


204104511 
LM410 THEOREMS IN LOGIC 1  Module A
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems.

6

MAT/01

32

16





Core compulsory activities

ITA 
204104512 
LM410 THEOREMS IN LOGIC 1  Module B
(objectives)
To acquire a good knowledge of first order classical logic and its fundamental theorems.

3

MAT/01

16

8





Core compulsory activities

ITA 
20410428 
CR510 – ELLIPTIC CRYPTOSYSTEMS
(objectives)
Acquire a basic knowledge of the concepts and methods related to the theory of public key cryptography using the group of points of an elliptic curve on a finite field. Apply the theory of elliptic curves to classical problems of computational number theory such as factorization and primality testing.

6

MAT/02

48

12





Core compulsory activities

ITA 
20410593 
AC310  Complex analysis
(objectives)
To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals.

Derived from
20410593 AC310ANALISI COMPLESSA in Matematica L35 BESSI UGO
( syllabus)
Complex numbers; holomorphic functions and the Cauchy_Riemann formula. Some examples of holomorphic functions; the Riemann sphere and e the point at infinity. The linear fractional transformations. Integral of a complex function along a curve; index of a point with respect to a curve. Cauchy theorem; Cauchy formula. The Liouville theorem. The mean value thorem, the maximum principle and the principle of identity of holomorphic functions. The almostuniform limit of holomorphic functions is holomorphic. Shcwarz lemma and the automorphisms of the disc. The metric of Poincare' on the disc and its geodesics. Laurent series; the general form of Cauchy theorem. Removable singularities; poles and essential singularities; the CasoratiWeierstrass theorem. Euler's product for the sine. Meromorphic functions. The argument principle and the theorem of Rouche'. Holomorphic maps are open; the almost uniform limit of univalent functions is eithe univalent or constant; Lagrange inversion formula. Harmonic functions; the mean value property, the maximum principle and Dirichlet problem; Poisson kernel; continuous functions with the mean value property are harmonic. Schwarz reflection principle. Analytic extension. Jensen's formula for the zeroes of a holomorphic function. Normal families and compactness for the almostuniform topology. The Riemann mapping theorem. When two rings are conformally equivalent. The small theorem of Picard. Holomorphic functions and fluidodynamics.
( reference books)
W. Rudin, Real and complex Analysis, McGrawHill.

9

MAT/05

48

24





Core compulsory activities

ITA 
20410625 
CR410Public Key Criptography


204106251 
CR410  Public Key Criptography  MODULE A

6

MAT/02

48

12





Core compulsory activities

ITA 
204106252 
CR410Public Key Criptography  MODULE B

Derived from
204106252 CR410CRITTOGRAFIA A CHIAVE PUBBLICA  MODULO B in Scienze Computazionali LM40 Onofri Elia
( syllabus)
The course is made of 6 lab lectures and will cover the implementation of some of the algorithm introduced in lectures from Module A, as well as some topics in realworld cryptography from recent literature. The course also covers the basics of scientific programming in C language and, in particular, it introduces the usage of the multiple precision arithmetic library GMP.
( reference books)
See Bibliography from Module A for further details, in particular:  Stinson: Cryptography  theory and practice  Languasco, Zaccagnini: Manuale di crittografia  Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici

3

MAT/02



12





Core compulsory activities

ITA 
20410613 
LM430Logic and mathematical foundations
(objectives)
To acquire the basic notions of ZermeloFraenkel's axiomatic set theory and present some problems related to that theory.

