Optional group:
CURRICULUM TEORICO SCEGLIERE QUATTRO INSEGNAMENTI (30 CFU) NEI SEGUENTI SSD MAT/01,02,03,05 TRA LE ATTIVITÀ CARATTERIZZANTI (B)  (show)

30








20410407 
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
20410334 AC310ANALISI COMPLESSA in Matematica L35 BESSI UGO
( syllabus)
Complex numbers; some complex maps; the Riemann sphere; fractionallinear transformations; they preserve the cross ratio and the set of circles and straight lines. The apollonius circles. Integral of a complex function along a curve; length of a curve; properties of the integral; indicator function; formal sum of curves. Cauchy's theorem on rectangles and on all curves; Caucgy's formula. Liouville's principle. The identity principle for holomorphic functions; removable singularities; almost uniform convergence and its properties. Morera's lemma; mean value theorem and maximum principle; harmonic functions locally are the real part of holomorphic functions. Hadamard's three circles theorem. Euler's product for the sine. Harmonic functions and the electric potential; mean value theorem for harmonic functions, maximum principle and uniqueness for the Dirichlet problem. Dirichlet's kernel; the functions with the mean value property are harmonic. Schwarz's reflection principle. Laurent series; residues and residue calculus. The argument principle and Rouche's theorem. Holomorphic maps are open; a variant of the argument principle and Lagrange's inversion formula; the almost uniform limit of univalent functions is either univalent or constant. Schwarz's lemma; automorphisms of the disc; hyperbolic metric on the disc and its geodesics; the automorphisms of the disc preserve the hyperbolic metric. Analytic extension and monodromy theorem; some examples of Riemann surfaces. The Riemann mapping theorem; Picard's small theorem.
( reference books)
W. Rudin, Analisi reale e complessa.
J. B. Conway, Functions of one complex variable.

9

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Core compulsory activities

ITA 
20410445 
AL410  COMMUTATIVE ALGEBRA
(objectives)
Acquire a good knowledge of some methods and fundamental results in the study of the commutative rings and their modules, with particular reference to the study of ring classes of interest for the algebraic theory of numbers and for algebraic geometry.

Derived from
20410445 AL410  ALGEBRA COMMUTATIVA in Matematica LM40 LELLI CHIESA MARGHERITA
( syllabus)
Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.
( reference books)
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. AddisonWesley, 1996. M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988. D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, SpringerVerlag, 1995. A. Gathmann, Commutative Algebra, Lecture notes.

9

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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.


20410455 
LM420  THEOREMS IN LOGIC 2
(objectives)
To support the students into an indepth analysis of the main results of first order classical logic and to study some of their remarkable consequences.

Derived from
20710122 TEOREMI SULLA LOGICA, 2 in Scienze filosofiche LM78 TORTORA DE FALCO LORENZO
( syllabus)
Logic and Arithmetic: incompleteness
Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semidecidability, undecidability.
Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.
( reference books)
V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

6

MAT/01

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Core compulsory activities

ITA 
20410444 
GE430  RIEMANNIAN GEOMETRY
(objectives)
Introdue to the study of Riemannian geometry, in particular by addressing the theorems of GaussBonnet and HopfRinow.

6

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48

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Core compulsory activities

ITA 
20410458 
LM430  LOGICAL THEORIES 2
(objectives)
To acquire the basic notions of ZermeloFraenkel's axiomatic set theory and present some problems related to that theory.

Derived from
20710092 TEORIE LOGICHE 2  LM in Scienze filosofiche LM78 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

36







Core compulsory activities

ITA 
20410460 
AM450  FUNCTIONAL ANALYSIS
(objectives)
To acquire a good knowledge of functional analysis: Banach and Hilbert spaces, weak topologies, linear and continuous operators, compact operators, spectral theory.

Derived from
20410460 AM450  ANALISI FUNZIONALE in Matematica LM40 BATTAGLIA LUCA
( syllabus)
Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems. HahnBanach theorem, analytic and geometric form, consequences. First and second category spaces, Baire's Theorem, BanachSteinhaus Theorem, open map and closed graph theorem, applications. Weak topologies, closed and convex sets, BanachAlaoglu theorem, separability and reflexivity. Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems. Spectral theory, Fredholm alternative, spectral theorem for compact and selfadjoint operators, application to variational problems.
( reference books)
H. Brezis  Analisi Funzionale  Liguori (1986) H. Brezis  Functional Analysis, Sobolev Spaces and Partial Differential Equations  Springer (2010) W. Rudin  Functional Analysis  McGrawHill (1991)

6

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Core compulsory activities

ITA 
20410425 
GE460 GRAPH THEORY
(objectives)
Provide tools and methods for graph theory.

Derived from
20410425 GE460  TEORIA DEI GRAFI in Scienze Computazionali LM40 MASCARENHAS MELO ANA MARGARIDA
( syllabus)
Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''. Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length. Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semiEulerian graphs. Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac. the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph. Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem. Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations. Euler's formula for planar graphs. The dual of a planar graph. Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties. Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces. Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem. Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability. The MaxFlow MinCut theorem and Menger's theorem. Complexity of algorithms and applications to the theory of graphs. Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability. Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids. Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids. Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix. Basis for the space of the cycles and for the space of the cuts of a graph. The RiemannRoch theorem for graphs. Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph. Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications. Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank. Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected. Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.
( reference books)
R. Diestel: Graph theory, Spriger GTM 173. R. Wilson: Introduction to Graph theory, Prentice Hall. B. Bollobas: Modern Graph theory, Springer GTM 184. J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244. N. Biggs: Algebraic graph theory, Cambridge University Press. C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207. J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

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Core compulsory activities

ITA 
20410518 
AM420  SOBOLEV SPACES AND PARTIAL DERIVATIVE EQUATIONS
(objectives)
To acquire a good knowledge of the general methods andÿclassical techniques necessary for the study ofÿweak solutions of partial differential equations.

