Courses list (2018/2019, all courses) 

66453 - Basic projective algebraic geometry

General information

Basic projective algebraic geometry ( IGS 66453 ), 7 CFU, second semester, years: 3° LT; 1° LM

Course content

Affine and projective algebraic sets. Non-singular points and  tangent space. Dimension and degree of a projective variety. Hypersurfaces in projective space. Linear systems of hypersurfaces. The Jacobian group of a linear system on a line. Algebraic correspondences between lines. Birational plane model of a projective curve. Plane curves. Cusps, flexes and ordinary multiple points. Bézout’s  theorem. The genus of a plane curve (according to Riemann). Rational curves. Rational normal curves. Elliptic normal curves. Plane cubic curves. Modulus  of a plane cubic. Structure of abelian group  on such curves. Modulus  of a plane cubic. Classification of degree 3 projective curves.

Language

Italian

Teacher

Stefano Vigni

Teaching style

In presence

Attendance

Suggested

61683 - Advanced Analysis 1

General information

Advanced Analysis 1 ( AnSup1 61683 ), 8 CFU, second semester, years: 1°, 2° LM

Course content

TO BE UPDATED

 

Introduction about topological vector spaces and locally convex topological vector spaces; metrizability. Frechet spaces. Spaces of test functions. Results on density. Definition of distribution. Order of a distribution. Positive distributions. Finite density distributions. Distributions with compact support. Calculus with distributions. Tensorial product of two distributions. Derivative. Study of some differential equations. Topology of D’. Convolution. Fourier transform. Sobolev spaces.

Language

English

Teacher

Andrea Bruno Carbonaro

Teaching style

In presence

Attendance

Suggested

98825 - Advanced topics in mathematical physics

General information

Advanced topics in mathematical physics ( CFM 98825 ), 5 CFU, second semester, years: 1°, 2° LM

Language

Italian

Teacher

Pierre Martinetti

Teaching style

In presence

Attendance

Suggested

25897 - Algebra 1

General information

Algebra 1 ( Alg1 25897 ), 9 CFU, first semester, years: 1° LT

Course content

- Introduction to the basic notions: sets, maps, surjectiive, , injective and bijective.

- Binary operation and their properties. Equivalence relations and induced quotients.

- Cardinality: countsble and uncountable sets. Permutations. Induction. Newton binomial formula.

- The integers: Euclidean algorithm and its applications. Prime numbers and unique factorization. The integers mod n. 

- Complex numbers.

- Polinomials: with rational, real or complex coefficients. Unique factorization property for polynomials. Irreducibility criteria.  Quotients and their properties: zero-divisors, nilpotents and invertible elements.

- Introduction to algebraic structures. Abelian groups. Subgroups, homomorphisms and quotients.

Language

Italian

Teacher

Maria Evelina Rossi

Other teachers

Aldo Conca

Teaching style

In presence

Attendance

Not required

25905 - Algebra 2

General information

Algebra 2 ( Alg2 25905 ), 8 CFU, first semester, years: 2° LT

Course content

Groups, homomorphisms of groups, subgroups and quotient groups. Linear groups, permutation groups, finite groups of low order. Group actions on sets.  Rings, subrings  and ideals. Euclidean rings and factorial rings. Polynomials rings. Filed extensions. Modules. Structure theorem of fg modules over PID

Language

Italian

Teacher

Matteo Varbaro

Other teachers

Emanuela De Negri

Teaching style

In presence

Lesson timetable

Monday: 9:00 - 11:00, room 706

Attendance

Suggested

25911 - Analytical mechanics

General information

Analytical mechanics ( MA 25911 ), 8 CFU, second semester, years: 2° LT

Course content

Mathematical preliminaries. Mechanics of a particle. Mechanics and relative motions. Discrete systems and rigid bodies. Analytical mechanics of holonomic systems. Introduction to the study of dynamical systems and of stability. Hamiltonian mechanics for mechanical systems.

Language

Italian

Teacher

Nicola Pinamonti

Other teachers

Pierre Martinetti

Teaching style

In presence

Attendance

Not required

42916 - Application of Mathematics to Medicine

General information

Application of Mathematics to Medicine ( AMM 42916 ), 7 CFU, second semester, years: 1°, 2° LM

Course content

Part I: X-ray tomography (overview); Radon transform,formulas for the inversion of the Radontransform (as back projection and filtered back projection), issues of uniqueness.

Part II: positron emission tomography (overview); on the two inverse problems related to positronemission tomography: an imaging problem (inversion of the Radon transform) and a compartmentalone (Gauss-Newton optimization scheme)

Part III: magnetic resonance imaging (overview); models for data acquisition and magnetic fielddistortion, Fourier transform, inversion of the Fourier transform from undersampled data.