Derived from
20410613 LM430  LOGICA E FONDAMENTI DELLA MATEMATICA
in Matematica LM40 TORTORA DE FALCO LORENZO
( syllabus)
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 ZermeloFraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and ZermeloFraenkel’s theory, extensions of the language by definition. Ordinals: orders, wellorders and wellfoundedness, wellfoundedness and induction principle, the ordinal numbers, wellorders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in prooftheory. 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.
( reference books)
V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

6

MAT/01

48

12





Core compulsory activities

ITA 
20410757 
AM410  AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS



AM410 MODULE A  AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

3

MAT/05

24

6





Core compulsory activities

ITA 

AM410  MODULE B  AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

3

MAT/05

24

6





Core compulsory activities

ITA 
20410756 
AM420  PARTIAL DIFFERENTIAL EQUATIONS
(objectives)
To acquire a good knowledge of advanced techniques necessary for the study of partial differential equations

Derived from
20410756 AM420  EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM40 HAUS EMANUELE, FEOLA ROBERTO
( syllabus)
1. Theory of distributions
Definition and convergence of distributions. Derivative of a distribution. Spaces of distributions. Convolutions. Schwartz space, tempered distributions. Fourier transform of tempered distributions. Sobolev spaces and pseudodifferential operators.
2. Application to Schrödinger equations
Linear Schrödinger equation, local and global smoothing effects. Nonlinear Schrödinger equation, local theory in L^2, H^1, H^2. Asymptotic results and singularity formation for NLS.
( reference books)
Bony  Cours d'analyse, Théorie des distributions et analyse de Fourier Ponce, Linares  Introduction to nonlinear dispersive equations

6

MAT/05

48

12





Core compulsory activities

ITA 
20410465 
GE450  ALGEBRAIC TOPOLOGY
(objectives)
To explain ideas and methods of algebraic topology, among which cohomology, homology and persistent homology. To understand the application of these theories to data analysis (Topological Data Analysis).

6

MAT/03

48

12





Core compulsory activities

ITA 
20410567 
GE470  Riemann surfaces
(objectives)
Acquire a sufficiently broad knowledge of the topological, analytical and geometric aspects of the theory of Riemann surfaces.

6

MAT/03

48

12





Core compulsory activities

ITA 

Optional group:
CURRICULUM MODELLISTICA FISICA E SIMULAZIONI NUMERICHE: scegliere 3 Insegnamenti (24 CFU) nei seguenti SSD MAT/06, MAT/07, MAT/08, MAT/09 tra le attività caratterizzanti (B), di cui almeno 1 Insegnamento (6 CFU) nel SSD MAT/06, 1 Insegnamento (6 CFU) nel SSD MAT/07 e 1 Insegnamento (6 CFU) nel SSD MAT/08  (show)

24








20410410 
FM310  Equations of Mathematical Physics
(objectives)
To acquire a good knowledge of the elementary theory of partial differential equations and of the basic methods of solution, with particular focus on the equations describing problems in mathematical physics.

9

MAT/07

48

24





Core compulsory activities

ITA 
20410413 
AN410  NUMERICAL ANALYSIS 1
(objectives)
Provide the basic elements (including implementation in a programming language) of elementary numerical approximation techniques, in particular those related to solution of linear systems and nonlinear scalar equations, interpolation and approximate integration.

Derived from
20410413 AN410  ANALISI NUMERICA 1 in Matematica L35 FERRETTI ROBERTO
( syllabus)
Linear Systems Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, GaussSeidel, 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 NewtonCotes formulae. Stability results and error estimation. Generalized NewtonCotes 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.
( reference books)
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

9

MAT/08

48

24





Core compulsory activities

ITA 
20410416 
FM410Complements of Analytical Mechanics
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.


20410421 
AN430 Finite Element Method
(objectives)
Introduce to the finite element method for the numerical solution of partial differential equations, in particular: computational fluid dynamics, transport problems; computational solid mechanics.

Derived from
20410421 AN430  METODO DEGLI ELEMENTI FINITI in Scienze Computazionali LM40 TERESI LUCIANO
( syllabus)
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.
( reference books)
1) Integral Form at a Glance, note a cura del docente
2) When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes, 2011
3) Understanding and Implementing the Finite Elements Method Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s Cap. 2 The weak form 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)

6

MAT/08

48

12





Core compulsory activities

ITA 
20410447 
CP410  Theory of Probability
(objectives)
Foundations of modern probability theory: measure theory, 0/1 laws, independence, conditional expectation with respect to sub sigma algebras, characteristic functions, the central limit theorem, branching processes, discrete parameter martingale theory.