Derived from
20410518 AM420  SPAZI DI SOBOLEV ED EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM40 HAUS EMANUELE, FEOLA ROBERTO
( syllabus)
Preliminaries  Weak topologies and weak convergence, weak lower semicontinuity of the norm  L^P spaces: reflexivity, separability, criteria for strong compactness.
Sobolev spaces and variational formulation of boundary value problems in dimension one  Motivations  The Sobolev space W^{1,p} (I)  The space W^{1,p}_0 (I)  Some examples of boundary value problems  Maximum principle
Sobolev spaces and variational formulation of boundary value problems in dimension N  Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)  Extension operators  Sobolev inequalities  The space W^{1,p}_0 (Omega)  Variational formulation of some elliptic boundary value problems  Existence of weak solutions  Regularity of weak solutions  Maximum principle
( reference books)
Functional analysis, H. Bre'zis

6

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48

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Core compulsory activities

ITA 
20410520 
AL420  ALGEBRAIC THEORY OF NUMBERS
(objectives)
Acquire methods and techniques of modern algebraic theory of numbers through classic problems initiated by Fermat, Euler, Lagrange, Dedekind, Gauss, Kronecker.

6

MAT/02

48

12





Core compulsory activities

ITA 

Optional group:
GRUPPO UNICO: SCEGLIERE QUATTRO INSEGNAMENTI (30 CFU) TRA LE ATTIVITÀ AFFINI INTEGRATIVE (C)  (show)

30








20410407 
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
20410334 AC310ANALISI COMPLESSA in Matematica L35 BESSI UGO
( syllabus)
Complex numbers; some complex maps; the Riemann sphere; fractionallinear transformations; they preserve the cross ratio and the set of circles and straight lines. The apollonius circles. Integral of a complex function along a curve; length of a curve; properties of the integral; indicator function; formal sum of curves. Cauchy's theorem on rectangles and on all curves; Caucgy's formula. Liouville's principle. The identity principle for holomorphic functions; removable singularities; almost uniform convergence and its properties. Morera's lemma; mean value theorem and maximum principle; harmonic functions locally are the real part of holomorphic functions. Hadamard's three circles theorem. Euler's product for the sine. Harmonic functions and the electric potential; mean value theorem for harmonic functions, maximum principle and uniqueness for the Dirichlet problem. Dirichlet's kernel; the functions with the mean value property are harmonic. Schwarz's reflection principle. Laurent series; residues and residue calculus. The argument principle and Rouche's theorem. Holomorphic maps are open; a variant of the argument principle and Lagrange's inversion formula; the almost uniform limit of univalent functions is either univalent or constant. Schwarz's lemma; automorphisms of the disc; hyperbolic metric on the disc and its geodesics; the automorphisms of the disc preserve the hyperbolic metric. Analytic extension and monodromy theorem; some examples of Riemann surfaces. The Riemann mapping theorem; Picard's small theorem.
( reference books)
W. Rudin, Analisi reale e complessa.
J. B. Conway, Functions of one complex variable.

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Related or supplementary learning activities

ITA 
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.

Derived from
20410342 FM310  ISTITUZIONI DI FISICA MATEMATICA in Matematica L35 PELLEGRINOTTI ALESSANDRO, CORSI LIVIA
( syllabus)
Classification of the semilinear partial differential equations in any dimension. Classification in 2 dimensions and canonical forms. Wave equation in an interval using the separation of variables method: homogeneous case, general case. Wave equation on the line: D’Alambert’s solution. The half line case. Wave equation in R3. Heat equation: deduction from a random walk on the line. Solution of the heat equation on the line. Maximum principle. Applications of the maximum principle to unicity theorems. Heat equation in a interval using the separation of variables method. Different generalizations of this problem. Introduction to the elliptic equations. Laplacian in spherical and cylindrical coordinate systems. Representation formula via Green’s formula. Properties of harmonic fuc tions. Maximum principle. Unicity results for the inner problems. Problem of existence for a circular domain. Poisson formula. External problems. Unicity theorems in the plane and in the space. External problem for the circle. Green functions: appllications to the sphere case and in the semispace case.Potential theory. Introduction to Quantum Mechanics. Schroedinger equa tion. Separation of variables. Free particle in a interval. Potential barrier. Harmonic oscillator. Hydrogen atom.
( reference books)
A.N. Tichonov, A.A. Samarskij Equazioni della Fisica Matematica Edizioni MIR

9

MAT/07

48

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Related or supplementary learning activities

ITA 
20410445 
AL410  COMMUTATIVE ALGEBRA
(objectives)
Acquire a good knowledge of some methods and fundamental results in the study of the commutative rings and their modules, with particular reference to the study of ring classes of interest for the algebraic theory of numbers and for algebraic geometry.

Derived from
20410445 AL410  ALGEBRA COMMUTATIVA in Matematica LM40 LELLI CHIESA MARGHERITA
( syllabus)
Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the spectrum of a ring. Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars. Rings and modules of fractions, localization. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.
( reference books)
M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. AddisonWesley, 1996. M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988. D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, SpringerVerlag, 1995. A. Gathmann, Commutative Algebra, Lecture notes.

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Related or supplementary learning 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.


204104161 
FM410Complements of Analytical Mechanics  MODULE A
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.

Derived from
20410084 COMPLEMENTI DI MECCANICA ANALITICA  MOD A in Fisica L30 GENTILE GUIDO
( syllabus)
Linear dynamic systems. Forced harmonic oscillation in the presence or absence of dissipation. Limit sets and limit cycles. Planar systems. Gradient systems. Stability theorems. LotkaVolterra equations. Van der pol equation. Euler angles. Euler's equations describing the dynamics of a rigid body.
( reference books)
G. Gentile, Introduzione ai sistemi dinamici. 1. Equazioni differenziali ordinarie, analisi qualitativa e alcune applicazioni, available onlineG. Gentile, Introduzione ai sistemi dinamici. 2. Meccanica lagrangiana e hamiltoniana, available online

3

MAT/07

30







Related or supplementary learning activities

ITA 
204104162 
FM410Complements of Analytical Mechanics  Module B
(objectives)
To deepen the study of dynamical systems, with more advanced methods, in the context of Lagrangian and Hamiltonian theory.