Language

English

Teacher

Michele Piana

Other teachers

Federico Benvenuto

Teaching style

In presence

Attendance

Suggested

39474 - Basic Algebraic Geometry

General information

Basic Algebraic Geometry ( GeoSup1 39474 ), 7 CFU, second semester, years: 2° LM

Course content

Local invariants of an algebraic variety in a point. The tangent space. Singular and nonsingular points of an algebraic variety. Topological properties of projective algebraic varieties and applications. The Segre embedding and products of algebraic varieties. Applications. The dimension theory of algebraic varieties and applications. Power series and the Weierstrass Preparation Theorem. Applications. The factoriality of the local ring of an algebraic variety and applications. Divisors and differential forms on a nonsingular algebraic variety. The Riemann-Roch theorem for nonsingular projective curves.

Language

English

Teacher

Stefano Vigni

Teaching style

In presence

Attendance

Not required

90694 - Basics of Higher Algebra

General information

Basics of Higher Algebra ( IALS 90694 ), 7 CFU, second semester, years: 3° LT; 1° LM

Course content

I - Rings, ideals and modules. Noetherian rings and the Hilbert basis theorem. Polynomials: The ring K [x_1, ..., x_n] of polynomials with coefficientsin a field. Grobner Bases  and the Buchberger algorithm. Systems of polynomial equations and elimination theory.

II - Review of field extensions. Splitting fields of polynomials with coefficients in a field of characteristic 0, normal extensions and their basic properties. Fundamental Theorem of Galois theory. The Galois group of a polynomial. Applications: cyclotomic fields, solvability by radicals of algebraic equations.

Language

English

Teacher

Aldo Conca

Stefano Vigni

Other teachers

Anna Maria Bigatti

Teaching style

In presence

Attendance

Suggested

61707 - Basics of Higher Geometry 2

General information

Basics of Higher Geometry 2 ( IGS2 61707 ), 8 CFU, first semester, years: 1°, 2° LM

Course content

Affine algebraic sets and affine varieties. Irreducible components. Hilbert Nullstellensatz. Regular and rational functions on an affine algebraic variety. Regular and rational maps between two affine varieties. Examples. Projective algebraic varieties and projective Nullstellensatz. Rational functions on a projective variety. Regular and rational maps between two projective varieties. Examples of projective varieties. Sheaves in algebraic geometry, (quasi‐)coherent sheaves. Definitions of schemes. The Picard variety. The tangent space at an algebraic variety in a point. The Zariski cotangent space. General properties. Singular and nonsingular points. Divisors on a projective nonsingular curve. The formulation of Riemann‐Roch for curves. Examples.

Language

English

Teacher

Arvid Perego

Teaching style

In presence

Attendance

Not required

66449 - Complementary Mathematics 1

General information

Complementary Mathematics 1 ( MC1 66449 ), 7 CFU, first semester, years: 1°, 2° LM

Course content

Mathematical modeling (differential e stochastic models) and related teaching problems in secondary school: from historical and epistemological reflections on modeling processes to educational choises (through examples of reasoned comparison between deterministic and probabilistic models).

Language

Italian

Teacher

Elda Guala

Teaching style

In presence

Attendance

Suggested

84039 - Complex Analysis

General information

Complex Analysis ( AC 84039 ), 7 CFU, first semester, years: 3° LT

Course content

Power series: formal power series; convergent power series; analytic functions. Complex differentiation: holomorphic functions; Cauchy-Riemann equations; conformal transformations; elementary functions. Complex integration: integration along paths; primitives; Cauchy's theorem. Consequences of Cauchy's theorem: Cauchy's integral formula; holomorphic functions are analytic; further consequences (theorems by Morera and Liouville, mean value, maximum modulus, Weierstrass' convergence theorems). Singularities and residues: Laurent series; behavior near singularities; residue theorem and applications. Miscellanea: zeros and poles of meromorphic functions; Euler's Gamma function; analytic continuation.

Language

Italian

Teacher

Alberto Perelli

Teaching style

In presence

Attendance

Suggested

29032 - Differential equations

General information

Differential equations ( ED 29032 ), 7 CFU, first semester, years: 3° LT; 1°, 2° LM

Course content

First order quasilinear equations. Classification of second order equations. Some classical linear equations of Mathematical Physics: the equations of Laplace, Poisson, the heat and the wave equation. General properties of the solutions: mean value property, maximum principle, energy estimates and their consequences. Some general techniques to obtain explicit formulas for solutions: separation of variables, Green’s function, reflection method, Duhamel’s principle, spherical means, method of descent.