Derived from
20410414 CP410  TEORIA DELLA PROBABILITÀ in Matematica L35 CANDELLERO ELISABETTA
( syllabus)
Branching processes, introduction to Sigmaalgebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pisystems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. BorelCantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 01 law. Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws. Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers. Doob's inequalities for martingales and submartingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.
( reference books)
D. Williams, Probability with martingales R. Durrett, Probability: Theory and examples

9

MAT/06

48

24





Core compulsory activities

ITA 
20410555 
ST410 Statistics
(objectives)
Introduction to the basics of mathematical statistics and data analysis, including quantitative numerical experiments using suitable statistical software.

Derived from
20410555 ST410STATISTICA in Scienze Computazionali LM40 MARTINELLI FABIO
( syllabus)
Random variables and their distribution, moment generating function, mean variance and covariance. Random sampling model and statistical model. Statistics: concept, examples, sufficient statistics. Point estimators: definition and desired properties, moments, maximum likelihood and Bayes. Computational methods: NewtonRaphson, EM algorithm Improving an estimator: RaoBlackwell, UMVU estimator, full statistic, LehmanScheff ́e II and CramerRao Confidence intervals: intuitive, pivotal quantity, IC for Bayes and asymptotic IC. Hypothesis testing: likelihood ratio, pivotal quantity test (Z and T test), duality with IC, UMP, NeymanPearson and KarlinRubin tests. Nonparametric methods: goodnessoffit, contingency table, KolmogorovSmirnov and ranking tests. Analysis of variance (ANOVA) and F. Regression: linear, multiple linear, generalized linear and Logistic / Poisson
( reference books)
Statistical Inference, Casella e Berger, 2nd Edition, Duxbury Advanced Series.

6

MAT/06

48

12





Core compulsory activities

ITA 
20410773 
IN570 – Quantum Computing
(objectives)
This course introduces basic concepts of quantum computation through the study of those physical phenomena that characterize this paradigm by comparing to the classical one. The course is divided into three main parts: the study of the quantum circuit model and its universality, the study of the most important quantum techniques for the design of algorithms and their analysis, and the introduction of quantum programming languages and software platforms for the specification of quantum computations.

Derived from
20410773 IN570 – QUANTUM COMPUTING in Scienze Computazionali LM40 PEDICINI MARCO
( syllabus)
Basic Linear Algebra: Hilbert Spaces, Products and Tensor Products, Matrices, Complex Spaces and Inner Products, Matrices, Graphs, and Sums Over Paths. Boolean Functions, Quantum Bits, and Feasibility: Feasible Boolean Functions, Quantum Representation of Boolean Arguments Quantum Feasibility. Special Matrices: Hadamard Matrices, Fourier Matrices, Reversible Computation and Permutation Matrices, Feasible Diagonal Matrices, Reflections. Tricks: Start Vectors, Controlling and Copying Base States, The CopyUncompute Trick, Superposition Tricks, Flipping a Switch, Measurement Tricks, Partial Transforms. Algorithms: Phil’s Algorithm: Phil Measures Up, Quantum Mazes versus Circuits versus Matrices. Deutsch’s Algorithm: Superdense Coding and Teleportation. The DeutschJozsa Algorithm. Simon’s Algorithm. Shor’s Algorithm, Quantum Part of the Algorithm, Analysis of the Quantum Part, Continued Fractions. FactoringIntegers: Basic Number Theory, Periods Give the Order, Factoring. Grover’s Algorithm: The binary case, the general case, with k Unknowns, Grover Approximate Counting. QuantumWalks: Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.
( reference books)
Richard J. Lipton, Kenneth W. Regan Introduction to Quantum Algorithms via Linear Algebra, Second Edition, ISBN 9780262045254, (2021), MIT Press;

9

MAT/09

48

24





Core compulsory activities

ITA 
20410457 
CP430  STOCHASTIC CALCULUS
(objectives)
Elements of stochastic analysis: Gaussian processes, Brownian motion, probabilistic representation for the solution to partial differential equations, stochastic integration and stochastic differential equations.