Derived from
20410085 COMPLEMENTI DI MECCANICA ANALITICA  MOD. B in Fisica L30 GENTILE GUIDO
( syllabus)
Trottola di Lagrange. Trasformazione canoniche. Parentesi di Poisson e condizione di Lie. Funzioni generatrici. Teoria delle perturbazioni. Equazione omologica. Sistemi iscocroni e anisocroni. Serie di Birkhoff. Teoria perturbativa a tutti gli ordini per sistemi isocroni e teorema di Nekhoroshev. Teorema KAM.
( reference books)
G. Gentile, Introduzione ai sistemi dinamici. 1. Equazioni differenziali ordinarie, analisi qualitativa e alcune applicazioni, available onlineG. Gentile, Introduzione ai sistemi dinamici. 2. Meccanica lagrangiana e hamiltoniana, available online

3

MAT/07

30







Related or supplementary learning activities

ITA 
20410448 
FS410  DIDACTICS OF PHYSICS WORKSHOP
(objectives)
Learn statistical and laboratory techniques for the preparation of didactic physics experiments.

Derived from
20410448 FS410  LABORATORIO DI DIDATTICA DELLA FISICA in Matematica LM40 DI NARDO ROBERTO, Postiglione Adriana
( syllabus)
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, semilogarithmic graphs, Doublelogarithmic. 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
( reference books)
notes distributed during the classes

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48

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Related or supplementary learning activities

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


20410418 
MA410  APPLIED AND INDUSTRIAL MATHEMATICS
(objectives)
Present a number of problems, of interest for application in various scientific and technological areas. Deal with the modeling aspects as well as those of numerical simulation, especially for problems formulated in terms of partial differential equations.

Derived from
20410418 MA410  MATEMATICA APPLICATA E INDUSTRIALE in Scienze Computazionali LM40 FERRETTI ROBERTO
( syllabus)
Fundamentals of the approximation of Ordinary Differential Equations systems. Onestep methods of forward/backward Euler type and their convergence. Convergence of numerical methods for timedependent Partial Differential Equations, LaxRichtmyer theorem. Advection equation: analytical aspects. Representation formula via characteristics. Monotone methods for the advection equation: Upwind, LaxFriedrichs. Hyperbolic scalar conservation laws in one dimension: analytical aspects, basics on entropic solutions, RankineHougoniot condition. Fundamentals on convergence theory for finitevolume approximations. Monotone finitevolume methods: Godunov, LaxFriedrichs, Rusanov. Linear and nonlinear hyperbolic systems: analytical aspects, characteristic decomposition. Central schemes for hyperbolic systems. The Shallow Water Equations in one and two space dimensions. Central scheme approximation, basics on boundary conditions. The heat equation: analytical aspects, domain of dependence, regularity. Explicit and implicit approximation in one and two space dimensions via centered second differences and Euler time discretization. Modelling of incompressible fluids: the NavierStokes Equations. Approximate formulations (Euler, Stokes), derivation of the Shallow Water Equations. Finite difference numerical methods based on the VorticityStreamfunction formulation.
( reference books)
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press
Additional material provided by the teacher.

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Related or supplementary learning activities

ITA 
20410452 
ME410  ELEMENTARY MATHEMATICS FROM AN ADVANCED POINT OF VIEW
(objectives)
Illustrate, using a critical and unitary approach,ÿsome interesting and classical results and notions that are central for teaching mathematics in high school (focussing, principally, on arithmetics, geometry and algebra). The aim of the course is also to give a contribution to teachers training through the investigation on historical, didactic and cultural aspects of these topics.

Derived from
20410452 ME410  MATEMATICHE ELEMENTARI DA UN PUNTO DI VISTA SUPERIORE in Matematica LM40 MEROLA FRANCESCA
( syllabus)
the course touches on basic algebraic topics: divisibility, Euclid's algorithm, continuous fractions, Fibonacci numbers, combinatorics, straightedge and compass construction, cardinality, lattices and Boole Algebras.
( reference books)
testi consigliati (non da acquistare) Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici Papick, Algebra Connections Materiale e dispense online

6

MAT/04

48

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Related or supplementary learning activities

ITA 
20410438 
MF410  Computational Finance
(objectives)
Basic knowledge of financial markets, introduction to computational and theoretical models for quantitative finance, portoflio optimization, risk analysis. The computational aspects are mostly developed within the Matlab environment.

9

SECSS/06

60







Related or supplementary learning activities

ITA 
20410419 
MS410Statistical Mechanics
(objectives)
To acquire the mathematical basic techniques of statistical mechanics for interacting particle or spin systems, including the study of Gibbs measures and phase transition phenomena, and apply them to some concrete models, such as the Ising model in dimension d = 1,2 and in the mean field approximation.

Derived from
20410419 MS410MECCANICA STATISTICA in Scienze Computazionali LM40 GIULIANI ALESSANDRO
( syllabus)
INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES – Review of equilibrium thermodynamics. Convex functions and Legendre transform. – Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states. – Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles. – The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and nonuniqueness of the Gibbs states
THE ISING MODEL – Known results on the ferromagnetic Ising model in dimensions one or more – GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or  boundary conditions – The onedimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations. – The mean field Ising model (CurieWeiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, longranged, interactions (Kac interactions): the theorem of LebowitzPenrose  Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures. – The LeeYang theorem – Convergence of cluster expansion for the high and low temperature expansions for the pressure of the ddimensional Ising model  The exact solution of the 2D Ising model with h=0
( reference books)
S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017. Available online in preprint version on https://www.unige.ch/math/folks/velenik/smbook/index.html

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MAT/07

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Related or supplementary learning activities

ITA 
20410453 
TN410  INTRODUCTION TO NUMBER THEORY
(objectives)
Acquire a good knowledge of the concepts and methods of the elementary number theory, with particular reference to the study of the Diophantine equations and congruence equations. Provide prerequisites for more advanced courses of algebraic and analytical number theory.

9

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Related or supplementary learning activities

ITA 
20410420 
AN420  NUMERICAL ANALYSIS 2
(objectives)
Introduce to the study and implementation of more advanced numerical approximation techniques, in particular related to approximate solution of ordinary differential equations, and to a further advanced topic to be chosen between the optimization and the fundamentals of approximation of partial differential equations.

Derived from
20410420 AN420  ANALISI NUMERICA 2 in Scienze Computazionali LM40 CACACE SIMONE
( syllabus)
Ordinary Differential Equations Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order RungeKutta methods. Single step implicit methods: backward Euler and CrankNicolson methods. Convergence of single step methods. Multistep methods: general structure, complexity, absolute stability. Stability and consistency of multistep methods. Adams methods, BDF methods, PredictorCorrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")
Partial Differential Equations Finite difference approximation for partial differential equations. Semidiscrete approximations and convergence. The LaxRichtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semidiscrete and fullydiscrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)
( reference books)
Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf
Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf
Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html
Additional notes given by the teacher

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Related or supplementary learning activities

ITA 
20410441 
CP420Introduction to Stochastic Processes
(objectives)
Introduction to the theory of stochastic processes. Markov chains: ergodic theory, coupling, mixing times, with applications to random walks, card shuffling, and the Monte Carlo method. The Poisson process, continuous time Markov chains, convergence to equilibrium for some simple interacting particle systems.