Language

Italian

Teacher

Gianfranco Bottaro

Teaching style

In presence

Attendance

Suggested

61467 - Differential Geometry

General information

Differential Geometry ( GD 61467 ), 7 CFU, second semester, years: 3° LT

Course content

Topological, differentiable and riemannian manifolds. Differentiable maps. Tangent and cotangent bundles. Tensor algebra. Vector fields and their flows; Lie derivative. Differential forms; de Rham complex. Orientable manifolds.  Riemannian metrics.Curves and surfaces in tridimentional Euclidean space.Gauss curvature and theorema egregium. Surfaces of constant curvature. Covariant derivative and Riemann curvature tensor; geodesics. Normal coordinates.  Lie groups and their Lie algebras; exponenential maps.

Language

Italian

Teacher

Claudio Bartocci

Teaching style

In presence

Attendance

Suggested

42925 - Elementary Mathematics from an Advanced Standpoint

General information

Elementary Mathematics from an Advanced Standpoint ( MEDPVS 42925 ), 7 CFU, first semester, years: 1°, 2° LM

Course content

With references to epistemological and historical issues, the relationships between some fields of the mathematics (numerical and algebraic structures, mathematization of the space, algorithms, mathematical analysis, probability), and the problem of setting up their teaching in our days, are discussed: comparison between various ways to introduce and to formalize the concepts, location of "unifying" or “synergical” didactical itineraries, role of computers, relationships between "experiential", "constructive" and "deductive" aspects, relationships with other strongly mathematizated disciplines, etc.

Language

Italian

Teacher

Carlo Eugenio Dapueto

Teaching style

In presence

Attendance

Required

90693 - Elements and Applications of Modern Physics

General information

Elements and Applications of Modern Physics ( EAFM 90693 ), 7 CFU, second semester, years: 1° LM

Course content

  1. Introduction to Special Relativity

    Classical relativity and the wave equation for electromagnetic fields
    The Michelson and Morley experiment
    Einstein postulates
    Lorentz transformations
    Relativistic dynamics
    Applications

  2. Quantum mechanics

    2.1 Light as particles, matter as waves 
    Blackbody radiation
    Photoelectric effect
    The Bohr atomic model
    The De Broglie hypotesis
    The Heisenberg uncertainty principle
    Wave packets
    Examples of quantum interference of matter waves

    2.2 The Schroedinger equation
    General and stationary equation
    Probabilistic interpretation of the wave function
    Particle in a box
    Harmonic oscillator
    Tunnelling

    2.3 Atomic models and particle statistics
    Hydrogen atom, spin, transitions
    Many-electrons atoms
    The Pauli exclusion principle
    The Fermi-Dirac and Bose-Einstein statistics (in brief)

    2.4 Lattice structures
    X-ray diffraction
    Band theory (in brief)
    Metals, insulators, semiconductors

    2.5 Applicazioni 
    Scanning tunnel microscope (STM)
    Atomic force microscope (AFM)
    Semiconducting devices
    Laser
    Quantum computation (in brief)

Language

Italian

Teacher

Fabio Cavaliere

Maura Sassetti

Teaching style

In presence

Attendance

Not required

29024 - Elements of Advanced Analysis 1

General information

Elements of Advanced Analysis 1 ( IAS1 29024 ), 7 CFU, first semester, years: 3° LT

Course content

Normed and Banach spaces; continuous operators. Hahn Banach, uniform boundedness, open map and closed graph theorems; Hilbert spaces, Riesz and projection theorems, L^p spaces. Convergences of measurable functions..Radon-Nikodym theorem.

Language

Italian

Teacher

Giovanni Alberti

Other teachers

Ada Aruffo

Teaching style

In presence

Attendance

Suggested

61705 - Elements of Advanced Analysis 2

General information

Elements of Advanced Analysis 2 ( IAS2 61705 ), 8 CFU, second semester, years: 1°, 2° LM

Course content

Quotient normed spaces; reflexivity; weak topologies; compactness in spaces of continuous functions: the Ascoli-Arzelà theorem; elements of operator theory: spectrum, adjoint operator, compact operators, the spectral theorem for compact selfadjoint operators, unbounded operators, closed operators; elements of spectral theory in Banach algebras; Gelfand theory for commutative Banach algebras; introduction to C*-algebras; the Gelfand representation theorem for commutative C*-algebras.

Language

Italian

Teacher

Laura Burlando

Other teachers

Ada Aruffo

Teaching style

In presence

Attendance

Suggested

66454 - Foundations of Numerical Analysis

General information

Foundations of Numerical Analysis ( FCN 66454 ), 8 CFU, second semester, years: 2° LT

Course content

Error theory. Solution of linear systems: conditioning, Gauss' method and pivoting, matrix factorization: LU and Cholsky; applications. Eigenvalues: power method and extensions, similarity transformations, Householder transformation; QR factorization; reduction to Hessenberg and tridiagonal forms, QR method. Approximation of functions: discrete least-squares: solution by means of normal equations. Singular Value Decomposition and application to the least-squaresproblem. Numerical solution of differential equations by means one step and multistep methods.