Derived from
20410457 CP430  CALCOLO STOCASTICO in Matematica LM40 CANDELLERO ELISABETTA
( syllabus)
Brownian motion: definition and property of BM, continuity and nondifferentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multidimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.
Stochastic integration: PaleyWienerZygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.
( reference books)
Brownian Motion (Moerters and Peres): http://www.mi.unikoeln.de/~moerters/book/book.pdf
An introduction to Stochastic Differential Equations (Evans)

6

MAT/06









Core compulsory activities

ITA 

Optional group:
GRUPPO UNICO: Scegliere 4 insegnamenti (30 CFU) nei seguenti SSD FIS, INF/01, INGINF/03, INGINF/04, INGINF/05, MAT/04,06,07,08,09, SECSS/01,SECSS/06 TRA LE ATTIVITA’ AFFINI INTEGRATIVE (C), di cui almeno 1 Insegnamento (6 CFU) nel SSD INF/01 nei curricula MODELLISTICA FISICA E SIMULAZIONI NUMERICHE e almeno 2 Insegnamenti (12 CFU) nel SSD INF/01 nei curricula GESTIONE E PROTEZIONE DEI DATI e ANALISI DEI DATI E STATISTICA  (show)

30








20410413 
AN410  NUMERICAL ANALYSIS 1
(objectives)
Provide the basic elements (including implementation in a programming language) of elementary numerical approximation techniques, in particular those related to solution of linear systems and nonlinear scalar equations, interpolation and approximate integration.

Derived from
20410413 AN410  ANALISI NUMERICA 1 in Matematica L35 FERRETTI ROBERTO
( syllabus)
Linear Systems Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, GaussSeidel, 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 NewtonCotes formulae. Stability results and error estimation. Generalized NewtonCotes 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.
( reference books)
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

9

MAT/08

48

24





Related or supplementary learning activities

ITA 
20410447 
CP410  Theory of Probability
(objectives)
Foundations of modern probability theory: measure theory, 0/1 laws, independence, conditional expectation with respect to sub sigma algebras, characteristic functions, the central limit theorem, branching processes, discrete parameter martingale theory.

Derived from
20410414 CP410  TEORIA DELLA PROBABILITÀ in Matematica L35 CANDELLERO ELISABETTA
( syllabus)
Branching processes, introduction to Sigmaalgebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pisystems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. BorelCantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 01 law. Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws. Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers. Doob's inequalities for martingales and submartingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.
( reference books)
D. Williams, Probability with martingales R. Durrett, Probability: Theory and examples

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20410416 
FM410Complements of Analytical Mechanics
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.


20410421 
AN430 Finite Element Method
(objectives)
Introduce to the finite element method for the numerical solution of partial differential equations, in particular: computational fluid dynamics, transport problems; computational solid mechanics.

Derived from
20410421 AN430  METODO DEGLI ELEMENTI FINITI in Scienze Computazionali LM40 TERESI LUCIANO
( syllabus)
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.
( reference books)
1) Integral Form at a Glance, note a cura del docente
2) When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes, 2011
3) Understanding and Implementing the Finite Elements Method Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s Cap. 2 The weak form 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)

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20410436 
FS420  QUANTUM MECHANICS
(objectives)
Provide a basic knowledge of quantum mechanics, discussing the main experimental evidence and the resulting theoretical interpretations that led to the crisis of classical physics, and illustrating its basic principles: notion of probability, waveparticle duality, indetermination principle. Quantum dynamics, the Schroedinger equation and its solution for some relevant physical systems are then described.