Derived from
20410441 CP420INTRODUZIONE AI PROCESSI STOCASTICI in Scienze Computazionali LM40 MARTINELLI FABIO
( syllabus)
1. Random walks and Markov Chains. Sequence of random variables, random walks, Markov chains in discrete and continuous time. Invariant measures, reversibility. 2. Classical examples. Random walks on graphs, Birth and death chains, exclusion process. Markov Chain Monte Carlo: Metropolis and Glauber dynamics for the Ising model, colorings and other interacting particle systems. 3. Convergence to equilibrium I. Variation distance and mixing time. Ergodic theorems and coupling techniques. Strong stationary times. The coupon collector problem and card shuffling. 4. Convergence to equilibrium II. Spectral gap and relaxation times. Cheeger inequality, conductance and canonical paths. Comparison method and spectral gap for the exclusion process. Logarithmic Sobolev inequality. 5. Other topics: Glauber dynamics for the Ising model, phase transition, cutoff phenomenon, perfect simulation.
( reference books)
D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

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ITA 
20410442 
IN420  Information Theory
(objectives)
Introduce key questions in the theory of signal transmission and quantitative analysis of signals, such as the notions of entropy and mutual information. Show the underlying algebraic structure. Apply the fundamental concepts to code theory, data compression and cryptography.

Derived from
20410442 IN420  TEORIA DELL'INFORMAZIONE in Scienze Computazionali LM40 BONIFACI VINCENZO
( syllabus)
1. Introduction to information theory. Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.
2. Source coding and data compression Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. ZivLempel compression.
3. Channel coding Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.
4. Further codes and applications Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.
( reference books)
David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2004.

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ITA 
20410454 
GL420Elements of Geology II
(objectives)
The course aims to provide an adequate overview of the scientific contents of Earth Sciences. The course deals with the modern aspects of Earth Sciences, framing geological phenomena in the framework of the most modern theories and illustrating the hazards and risks associated with natural phenomena such as, for example, seismic and volcanic phenomena, also referring to the geology of the Italian territory. The course also aims to provide the basis for understanding the rocks cycle and their rocks genetic processes through laboratory and field experiences. During the didactical laboratories and field excursions students will learn to understand the different aspects of Italian territory, with particular regard to its environmental value e fragility.

Derived from
20410328 ELEMENTI DI GEOLOGIA II in Geologia del Territorio e delle Risorse LM74 CIFELLI FRANCESCA
( syllabus)
The materials of the Earth: minerals, the lithogenetic processes, the lithogenetic cycle, the magmatic rocks, the sedimentary rocks, the metamorphic rocks, the bedding and the deformation of the rocks. Volcanic phenomena: magma and volcanic activity, the main types of eruptions, shape of volcanic buildings, products of volcanic activity, the geographic distribution of volcanoes, volcanoes and man (the volcanic risk). Seismic phenomena: the theory of elastic rebound, the seismic cycle, types of seismic waves and their propagation and registration, the force of an earthquake (scales of intensity and magnitude), the geographic distribution of earthquakes, the seismic activity and the man (seismic risk) Plate tectonics: the internal structure of the Earth, the structure of the crust, the Earth's magnetic field, Earth’s internal heat, the convective mantle, from the hypothesis of the drift of the continents to the formulation of the theory of plate tectonics. The Earth as an integrated system: interaction between the different systems of the planet (biosphere, atmosphere, hydrosphere, lithosphere, cryosphere), the earth's atmosphere, climate and meteorological phenomena, renewable and nonrenewable natural resources. Field trip in Caffarella Valley Field trip in the city of Rome
( reference books)
Capire la Terra J.P. Grotzinger, TH Jordan (Terza edizione italiana condotta sulla settima edizione americana)
Il Globo Terrestre e la sua evoluzione E. L. Palmieri e M. Parotto Sesta Edizione (2008)
Educational material distributed during the course

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Related or supplementary learning activities

ITA 
20410455 
LM420  THEOREMS IN LOGIC 2
(objectives)
To support the students into an indepth analysis of the main results of first order classical logic and to study some of their remarkable consequences.

Derived from
20710122 TEOREMI SULLA LOGICA, 2 in Scienze filosofiche LM78 TORTORA DE FALCO LORENZO
( syllabus)
Logic and Arithmetic: incompleteness
Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semidecidability, undecidability.
Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.
( reference books)
V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

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Related or supplementary learning activities

ITA 
20410456 
MC420Dydactics of Mathematics
(objectives)
1. Critical analysis of the evolution of ideas and methodologies in teaching mathematics, with particular emphasis on the role of the teacher. 2. The mathematics curriculum in compulsory schooling and in the various secondary schools (high schools, technical schools and trade schools), in an international context. 3. Didactic planning and methodologies for teaching mathematics: programming and rhythm, principles and methods for the construction of activities, classroom management. 4. Problem solving. Logic, intuition and history in teaching mathematics.

Derived from
20410456 MC420DIDATTICA DELLA MATEMATICA in Matematica LM40 MILLAN GASCA ANA MARIA
( syllabus)
The course is aimed at introducing students to the teaching of mathematics in 6th to 12th grades. Contents include a historical, epistemological approach to the basic concepts in elementary mathematics (numbers, geometry, algebra, probability and functions); a discussion of the origins and present situations of mathematical education in compulsory education and secondary education; and examples regarding the mathematical biography of pupils from preschool, the mathematical anxiety and difficulties in understanding and appropriation of mathematical concepts and vision (intuition, error, deduction and argumentation, math draws, math conversation, meaning and the role of history), and the main elements of a good didactical approach in the classroom.
( reference books)
GIORGIO ISRAEL, ANA MILLÁN GASCA, Pensare in matematica, Zanichelli, 2012. FEDERIGO ENRIQUES 1921, “Insegnamento dinamico”, Periodico di Matematiche, s. IV, 1, pp. 616. http://www.mat.uniroma2.it/mep/Articoli/Enri/Enri.html Altri riferimenti saranno forniti durante il corso.