Laboratory: 5 exercises about topics addressed during the semester (the use of Matlab is required)

 

Language

Italian

Teacher

Claudia Fassino

Other teachers

Michele Piana

Teaching style

In presence

Attendance

Suggested

61682 - Fourier Analysis

General information

Fourier Analysis ( AnFour1 61682 ), 8 CFU, first semester, years: 1°, 2° LM

Course content

PROGRAM :

Fourier series. The space of periodic square summable functions. Orthorormal basis. Fourier series. Gibbs phenomenon. Fourier transform of periodic absolute integrable functions. Applications: spectral methods for partial differential equations.

Fourier integrals Fourier integral of absolute integrable functions on R. Fourier transform of elementary fiunzions. Convolution. Approximate indentities. Inversion formula. Fourier transform of square integrable functions. Poisson summation formula. The Paley-Wiener theorem. 
Discrete Fourier transform. Fast Fourier Transform. Cosine transform.  


Signal analysis. Shannon theorem. Hilbert transform. Gabor transform

SUGGESTED  BOOKS:

V. Del Prete Introduzione all'analisi di Fourier Dispense on line.

Y. Katznelson An introduction to harmonic analysis Collocaz Bibl. DIMA 43-1968-07.

E. O. Brigham, The Fast Fourier Transform, Prentice Hall Englewood Cliffs, Boston,1974.
-

H. Dym - H. P. Mc Kean, Fourier Series and Integrals, Academic Press, 1972.

I. Korner, Fourier Analysis, 1995. 
- I. Korner, Exercises for Fourier Analysis, 1995.
E. Prestini, Applicazioni dell'analisi armonica. U.Hoepli,Milano, 1996I.

E. Prestini, The Evolution of Applied Harmonic Analysis. Models of the Real World Series, A Birkhäuser 2004.

G.B. Folland,  Fourier analysis and its applications Collocaz Bibl. DIMA 42-1992-01. 
The examination os oral

Language

Italian

Teacher

Filippo De Mari Casareto Dal Verme

Other teachers

Giovanni Alberti

Teaching style

In presence

Attendance

Suggested

38737 - Game Theory

General information

Game Theory ( TMG 38737 ), 7 CFU, second semester, years: 3° LT; 1°, 2° LM

Course content

Int_1) Very important topics in GT

a)    a)Taxonomy of games (cooperative, non-cooperativi, static and dynamic games), from finite games to infinite ones;b) the nash equilibrium and the problem of efficiency of solutions; c) strategic , extensive, with complete information,  with imperfect information games)

2) Game Theory and Mathematical Analysis: well-posedness problem, approximate solutions and convergence.

3) GT and Economics: oligopoly problems, from the static case

(Counot, Bertrand) to dynamic one (Stackelberg)

4) Refinment of Nash equilibria: dominated strategies , stability and perfectness in the subgames.

5)From vector Optimization to multicriteria games

6) Cooperative games and solutions

7) Partially cooperative games and applications to environmental and medicine problems

Language

English

Teacher

Angela Lucia Pusillo

Teaching style

In presence

Attendance

Not required

66452 - General Physics (Mechanics , Thermodynamics)

General information

General Physics (Mechanics , Thermodynamics) ( FisI 66452 ), 9 CFU, first semester, years: 2° LT

Course content

 This one semester course will provide students with an in-depth study of the fundamental laws of Mechanics and with an introduction to Thermodynamics. The course is designed for first year students who are assumed to have little or no background knowledge in Physics. It is assumed however that the students have some fundamental knowledge of Mathematics: algebra, geometry, trigonometry and calculus. The basic and practical aspects of Mechanics will be introduced with an emphasis on the energy balance of a mechanical process. The final part of the course deals with  systems of particles and solid bodies by studying their collective motion, through the fundamental laws involving momentum and angular momentum, and their internal Energy and related transformations through an introduction to the fundamental laws of Thermodynamics.

Language

Italian

Teacher

Fabio Cavaliere

Teaching style

In presence

Attendance

Not required

57191 - General Physics - Electromagnetism and Optics

General information

General Physics - Electromagnetism and Optics ( FisII 57191 ), 7 CFU, second semester, years: 3° LT