Derived from
20410015 MECCANICA QUANTISTICA in Fisica L30 LUBICZ VITTORIO, TARANTINO CECILIA
( syllabus)
The crisis of classical physics. Waves and particles. State vectors and operators. Measurements, observables and uncertainty relation. The position operator. Translations and momentum. Time evolution and the Schrödinger equation. Onedimensional problems. Parity. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.
( reference books)
Lecture notes available on the course website
J.J. Sakurai, Jim Napolitano  Meccanica Quantistica Moderna  Zanichelli An english version of the book is also available: Sakurai J.J., Modern Quantum Mechanics  AddisonWesley

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20410437 
FS430 Theory of Relativity
(objectives)
Make the student familiar with the theoretical underpinnings of General Relativity, both as a geometric theory of spacetime and by stressing analogies and differences with the field theories based on local symmetries that describe the interactions among elementary particles. Illustrate the basic elements of differential geometry needed to correctly frame the various concepts. Introduce the student to extensions of the theory of interest for current research.

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20410435 
FS440  Data Acquisition and Experimental Control
(objectives)
The lectures and laboratories allow the student to learn the basic concepts pinpointing the data acquisition of a high energy physics experiment with specific regard to the data collection, control of the experiment and monitoring.

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20410424 
IN450  ALGORITHMS FOR CRYPTOGRAPHY
(objectives)
Acquire the knowledge of the main encryption algorithms. Deepen the mathematical skills necessary for the description of the algorithms. Acquire the cryptanalysis techniques used in the assessment of the security level provided by the encryption systems.

PEDICINI MARCO
( syllabus)
1. Classic Cryptography
 Basic cryptosystems: encryption by substitution, by translation, by permutation, affine cryptosystem, by Vigenère, by Hill. Stream encryption (synchronous and asynchronous), Linear feedback shift registers (LFSR) on finite fields, Autokey cypher. Product cyphers. Basic cryptanalysis: classification of attacks; cryptoanalysis for affine cyphers, for substitution cypher (frequency analysis), for Vigenere cypher: Kasiski test, coincidence index; cryptoanalysis of Hill's cypher and LFSR: algebraic attacks, cube attack.
2. Application of Shannon theory to cryptography
 Security of cyphers: computational security, provable security, unconditional security. Basics of probability: discrete random variables, joint probability, conditional probability, independent random variables, Bayes' theorem. Random variables associated with cryptosystems. Perfect secrecy for encryption systems. Vernam cryptosystem. Entropy. Huffman codes. Spurious Keys and Unicity distance.
3. Block cyphers
 iterative encryption schemes; SubstitutionPermutation Networks (SPN); Linear cryptanalysis for SPN: PilingUp Lemma, linear approximation of Sboxes, linear attacks on Sboxes; Differential cryptanalysis for SPN; Feistel cyphers; DES: description and analysis; AES: description; Notes on finite fields: operations on finite fields, Euclid's generalized algorithm for the computation of the GCD and inverse; Operating modes for block cyphers.
4. Hash functions and codes for message authentication
 Hash functions and data integrity. Safe hash functions: resistance to the preimage, resistance to the second preimage, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second preimage, collision on the problem of the preimage. Iterated hash functions; the construction of MerkleDamgard. Safe Hash Algorithm (SHA1). Authentication Codes (MAC): nested authentication codes (HMAC).
( reference books)
[1] Antoine Joux, Algorithmic Cryptanalysis, (2010) CRC Press. [2] Douglas Stinson, Cryptography: Theory and Practice, 3rd edition, (2006) Chapman and Hall/CRC. [3] Delfs H., Knebl H., Introduction to Cryptography, (2007) Springer Verlag.

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20410426 
IN480  PARALLEL AND DISTRIBUTED COMPUTING
(objectives)
Acquire parallel and distributed programming techniques, and know modern hardware and software architectures for highperformance scientific computing. Parallelization paradigms, parallelization on CPU and GPU, distributed memory systems. Dataintensive, Memory Intensive and Compute Intensive applications. Performance analysis in HPC systems.