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Related or supplementary learning activities

ITA 
20410444 
GE430  RIEMANNIAN GEOMETRY
(objectives)
Introdue to the study of Riemannian geometry, in particular by addressing the theorems of GaussBonnet and HopfRinow.

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Related or supplementary learning activities

ITA 
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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Related or supplementary learning activities

ITA 
20410422 
IN430  ADVANCED COMPUTING TECHNIQUES
(objectives)
Acquire the conceptualskills in structuring problems according to the objectoriented programming paradigm. Acquire the ability to design algorithmic solutions based on the objectoriented paradigm. Acquire the basic concepts related to programming techniques based on the objectoriented paradigm. Introduce the fundamental notions of parallel and concurrent programming.

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Related or supplementary learning activities

ITA 
20410458 
LM430  LOGICAL THEORIES 2
(objectives)
To acquire the basic notions of ZermeloFraenkel's axiomatic set theory and present some problems related to that theory.

Derived from
20710092 TEORIE LOGICHE 2  LM in Scienze filosofiche LM78 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

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Related or supplementary learning activities

ITA 
20410459 
MC430  LABORATORY: DIDACTICS FOR MATHEMATICS
(objectives)
1. Mathematics software, with particular attention to their use for teaching mathematics in school. 2. Analysis of the potential and criticality of the use of technological tools for teaching and learning mathematics.

Derived from
20410459 MC430  LABORATORIO DI DIDATTICA DELLA MATEMATICA in Matematica LM40 FALCOLINI CORRADO
( syllabus)
TEACHING MATHEMATICS WITH THE HELP OF A COMPUTER: GEOGEBRA AND MATHEMATICA SOFTWARES. COMMANDS FOR NUMERICAL AND SYMBOLIC CALCULUS, GRAPHICS VISUALIZATION, PARAMETRIC SURFACES AND CURVES WITH ANIMATIONS IN CHANGING PARAMETERS. SOLVING PROBLEMS: TRIANGLE'S PROPERTIES IN EUCLIDEAN AND NONEUCLIDEAN GEOMETRY WITH EXAMPLES, APPROXIMATION OF PI AND OTHER IRRATIONAL NUMBERS, SOLUTIONS OF EQUATIONS AND INEQUALITIES,SYSTEMS OF EQUATIONS, DEFINING AND VISUALIZING GEOMETRICAL LOCI, FUNCTION INTEGRAL AND DERIVATIVES, APPROXIMATION OF SURFACE AREA.
( reference books)
LIST OF PROBLEMS GIVEN IN CLASS WITH VISUALIZATION AND SOLUTIONS WITH THE HELP OF SOFTWARE MATHEMATICA OR GEOGEBRA.

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Related or supplementary learning activities

ITA 
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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Related or supplementary learning activities

ITA 
20410423 
IN440  COMBINATORIAL OPTIMISATION
(objectives)
Acquire skills on key solution techniques for combinatorial optimization problems; improve the skills on graph theory; acquire advanced technical skills for designing, analyzing and implementing algorithms aimed to solve optimization problems on graphs, trees and flow networks.

Derived from
20410423 IN440  OTTIMIZZAZIONE COMBINATORIA in Scienze Computazionali LM40 LIVERANI MARCO
( syllabus)
Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits. Review of algorithm theory and computational complexity: complexity classes, the class of NP, NPcomplete, NPhard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics. Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and BellmanFord algorithms, dynamic programming technique, FloydWarshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, FordFulkerson algorithm, EdmondsKarp algorithm, preflow algorithms, "pushrelabel" algorithms. Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques. Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm. Huffman codes. Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.
( reference books)
T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"
Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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ITA 
20410460 
AM450  FUNCTIONAL ANALYSIS
(objectives)
To acquire a good knowledge of functional analysis: Banach and Hilbert spaces, weak topologies, linear and continuous operators, compact operators, spectral theory.

Derived from
20410460 AM450  ANALISI FUNZIONALE in Matematica LM40 BATTAGLIA LUCA
( syllabus)
Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems. HahnBanach theorem, analytic and geometric form, consequences. First and second category spaces, Baire's Theorem, BanachSteinhaus Theorem, open map and closed graph theorem, applications. Weak topologies, closed and convex sets, BanachAlaoglu theorem, separability and reflexivity. Sobolev spaces in dimensione one, immersion theorems, Poincaré inequality, application to variational problems. Spectral theory, Fredholm alternative, spectral theorem for compact and selfadjoint operators, application to variational problems.
( reference books)
H. Brezis  Analisi Funzionale  Liguori (1986) H. Brezis  Functional Analysis, Sobolev Spaces and Partial Differential Equations  Springer (2010) W. Rudin  Functional Analysis  McGrawHill (1991)

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Related or supplementary learning activities

ITA 
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.

Derived from
20410424 IN450 ALGORITMI PER LA CRITTOGRAFIA in Scienze Computazionali LM40 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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ITA 
20410461 
FS460  Dydactics of Physics
(objectives)
The objectives of the course are to enable the students to acquire the necessary skills to practice an affective teaching of Physics in the secondary school, with particular attention to: a) knowledgeÿof literature research on Physic teaching; the Italian educational system and school regulations; b) the design of culturally significant educational paths for Physics teaching; c) the production of materials for the measurement and verification of learning through the exercise of formative evaluation; d) the role of the "laboratory" as a way of working that involves students in an active and participated way, which encourages experimentation and planning.