Language

Italian

Teacher

Giovanni Ridolfi

Teaching style

In presence

Attendance

Suggested

52449 - General physics laboratory

General information

General physics laboratory ( LFG 52449 ), 6 CFU, second semester, years: 2° LM

Language

Italian

Teacher

Corrado Boragno

80412 - Geometric Modeling

General information

Geometric Modeling ( GM 80412 ), 6 CFU, first semester, years: 1°, 2° LM

Course content

Background Notions

Models of discrete geometric shapes

Representations for cell and simplicial complexes

Discrete differential geometry

Geometry processing

Language

English

Teacher

Chiara Eva Catalano

Enrico Puppo

Teaching style

In presence

Attendance

Suggested

25909 - Geometry 1

General information

Geometry 1 ( Geo1 25909 ), 8 CFU, first semester, years: 2° LT

Course content

Topological spaces, metric spaces and continuous maps: definitions, general properties, examples. Subspaces. Finite products of topological spaces. Quotient topological spaces: examples (projective spaces, spheres, tori,...). Hausdorff spaces. Connected spaces and path-connected spaces, compact spaces, examples: relation with some concepts of Analysis. The theorem of Tychonoff. Locally compact spaces and the compactification theorem of Alexandroff. Locally connected spaces. Complete metric spaces. Urysohn's lemma and Urysohn's metrizability theorem. Tietze's extension theorem. Baire spaces. 

Language

Italian

Teacher

Matteo Penegini

Other teachers

Arvid Perego

Teaching style

In presence

Attendance

Suggested

25910 - Geometry 2

General information

Geometry 2 ( Geo2 25910 ), 7 CFU, second semester, years: 2° LT

Course content

Homotopy of maps. Retractions and deformations. The fundamental group and simply connected spaces. Examples. Some slementary methods to compute the fundamental group. The fundamental group of spheres and some other spaces.

Language

Italian

Teacher

Matteo Penegini

Other teachers

Arvid Perego

39407 - Higher Algebra 1

General information

Higher Algebra 1 ( AlgSup1 39407 ), 7 CFU, first semester, years: 1°, 2° LM

Course content

Rings, Ideal, modules. Rings and modules of fractions. Tensor products of modules. Primary decomposition, integral dependence and valuations, Chain conditions, Noetherian and Artinian rings. Dimension theory. Krullhauptidealssatz

Language

English

Teacher

Aldo Conca

Maria Evelina Rossi

Teaching style

In presence

Attendance

Suggested

42911 - Higher Algebra 2

General information

Higher Algebra 2 ( AlgSup2 42911 ), 7 CFU, second semester, years: 1°, 2° LM

Course content

Homological algebra. Hilbert functions. Regular sequences. Grade and depth. The Koszul complex. Free resolutions. Regular rings. Cohen-Macaulay rings. Complete intersections. Gorenstein rings. Canonical modules and local cohomology. Stanley Reisner rings, determinanntal rings. 

Language

English

Teacher

Matteo Varbaro

Teaching style

In presence

Attendance

Not required

35288 - History of Mathematics

General information

History of Mathematics ( SM 35288 ), 7 CFU, second semester, years: 3° LT; 1°, 2° LM

Language

Italian

Teacher

Claudio Bartocci

Teaching style

In presence

Attendance

Suggested

62425 - Image processing

General information

Image processing ( EI 62425 ), 6 CFU, second semester, years: 1°, 2° LM

Course content

Basics of image processing: 1)  Image formation and image recording: blurring and noise. 2) Digital images: sampling and quantization. Basic relationships between pixels. 3) Two dimensional Discrete Fourier Transform, two dimensional Fast Fourier Transform algorithm (FFT 2D) and Fourier Transform of an image. 4) Basic operators for image processing. Edge detection. Hough Transform for feature extraction.

 

Astronomical image processing: Image reconstruction from visibilities: definition of visibility; inverse Fourier Transform from limited data; interpolation in Fourier space; deconvolution techniques; iterative methods for image reconstruction. The NASA RHESSI satellite.

Language

English

Teacher

Anna Maria Massone

Teaching style

In presence

Attendance

Not required

48384 - Inferential Statistics

General information

Inferential Statistics ( StInf 48384 ), 8 CFU, second semester, years: 3° LT; 1° LM

Language

Italian

Teacher

Elda Guala

Other teachers

Eva Riccomagno

Teaching style

In presence

Attendance

Suggested

62247 - Introduction to Cryptography and Code Theory

General information

Introduction to Cryptography and Code Theory ( ICCT 62247 ), 7 CFU, first semester, years: 3° LT; 1° LM

Course content

  1. Crittography: Elliptic Curves
  2. Aritmetics of Finite Fields
  3. Coding theory: introduction
  4. Protocols and Bitcoin

Language

Italian

Teacher

Teo Mora

Teaching style

In presence

Attendance

Suggested

44142 - Introduction to gauge theory

General information

Introduction to gauge theory ( MGFM 44142 ), 5 CFU, second semester, years: 1°, 2° LM

Language

English

Teacher

Claudio Bartocci

Teaching style

In presence

Attendance

Suggested

52473 - Introduction to imperative programming

General information

Introduction to imperative programming ( Prog1 52473 ), 8 CFU, second semester, years: 1° LT

Course content

The parts of the course concerned with programming will focus on the procedural part of C++ as an example of a high level programming language.