CIANFRIGLIA MARCO
( syllabus)
 Introduction to parallel and distributed computing  Base concepts: hardware architectures and memory hierarchy  The C language  Parallel programming models  Message Passing Interface (MPI)  OpenMP  Introduction to general purpose programming on Graphics Processing Unit (GPU)  CUDA and OpenCL
( reference books)
Introduction to Parallel Computing: From Algorithms to Programming on StateoftheArt Platforms, Trobec, Slivnik, Bulić, Robič, Springer
Programming on Parallel Machines, Norm Matloff

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20410427 
IN490  PROGRAMMING LANGUAGES
(objectives)
Introduce the main concepts of formal language theory and their application to the classification of programming languages. Introduce the main techniques for the syntactic analysis of programming languages. Learn to recognize the structure of a programming language and the techniques to implement its abstract machine. Study the objectoriented paradigm and another nonimperative paradigm.

LOMBARDI FLAVIO
( syllabus)
The objective of Linguaggi di Programmazione course is to introduce main formal language theory concepts and results as well as their application for programming language classification. Most relevant approaches for syntactic analysis of programming languages are introduced. Learning how to recognize the structure of a programming language and the implementation techniques for the abrstract machine. Understanding the Object Oriented paradigm together with other non imperative approaches.
( reference books)
[1] Maurizio Gabbrielli, Simone Martini,Programming Languages  Principles and paradigms, 2/ed. McGrawHill, (2011). [2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, 2 edizione. O’Reilly Media, (2014). [3] David Parsons, Foundational Java Key Elements and Practical Programming. Springer Verlag, (2012). Course Slides

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20410429 
FS510  MONTECARLO METHODS
(objectives)
Acquire the basic elements for dealing with mathematics and physics problems using statistical methods based on random numbers.

FRANCESCHINI ROBERTO
( syllabus)
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
Curvefitting, leastsquares, 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
( reference books)
Weinzierl, S.  Introduction to Monte Carlo methods arXiv:hepph/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

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20410432 
IN550 – MACHINE LEARNING
(objectives)
Learn to instruct a computer to acquire concepts using data, without being explicitly programmed. Acquire knowledge of the main methods of supervised and nonsupervised machine learning, and discuss the properties and criteria of applicability. Acquire the ability to formulate correctly the problem, to choose the appropriate algorithm, and to perform the experimental analysis in order to evaluate the results obtained. Take care of the practical aspect of the implementation of the introduced methods by presenting different examples of use in different application scenarios.

BONIFACI VINCENZO
( syllabus)
1. Machine learning. Types of learning. Loss functions. Empirical risk minimization. Generalization and overfitting. 2. Model optimization. Convex functions. Gradient descent. Stochastic gradient descent. 3. Regression. Linear regression. Basis functions. Feature selection. Regularization. 4. Classification. Generative models. Nearest neighbor. Logistic regression. Support vector machines. Neural networks. 5. Ensemble methods. Decision trees. Boosting. Bagging. 6. Unsupervised learning. Kmeans clustering. Hierarchical clustering. Principal component analysis. 7. Application of the methods using the Python language. Examples using the NumPy, Pandas, SciKitLearn, and TensorFlow libraries.
( reference books)
J. Watt, R. Borhani, A. Katsaggelos. Machine Learning Refined. Cambridge University Press, 2nd edition, 2020.

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20410410 
FM310  Equations of Mathematical Physics
(objectives)
To acquire a good knowledge of the elementary theory of partial differential equations and of the basic methods of solution, with particular focus on the equations describing problems in mathematical physics.

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20410555 
ST410 Statistics
(objectives)
Introduction to the basics of mathematical statistics and data analysis, including quantitative numerical experiments using suitable statistical software.

Derived from
20410555 ST410STATISTICA in Scienze Computazionali LM40 MARTINELLI FABIO
( syllabus)
Random variables and their distribution, moment generating function, mean variance and covariance. Random sampling model and statistical model. Statistics: concept, examples, sufficient statistics. Point estimators: definition and desired properties, moments, maximum likelihood and Bayes. Computational methods: NewtonRaphson, EM algorithm Improving an estimator: RaoBlackwell, UMVU estimator, full statistic, LehmanScheff ́e II and CramerRao Confidence intervals: intuitive, pivotal quantity, IC for Bayes and asymptotic IC. Hypothesis testing: likelihood ratio, pivotal quantity test (Z and T test), duality with IC, UMP, NeymanPearson and KarlinRubin tests. Nonparametric methods: goodnessoffit, contingency table, KolmogorovSmirnov and ranking tests. Analysis of variance (ANOVA) and F. Regression: linear, multiple linear, generalized linear and Logistic / Poisson
( reference books)
Statistical Inference, Casella e Berger, 2nd Edition, Duxbury Advanced Series.