Derived from
20410502 DIDATTICA DELLA FISICA in Fisica LM17 DE ANGELIS ILARIA, Postiglione Adriana
( syllabus)
Module 1. From common knowledge to scientific knowledge. The indicators of scientific knowledge; the contribution of formal education to the image of science; scientific communication. Module 2. Physics education, a research fieldOrigin and development of research in physics education in Italy; the constructivist paradigm; concepts and misconceptions; research on mental representations; conceptual change models; the conceptual nuclei of Physics. Module 3. Scientific teaching in secondary school Design the curriculum of Physics in the different orders and in the various study addresses; the teaching / learning process; orientation teaching and laboratory teaching as a didactic methodological approach; from content to programming by skills; training orientation. Module 4. The role of the "laboratory" in learning Physics Integration between theory and experimental verification; from observation of the phenomenon to the construction of the model; the different ways of "doing laboratory"; design of work units identifying the most appropriate experimental activities (demonstration; in the classroom with poor materials, in the instrumental laboratory, simulated through multimedia aids); implementation of laboratory operational skills for experiment management. Module 5. Flexible and modular design of content / knowledge, teaching methodologies and learning environments Core foundations of Physics; analysis and planning of didactic courses that respond to verticality criteria (evolution of concepts coherent and appropriate to students' cognitive development) and transversality (integration of Physics with other disciplines); simulations of teaching methodologies such as: dialogue lesson, microteaching, coplanning, peertopeer evaluation, cooperative learning activities, group work. Module 6. Learning evaluation of learning Modes and tools used in the various stages of monitoring, measurement, verification, evaluation and selfevaluation of learning; identification of learning contexts capable of developing and detecting skills; the National Evaluation System (SNV). Module 7. Modern and Contemporary Physics The role of modern and contemporary Physics in school curricula: what content / paths to propose that guarantee students a real understanding of them. The new State Exam and the role of Physics in the second written test in scientific high schools. Instrumental laboratory. Five instrumental laboratories of three hours each in the months of March, April and May.
( reference books)
Students can take advantage of the following teaching material on the Platform of the Department of Mathematics and Physics: Powerpointrelative presentations to the contents of the lessons, research articles, work material (texts to be analyzed taken from textbooks, articles of scientific dissemination, original memories , “tutorial” cards, videos, applets, cards for group work, grids for assessing learning, websites).
ESSENTIAL BIBLIOGRAFY Arons Arnold B. 1992, Guida all'insegnamento della fisica, Zanichelli•P. Guidoni, M. Arcà 2000 –Guardare per sistemi e guardare per variabili –l’educazione scientifica di base AIF Editore•Vicentini M., Mayer M. (a cura di) (1999). Didattica della Fisica, Loescher Editore.•Grimellini Tomasini N., Segré G. (a cura di) (1991). Conoscenze scientifiche: le rappresentazioni mentali degli studenti, La Nuova Italia, Firenze.•La fisica secondo il PSSC, 25 film del Physical Science StudyZanichelli•F. Bocci, Manuale per il laboratorio di fisica: introduzione all’analisi dei dati sperimentali Zanichell

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Related or supplementary learning activities

ITA 
20410425 
GE460 GRAPH THEORY
(objectives)
Provide tools and methods for graph theory.

Derived from
20410425 GE460  TEORIA DEI GRAFI in Scienze Computazionali LM40 MASCARENHAS MELO ANA MARGARIDA
( syllabus)
Graphs: basic definitions. Simple and non simple graphs, planarity, connectivity, degree, regularity, incidence and adjacency matrices. Examples of families of graphs. The '' handshaking lemma ''. Graphs obtained from others: complement, subgraph, cancellation and contraction. Isomorphisms and automorphisms of graphs. Connectivity: paths, cycles. A graph is bipartite if and only if each cycle has equal length. Connetivity and connected component. Connectivity for sides and vertices. Eulerian and semiEulerian graphs. Euler's theorem: a connected graph is Eulerian if and only if every vertex has an even degree. Hamiltonian graphs. Sufficient conditions to guarantee that a graph is Hamiltonian: the theorems of Ore and Dirac. the Ore theorem. Trees and forests. The cyclomatic number and the "cutset" rank of a graph. Fundamental system of cycles and cuts associated with a generating forest. Enumeration of generating forests. The Cayley theorem. Generating trees: the "greedy" algorithm for the "connector problem". Planar graphs. K3.3 and K5 are not planar. Statement of the theorem of Kuratovski and variations. Euler's formula for planar graphs. The dual of a planar graph. Correspondence between cycles and cuts for planar graphs and their dual. Dual abstract. A graph that admits a dual abstract is planar. Colorings: initial considerations and some properties. Colorings: the 5 colors theorem. Graphs on surfaces: classification of topological surfaces. Coloring of faces and duality between this problem and the coloring of vertices. Reduction of the proof of the 4color theorem to the coloring of cubic graph faces. '' The marriage problem ": Hall's theorem. Hall's theorem in the language of transversals. Criteria of existence of transversal and partial transversivities. Application to the construction of Latin squares. Direct graphs: basic notions and orientability. The MaxFlow MinCut theorem and Menger's theorem. Complexity of algorithms and applications to the theory of graphs. Introduction to the theory of matroids: definitions using bases and independent elements. Graphical and cographic matroids, vector matroids and the problem of representability. Definition of matroid using the cycles and the rank function. Minors of a matroid. Transverse matroids and the Rado Theorem for matroids. Union of matroids and applications: existence of disjoint bases in a matroid. Duality for matroids and applications to graphic and cographic matroids. Planar matroids and the generalization of Kuratovski's theorem for matroids. Elements of algebraic graph theory: the incidence matrix and the Laplacian matrix of an oriented graph. The vertex space and the space of edges of a graph. Subspaces of the cycles and subspace of the cuts of a defined oriented graph of the incidence matrix. Basis for the space of the cycles and for the space of the cuts of a graph. The RiemannRoch theorem for graphs. Proof of the "generalized Matrix Tree theorem". The contraction / restriction algorithm for matroids. Examples. The number of acyclical orientations of a graph. Graph polynomials: the chromatic polynomial, the "reliabiliaty" polynomial. Examples. The polynomial rank of a matroid. Properties and first applications. Proof of the structure theorem for functions on matroids that satisfy contraction/restriction properties. Their incarnation through the polynomial rank. Whitey's moves and two isomorphisms for graphs. Isomorphism between graphical matroids implies isomorphism between graphs in case the graphs are 3 connected. Whitney's Theorem for graphic matroids: sketch of the demonstration. Characterization for minors excluded from binary and regular matroids. The theorem of Seymour.
( reference books)
R. Diestel: Graph theory, Spriger GTM 173. R. Wilson: Introduction to Graph theory, Prentice Hall. B. Bollobas: Modern Graph theory, Springer GTM 184. J. A. Bondy, U.S.R. Murty: Graph theory, Springer GTM 244. N. Biggs: Algebraic graph theory, Cambridge University Press. C. D. Godsil, G. Royle: Algebraic Graph theory, Springer GTM 207. J. G. Oxley: Matroid theory. Oxford graduate texts in mathematics, 3.