    Basic notions about computers: data representation; computer arithmetic; overview of operating systems and the Von Neumann architecture (for understanding program execution).
    Introduction to programming: high-level programming languages; state and state change; declarations; basic statements; typed languages: basic and structured types; simple algorithms and their implementation; structured programming and modularity.
    Small Scale Programming: design of small sequential programs starting from an informal specification of the problem; use of an integrated development environment; program compilation/interpretation; code execution; simple testing.  

The parts of the course concerned with programming will focus on the procedural part of C++ as an example of a high level programming language.    Basic notions about computers: data representation; computer arithmetic; overview of operating systems and the Von Neumann architecture (for understanding program execution).
Introduction to programming: high-level programming languages; state and state change; declarations; basic statements; typed languages: basic and structured types; simple algorithms and their implementation; structured programming and modularity.
Small Scale Programming: design of small sequential programs starting from an informal specification of the problem; use of an integrated development environment; program compilation/interpretation; code execution; simple testing.

The parts of the course concerned with programming will focus on the procedural part of C++ as an example of a high level programming language.

Language

Italian

Teacher

Patrizia Boccacci

Other teachers

Francesco Masulli

Gianna Reggio

Teaching style

In presence

Attendance

Suggested

38754 - Inverse problems and applications

General information

Inverse problems and applications ( PbInv 38754 ), 7 CFU, first semester, years: 1° LM

Course content

Linear operators in Hilbert spaces:closed and non closed range operators. Ill-posed problems, generalized solution. Compact operators. Singular system and regularization methods: regularization algorithms in the sense of Tikhonov. 

Iterative methods: the Landweber method and the conjugate gradient method. Choice of the regularization parameter.

Problems of image reconstruction and of image deconvolution.Regularization methods are analyzed using the tools already exposed adapted its Fourier analysis

Statistical approach to inverse problems: Maximum Likelihood and Bayes Theorem.

Monte Carlo methods for non-linear inverse problems: importance sampling and Markov Chain Monte Carlo.

Methods for dynamic inverse problems: Kalman and particle filtering.

The course also includes numerical experiments with Matlab.

Language

English

Teacher

Claudio Estatico

Alberto Sorrentino

Other teachers

Federico Benvenuto

Teaching style

In presence

Attendance

Not required

42924 - Laboratory of Mathematics Education

General information

Laboratory of Mathematics Education ( LDM 42924 ), 5 CFU, second semester, years: 1° LM

Language

Italian

Teacher

Elda Guala

Other teachers

Francesca Morselli

Teaching style

In presence

Attendance

Required

80275 - Linear algebra and Geometry

General information

Linear algebra and Geometry ( ALGA 80275 ), 16 CFU, first and second semester, years: 1° LT

Language

Italian

Teacher

Emanuela De Negri

Other teachers

Anna Oneto

Arvid Perego

Teaching style

In presence

Lesson timetable

Monday: 9:00 - 11:00, room 508

Attendance

Suggested

25900 - Mathematical Analysis 2

General information

Mathematical Analysis 2 ( An2 25900 ), 8 CFU, first semester, years: 2° LT

Course content

Functions of n real variables: limits, continuity, differential calculus; ordinary differential equations; sequences and series of functions; power series;integrals of fields and scalar.

Language

Italian

Teacher

Marco Baronti

Other teachers

Andrea Bruno Carbonaro

Teaching style

In presence

Attendance

Suggested

25907 - Mathematical Analysis 3

General information

Mathematical Analysis 3 ( An3 25907 ), 7 CFU, second semester, years: 2° LT

Course content

Implicit functions, Dini theorem, local invertibility. Notion of sigma-algebra and measure. Lebesgue integral and theorems of convergence under sign of integral. Riesz extension of Riemann integral for continuous functions with compact support. Lebesgue measurable sets and their measure. Fubini theorem. Integrability criteria. Integrals depending by a parameter. Curves and surfaces; length and area; integration on curves and surfaces. Differential forms of degree 1; integration of 1-differential forms on oriented curves; closed and exact 1-differential forms.

Language

Italian

Teacher

Ada Aruffo

Other teachers

Laura Burlando

Teaching style

In presence

Attendance

Suggested

52474 - Mathematical Analysis I

General information

Mathematical Analysis I ( An. Mat. I 52474 ), 16 CFU, first and second semester, years: 1° LT

Language

Italian

Teacher

Filippo De Mari Casareto Dal Verme

Other teachers

Sandro Bettin

Emanuela Sasso

Alberto Sorrentino

Teaching style

In presence

Lesson timetable

Monday: 11:00 - 13:00, room 508

Attendance

Not required

90705 - Mathematical Logic 1

General information

Mathematical Logic 1 ( Log1 90705 ), 7 CFU, first semester, years: 3° LT; 1° LM

Course content

Logic for the mathematical practice. Formal theories and the deduction calculi.