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20410560 
IN400 Python and MATLAB programming
(objectives)
Acquire the ability to implement highlevel programs in the interpreted languages Python and MATLAB. Understand the main constructs used in Python and MATLAB and their application to scientific computing and data processing scenarios.


204105601 
MODULO A  PYTHON programming
(objectives)
Acquire the ability to implement highlevel programs in the interpreted language Python . Understand the main constructs used in Python and its application to scientific computing and data processing scenarios.

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204105602 
MODULO B  MATLAB programming

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20410773 
IN570 – Quantum Computing
(objectives)
This course introduces basic concepts of quantum computation through the study of those physical phenomena that characterize this paradigm by comparing to the classical one. The course is divided into three main parts: the study of the quantum circuit model and its universality, the study of the most important quantum techniques for the design of algorithms and their analysis, and the introduction of quantum programming languages and software platforms for the specification of quantum computations.

Derived from
20410773 IN570 – QUANTUM COMPUTING in Scienze Computazionali LM40 PEDICINI MARCO
( syllabus)
Basic Linear Algebra: Hilbert Spaces, Products and Tensor Products, Matrices, Complex Spaces and Inner Products, Matrices, Graphs, and Sums Over Paths. Boolean Functions, Quantum Bits, and Feasibility: Feasible Boolean Functions, Quantum Representation of Boolean Arguments Quantum Feasibility. Special Matrices: Hadamard Matrices, Fourier Matrices, Reversible Computation and Permutation Matrices, Feasible Diagonal Matrices, Reflections. Tricks: Start Vectors, Controlling and Copying Base States, The CopyUncompute Trick, Superposition Tricks, Flipping a Switch, Measurement Tricks, Partial Transforms. Algorithms: Phil’s Algorithm: Phil Measures Up, Quantum Mazes versus Circuits versus Matrices. Deutsch’s Algorithm: Superdense Coding and Teleportation. The DeutschJozsa Algorithm. Simon’s Algorithm. Shor’s Algorithm, Quantum Part of the Algorithm, Analysis of the Quantum Part, Continued Fractions. FactoringIntegers: Basic Number Theory, Periods Give the Order, Factoring. Grover’s Algorithm: The binary case, the general case, with k Unknowns, Grover Approximate Counting. QuantumWalks: Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.
( reference books)
Richard J. Lipton, Kenneth W. Regan Introduction to Quantum Algorithms via Linear Algebra, Second Edition, ISBN 9780262045254, (2021), MIT Press;

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20410457 
CP430  STOCHASTIC CALCULUS
(objectives)
Elements of stochastic analysis: Gaussian processes, Brownian motion, probabilistic representation for the solution to partial differential equations, stochastic integration and stochastic differential equations.

Derived from
20410457 CP430  CALCOLO STOCASTICO in Matematica LM40 CANDELLERO ELISABETTA
( syllabus)
Brownian motion: definition and property of BM, continuity and nondifferentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multidimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.
Stochastic integration: PaleyWienerZygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.
( reference books)
Brownian Motion (Moerters and Peres): http://www.mi.unikoeln.de/~moerters/book/book.pdf
An introduction to Stochastic Differential Equations (Evans)

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Optional group:
12 CFU a scelta dello studente: Nei percorsi formativi proposti scegliere gli insegnamenti in base a precise esigenze formative nel seguente modo: 2 insegnamenti oppure 1 insegnamento e QLM. Si rinvia al regolamento per suggerimenti.  (show)

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