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Related or supplementary learning 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.

Derived from
20410428 CR510 – CRITTOSISTEMI ELLITTICI in Scienze Computazionali LM40 PAPPALARDI FRANCESCO
( syllabus)
1. Theory of Elliptic Curves Weierstrass Equation, The structure of the group on rational points, formulas for the addition and duplication. Generalities on the intersections between lines and curves in P2(K) Preparatory results for the proof of the associativity of points on elliptic curves. Proof of the associativity of the sum for the points of an elliptic curve. Other equations for elliptic curves, Legendre's equation, Cubic equations, Quartic equations, intersections of two cubic surfaces. The jinvariant, elliptic curve in characteristic 2, Endomorphisms, singular curves, elliptic curves module n.
2. Torsion points Torsion points, Division polynomials. Weil's pairing
3. Elliptic curve on finite fields Frobenius endomorphism. The problem of determining the order of the group. Curves on subfields, Legendre's symbols, Point orders, Shanks's "Baby Step, Giant Step" algorithm. Particular families of elliptic curves. Schoof's algorithm.
4. Cryptosystems on Elliptical Curves. The Discrete Logarithm Problem. Algorithms for calculating the discrete logarithm: BabyStep GiantStep and PoligHellman. MOV attack. Attack on anomalous curves. DiffieHellman Key Exchange. Cryptosystems by Massey Omura and ElGamal. El Gamal Signature Scheme. Cryptosystems on elliptic curves analysis on the factorization problem. A cryptosystem based on Weil coupling. Factorization of internal numbers using elliptic curves. Using Pari.
( reference books)
Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003.

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Related or supplementary learning activities

ITA 
20410470 
FM510  MATHEMATICAL PHYSICS APPLICATIONS
(objectives)
To apply methods and tools of mathematical physics to some classes of models of dynamical systems and statistical mechanics, through both theoretical lectures and numerous practical exercises carried out in the computer lab.

Derived from
20410470 FM510  APPLICAZIONI DELLA FISICA MATEMATICA in Scienze Computazionali LM40 SCOPPOLA ELISABETTA, TERESI LUCIANO, D'AUTILIA ROBERTO
( syllabus)
Part II Statistical Mechanics models  Stochastic dynamics and applications
Mathematical model of different problems are presented as spread of epidemics, sample problems, optimisation problems, physical problems, with numerical simulations. Laboratory exercises are an essential part of the course. Statistical Mechanics models, as the Ising model, and probability tools, as Markov Chain, are applied, with references to relative theory.
( reference books)
S.Freidli and Y.Velenik : Statistical Mechanics of Lattice Systems  A concrete mathematical introduction. In rete
O.H¨aggstr¨om: Finite Markov Chain and Algorithmic Applications, London Mathematical SocietyStudent Texts 52

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Related or supplementary learning activities

ITA 
20410462 
GE510  ALGEBRAIC GEOMETRY 2
(objectives)
Introduce to the study of algebraic geometry, with particular emphasis on beams, schemes and cohomology.

Derived from
20410462 GE510  GEOMETRIA ALGEBRICA 2 in Matematica LM40 LOPEZ ANGELO
( syllabus)
Sheaf theory and its use in on schemes
Preseheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks and similar properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products. Algebraic sheaves on a scheme. Quasicoherent sheaves and coherent sheaves.
Cohomology of sheaves
Homological algebra in the category of modules over a ring. Flasque sheaves. The cohomology of the sheaves using canonical resolution with flasque sheaves.
Cohomology of quasicoherent and coherent sheaves on a scheme.
Cech cohomology and ordinary cohomology. Cohomology of quasicoherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. EuleroPoincaré characteristic.
Invertible sheaves and linear systems
Glueing of sheaves. Invertible sheaves and their description. The Picard group. Morphisms in a projective space. Linear systems.
( reference books)
Notes from Prof. Lopez, Prof. Sernesi R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. SpringerVerlag, New YorkHeidelberg, 1977. D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000). U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).

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Related or supplementary learning activities

ITA 
20410463 
TN510  NUMBER THEORY
(objectives)
Provide a good knowledge of concepts and methods of analytical theory of numbers, with particular concern to the theory of prime numbers and prime numbers in arithmetic progression.ÿIntroduce to Riemann's zeta function theory.

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Related or supplementary learning activities

ITA 
20410464 
IN530  COMPUTER INFORMATION SYSTEMS
(objectives)
Introduce the basic concepts of security and then show how to acquire autonomy in updating the understanding in the data and networks security domain. Provide the basic concepts for understanding and evaluating a security solution. Provide the basic knowledge to produce security solutions for small/mediumsized system.

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Related or supplementary learning activities

ITA 
20410431 
IN540  COMPUTATIONAL TOPOLOGY
(objectives)
Introduce the study of computational topology and in particular the concepts, representations and algorithms for topological and geometric structures to support geometric modeling, construction of simulations meshes, and scientific visualization. Acquire techniques for parallel implementation in the representation and processing of largesized graphs and complexes. Application of sparse matrices, for the implementation of algorithms on graphs and complexes with linear algebraic methods.

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Related or supplementary learning activities

ITA 
20410434 
FS450  Elements of Statistical Mechanics
(objectives)
Gain knowledge of fundamental principles of statistical mechanics for classical and quantum systems.