Type theory. The problem of consistency. First order number theory. Recursive functions. The incompleteness theorems.

Hilbert tenth problem and its solution.

Language

Italian

Teacher

Giuseppe Rosolini

Teaching style

In presence

Attendance

Not required

61711 - Mathematical Logic 2

General information

Mathematical Logic 2 ( Log2 61711 ), 8 CFU, first semester, years: 1°, 2° LM

Course content

a) A proof of Gödel’s incompleteness theorem: computability and the theory of recursive functions, proof of the incompleteness theorem with an analysis of its consequences.

b) Mathematical proofs and formal proofs. Introduction to proof-theory. Gentzen’s natural deduction and the sequent calculus. The normalization theorem and the cut-elimination theorem.

c) Foundations revisited and analysis of the relevance for the teaching of mathematics.

Language

English

Teacher

Giuseppe Rosolini

Teaching style

In presence

Attendance

Not required

90700 - Mathematical Methods in General Relativity

General information

Mathematical Methods in General Relativity ( MMGR 90700 ), 5 CFU, first semester, years: 1°, 2° LM

Course content



Language

Italian

Teacher

Pierre Martinetti

Teaching style

In presence

Attendance

Suggested

90697 - Mathematical Methods in Quantum Mechanics

General information

Mathematical Methods in Quantum Mechanics ( MMMQ 90697 ), 5 CFU, second semester, years: 1°, 2° LM

Course content

TO BE UPDATED

General relativity

Introduction to quantum theories

Language

Italian

Teacher

Nicola Pinamonti

Teaching style

In presence

Attendance

Suggested

52503 - Mathematical Statistics

General information

Mathematical Statistics ( StMat 52503 ), 7 CFU, first semester, years: 1°, 2° LM

Course content

Classical statistical inference: samples, probabilistic and statistical models, statistics, point estimators. Finding estimators, evaluating estimators. Hypothesis testing. Likelihood theory. Introduction to Bayesian statistics.

Language

English

Teacher

Eva Riccomagno

Maria Piera Rogantin

Teaching style

In presence

Attendance

Not required

66446 - Mathematics education

General information

Mathematics education ( DM 66446 ), 7 CFU, first semester, years: 1° LM

Course content

The course covers the foundations of didactics of mathematics: the didactic transposition, the organization of the learning situations, the analysis of the processes and the difficulties of mathematics learning (with references to the "theories" of Wertheimer, Piaget and Vygotsky). In the course some specific characteristics of teaching / learning mathematics will be discussed and compared to the other sciences and disciplines, with reflections on the relationship between teaching / learning in the university and in the secondary school. Students will be involved through problem solving, cultural reflection about setting and didactic approach to mathematical topics, including in relation to various possible uses of the computer.

Language

Italian

Teacher

Francesca Morselli

Teaching style

In presence

Attendance

Required

98795 - Mathematics of machine learning

General information

Mathematics of machine learning ( MML 98795 ), 7 CFU, second semester, years: 1°, 2° LM

Language

English

Teacher

Ernesto De Vito

Other teachers

Lorenzo Rosasco

61712 - Modelling of continuous systems with applications

General information

Modelling of continuous systems with applications ( MSCA 61712 ), 8 CFU, first semester, years: 1°, 2° LM

Course content

TO BE UPDATED

Generalities on mathematic modelling of deformable continuous sytems. Perfect fluids. Viscous fluids. Linear elastic solids. Specific examples: vibrating string, surface tension, capillarity, coat of paint down a wall, etc. Dimensional analysis with applications.

Language

Italian

Teacher

Nicola Pinamonti

Maurizio Romeo

Teaching style

In presence

Attendance

Not required

52480 - Multivariate exploratory data analysis

General information

Multivariate exploratory data analysis ( SD 52480 ), 8 CFU, second semester, years: 1° LT

Language

Italian

Teacher

Maria Piera Rogantin

Other teachers

Alberto Sorrentino

Teaching style

In presence

Attendance

Suggested

84023 - Number Theory 1

General information

Number Theory 1 ( TN1 84023 ), 7 CFU, second semester, years: 3° LT; 1°, 2° LM

Course content

Arithmetical functions: arithmetical and algebraic aspects, asymptotic behavior. Elementary methods for the distribution of primes: Euler, Legendre and Chebyshev. Elements of cryptography. Complements of Analysis: Dirichlet series, Mellin transform and Poisson formula. Riemann zeta function: general properties and distribution of zeros. Prime Number Theorem: explicit formulae and PNT with remainder. Dirichlet L-functions: Dirichlet characters, general properties of L-functions and distribution of zeros. Dirichlet's theorem: explicit formulae and Dirichlet's theorem with remainder.

