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.
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.
- 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.
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
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
Classical relativity and the wave equation for electromagnetic fields
The Michelson and Morley experiment
Einstein postulates
Lorentz transformations
Relativistic dynamics
Applications
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)
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.
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.
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)
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
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
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.
Background Notions
Models of discrete geometric shapes
Representations for cell and simplicial complexes
Discrete differential geometry
Geometry processing
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.
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.
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
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.
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.
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.
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.
Functions of n real variables: limits, continuity, differential calculus; ordinary differential equations; sequences and series of functions; power series;integrals of fields and scalar.
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.
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.
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.
TO BE UPDATED
General relativity
Introduction to quantum theories
Classical statistical inference: samples, probabilistic and statistical models, statistics, point estimators. Finding estimators, evaluating estimators. Hypothesis testing. Likelihood theory. Introduction to Bayesian statistics.
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.
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.
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.
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.
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.
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.
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.