Derived from
20401806 ELEMENTI DI MECCANICA STATISTICA in Fisica L30 N0 RAIMONDI ROBERTO
( syllabus)
CONTENTS OF THE LECTURES: the numbers in round brackets refer to the chapter and section of the textbook adopted. Kinetic theory of gases. Boltzmann equation and H theorem. (1, Par.2.1,2.2,2.3,2.4) MaxwellBoltzmann distribution. (1, Par. 2.5) Phase space and Liouville theorem. (1, Par. 3.1,3.2) Gibbs ensembles. Micro canonical ensemble. Definition of entropy. (1, Par. 3.3,3.4) The ideal gas in the micro canonical ensemble. (1, Par. 3.6) The equipartition theorem. (1, Par. 3.5) The canonical ensemble. (1, Par.4.1). The partition function and the free energy. Fluctuations of energy in the canonical ensemble. (1 Par. 4.4) The grand canonical ensemble. The grand potential. The ideal gas in the grand canonical ensemble. (1 Par. 4.3). Fluctuations of the particle number. (1 Par. 4.4) Classical theory of the linear response and fluctuationdissipation theorem. (1, Par. 8.4). Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2). JohnsonNyqvist theory of thermal noise. (1 Par. 11.3). Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4) FermiDirac and Boseeinstein quantum statistics. ( 1, Par. 7.1) The Fermi gas. The Sommerfeld expansion and the electron specific heat. (1, Par. 7.2) The Bose gas. The BoseEinstein condensation. (1, Par. 7.3) Quantum theory of blackbody radiation. (1, Par. 7.5)
Web page with additional material about the lecture course
https://sites.google.com/a/personale.uniroma3.it/robertoraimondi/home/teaching/elementi
( reference books)
Suggested textbook: 1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.
Further reading: 2) K. Huang, Meccanica Statistica, Zanichelli, 1997. 3) L. Peliti, Appunti di Meccanica Statistica, Bollati Boringhieri, 2003. 4) Joel L. Lebowitz, Statistical mechanics: A selective review of two central issues, Reviews of Modern Physics, 71, S346 (1999).
Further information is available on https://sites.google.com/a/personale.uniroma3.it/robertoraimondi/home/teaching/elementi

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Related or supplementary learning activities

ITA 
20410443 
FS520  Complex systems
(objectives)
To understand algorithms related to complex systems, writing, executing and optimising simulation programs of such systems (Montecarlo and molecular dynamics programs) and analysing the data produced by simulations.

Derived from
20410571 FS520 – RETI COMPLESSE in Scienze Computazionali LM40 CAMISASCA GAIA
( syllabus)
NETWORKS AND GRAPHS  Graphs, trees and networks  Centrality measures and degree  Random graphs, the Erdős and Rényi model
SMALL WORLDS NETWORKS  Definition of Small World  Clustering Coefficient  The WattsStrogatz model
GENERALISED RANDOM GRAPHS  Statistical description of networks  Degree Distributions of real networks  Generalization of the Erdős–Rényi model  Radom graphs with powerlaw degree distributions
GROWING GRAPHS  Dynamical evolution of random graphs  The Barabási–Albert model
DEGREE CORRELATIONS  Correlated networks  Assortative and Disassortative Networks, "Rich Club" behavior
WEIGHTED NETWORKS  Beyond purely topological networks: tuning the interactions in a complex system  The BarratBarthélemyVespignani model
INTRODUCTION TO DYNAMICAL PROCESSES: THEORY AND SIMULATION
( reference books)
Main textbook: V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Cambridge University press (2017)
The course also follows selected parts of the book: A. Barrat, M. Barthelemy, A. Vespignani, "Dynamical processes on complex networks", Cambridge University Press (2008)

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FIS/03

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Related or supplementary learning activities

ITA 
20410518 
AM420  SOBOLEV SPACES AND PARTIAL DERIVATIVE EQUATIONS
(objectives)
To acquire a good knowledge of the general methods andÿclassical techniques necessary for the study ofÿweak solutions of partial differential equations.

Derived from
20410518 AM420  SPAZI DI SOBOLEV ED EQUAZIONI ALLE DERIVATE PARZIALI in Matematica LM40 HAUS EMANUELE, FEOLA ROBERTO
( syllabus)
Preliminaries  Weak topologies and weak convergence, weak lower semicontinuity of the norm  L^P spaces: reflexivity, separability, criteria for strong compactness.
Sobolev spaces and variational formulation of boundary value problems in dimension one  Motivations  The Sobolev space W^{1,p} (I)  The space W^{1,p}_0 (I)  Some examples of boundary value problems  Maximum principle
Sobolev spaces and variational formulation of boundary value problems in dimension N  Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)  Extension operators  Sobolev inequalities  The space W^{1,p}_0 (Omega)  Variational formulation of some elliptic boundary value problems  Existence of weak solutions  Regularity of weak solutions  Maximum principle
( reference books)
Functional analysis, H. Bre'zis

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MAT/05

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Related or supplementary learning activities

ITA 
20410520 
AL420  ALGEBRAIC THEORY OF NUMBERS
(objectives)
Acquire methods and techniques of modern algebraic theory of numbers through classic problems initiated by Fermat, Euler, Lagrange, Dedekind, Gauss, Kronecker.

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MAT/02

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Related or supplementary learning activities

ITA 
20410523 
MA430  MATHEMATICAL METHODS FOR APPLIED SCIENCES
(objectives)
The purpose of this Course is presenting a few typical topics of Mathematics especially useful for Physics and Engineering. These are the theory of analytic functions, Hilbert spaces, Fourier series, Fourier and Laplace transforms.

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MAT/05









Related or supplementary learning activities

ITA 
20410522 
CP450  DISCRETE PROBABILITY
(objectives)
Development of probabilistic techniques and advanced methods for the study of stochastic processes on graphs, randomized algorithms and random graphs, random walks and interacting particle systems.

Derived from
20410556 CP450  METODI PROBABILISTICI E ALGORITMI ALEATORI in Matematica LM40 DE OLIVEIRA STAUFFER ALEXANDRE
( syllabus)
The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them (for example, because it gives a more efficient algorithm, or because the solution is becomes more resilient against and adversary, or simply because it gives a simple and elegant solution to an apparently complicated problem).
Some of the topics seen in the course are the following: * Randomized algorithms * Can we really use perfect randomness in computer science? * Allocation process like balls into bins and random data structures via hash functions * Branching processes and spread of infections * Probabilistic method and its application to combinatorics and some games * Concentration of random variables and martingales, and their applications to routing and dimension reduction * Percolation, ErdosRenyi random graphs and random networks * Random walks on graphs and application to clustering data
( reference books)
"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press "The probabilistic method", Alon and Spencer, John Wiley & Sons

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MAT/06

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Related or supplementary learning activities

ITA 
20410524 
GE520  ADVANCED GEOMETRY
(objectives)
Acquire uptodate and advanced skills on topics chosen within the research themes of contemporary geometry

6

MAT/03

48

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Related or supplementary learning activities

ITA 
20410529 
LM510  LOGICAL THEORIES 1
(objectives)
Address some questions of the theory of the proof of the twentieth century, in connection with the themes of contemporary research

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MAT/01

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Related or supplementary learning activities

ITA 