Language

English

Teacher

Sandro Bettin

Teaching style

In presence

Attendance

Suggested

26938 - Numerical Analysis

General information

Numerical Analysis ( CN 26938 ), 8 CFU, first semester, years: 3° LT

Course content

Iterative methods for the solution of linear systems.Minimizing quadratic forms: gradient method and conjugate gradient method. Methods for the solution of nonliner systems.The interpolation polynomials. Numerical integration: Newton-cotes quadrature rules and composite quadrature formulae: trapeziodal rule and Cavalieri-Simpson rule.Interpolation with splines and trigonometric functions.  Least squares approximation: continuos case. Orthogonal polynomials and Gaussian quadrature.

Language

Italian

Teacher

Alberto Sorrentino

Other teachers

Federico Benvenuto

Teaching style

In presence

Attendance

Not required

42927 - Numerical Linear Algebra

General information

Numerical Linear Algebra ( MNAL 42927 ), 6 CFU, first semester, years: 1°, 2° LM

Course content

Numerical methods for large scale matrices: sparse matrices, structured matrices. Graph theory and permutation techniques for sparse matrices analysis.

Matrix inverse by low rank corrections. Matrix inverse by block partitioning. Schur complements and Woodbury-Sherman-Morrison formula.

Kronecker product. Sylvester matrix equation and Kronecker sum.

QR factorization of sparse matrices.

Integral equations, discretization and convolution. Fast Fourier Transform (FFT) and its applications in matrix algebra. Toeplitz matrices, generating function, spectrum, equidistribution and Szegö-Tyrtysnikov theorem.

Convergence theory for stationary iterative methods for linear systems. Perron-Frobenius theory for nonnegative matrices. Regular splitting. Spectral radius and localization of eigenvalues. 

Iterative methods of minimization for the solution of linear systems. Non-stationary methods. Methods with optimum step length. Method of steepest descent. Conjugate gradient. Convergence analysis via matrix spectrum analysis. 

Preconditioning techniques. 

Matlab laboratory exercises.

Language

English

Teacher

Claudio Estatico

Teaching style

In presence

Attendance

Suggested

61473 - Numerical Solution of Differential Equations

General information

Numerical Solution of Differential Equations ( TNED 61473 ), 8 CFU, second semester, years: 1° LM

Course content

Difference equations: the constant-coefficient linear case. Runge-Kutta and Multistep methods for initial value Cauchy problems: consistency, convergence, stability, automatic step control. Finite difference approximations of initial and boundary value problems for elliptic, parabolic and hyperbolic PDEs. Explicit and implicit methods. Consistency, stability, convergence. An outline of finite element and finite volume methods. Practical Matlab exercises about the studied methods.

Language

Italian

Teacher

Fabio Di Benedetto

Teaching style

In presence

Attendance

Not required

80155 - Operations Research

General information

Operations Research ( RO 80155 ), 7 CFU, first semester, years: 3° LT; 1° LM

Course content

 

Language

Italian

Teacher

Marcello Sanguineti

Teaching style

In presence

Attendance

Suggested

87081 - Probability

General information

Probability ( Prob 87081 ), 8 CFU, first semester, years: 3° LT

Course content

Definition of Probability: frequencies, classical definition and subjective definition. Axiomatic definition of probability space: events, sigma-algebra, probability, first calculation rules and continuity of the probability measure. Indipendence and conditioning: total probability and Bayes theorem. Borel-Cantelli lemma. Random variables: distribution function and its properties. Continuous and discrete random variables (Bernoulli, Binomial, Geometric, Negative Binomial, Ipergeometric, Normal, Uniform, Cauchy, Exponential, Gamma, Chi-Square, t  Student,...). Multidimensional random variables, indipendence. Moments. Moment generating function and characteristic function. Inequalities: Markov and Chebyshev. Asymptotics: convergence in law, convergence in probability, almost sure convergence, normal limit of the binomial distribution, law of large numbers, central limit theorem. Conditional expecation. Stochastic simulation.

Language

Italian

Teacher

Emanuela Sasso

Other teachers

Veronica Umanità

Teaching style

In presence

Attendance

Not required

57320 - Stochastic Processes

General information

Stochastic Processes ( ProcStoc 57320 ), 7 CFU, first semester, years: 1°, 2° LM

Course content

Discrete time Markov chain. Random walks and queues. Classification of states. Transience and recurrence. Invariant laws. Limit theorems. 
Continuous time Markov chain: Chapman-Kolmogorov equations, invariant laws, Jump chain and holding time, Poisson process, birth and death process.
Mathematical foundation of queueing theory.

Language

Italian

Teacher

Veronica Umanità

Teaching style

In presence

Attendance

Not required