Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's construction.

Splitting fields and normal extensions.

Separable, inseparable and purely inseparable extensions. Primitive element theorem.

Galois extensions. Galois group and Galois correspondence for finite extensions.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Algebraic varieties in affine snd projective spaces on an algebraically closed field. Rational maps and morphisms, Segre and Veronese varieties, products, projections. Local geometry of an algebraic variety. Normal varieties and normalization. Divisors, linear systems and morphisms of projective varieties.

1) R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2) I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1977.
3) J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.
4) Notes of the course by Lucia Caporaso

Algebraic varieties in affine snd projective spaces on an algebraically closed field. Rational maps and morphisms, Segre and Veronese varieties, products, projections. Local geometry of an algebraic variety. Normal varieties and normalization. Divisors, linear systems and morphisms of projective varieties.

1) R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2) I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1977.
3) J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.
4) Notes of the course by Lucia Caporaso

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst-case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. The transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms concerning alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Calculus in Banach spaces, derivation, Implicit Function Theorem;
Bifurcation theorem, Ljapunov-Schmidt method;
Energy-minimizing solutions, coercivity, lower semi-continuity;
Saddle point solutions, Mountain Pass Theorem;
Topological degree, Linking Theorems;
Ljusternik-Schnirelmann Category, existence of infinitely many solutions.

A. Ambrosetti, A. Malchiodi - "Nonlinear Analysis and Semilinear Elliptic Problem" - Cambridge
P. Rabinowitz - "Minimax methods in critical point theory with application to differential equations" - American Mathematical Society

Some techniques of Nonlinear Analysis.

P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations.

Lebesgue integration in Rn, Fourier transform, theory of differential equations

Luigi Chierchia, Analisi matematica due
Reed-Simon

Measure theory, outer measures, construction of Borel measures.
Integration theory, limit theorems, convergence in mean and in measure, integration on product spaces.
Radon measures, regularity, positive linear functionals, Riesz representation theorem.
Signed measures, decomposition theorems, differentiation, BV functions, fundamental theorem of calculus.
Lp spaces, basic properties, dual spaces, density theorems.
Introduction to geometric measure theory

G. Folland - "Real Analysis" - Wiley

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.

I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.

[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159

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

[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.

Evolution equations of Mathematical Physics: transport, wave and heat equations. Introduction to quantum mechanics. Fourier transform

[B] P. Butta', Note del corso di Fisica Matematica
[Cr] W. Craig, A course on Partial Differential Equations
[L1] V. Lubicz, Apppunti di Meccanica Quantistica

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available from the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available from the course page

Slides of the lessons, available from the course page

Branching processes, introduction to Sigma-algebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pi-systems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. Borel-Cantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 0-1 law.
Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws.
Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers.
Doob's inequalities for martingales and sub-martingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.

D. Williams, Probability with martingales
R. Durrett, Probability: Theory and examples

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak form of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Part I. Elements of ergodic theory.
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.

Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.

Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.

G. Gallavotti, F. Bonetto, G. Gentile
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.

Part I. Elements of ergodic theory.
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.

Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.

Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.

G. Gallavotti, F. Bonetto, G. Gentile
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.

Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's construction.

Splitting fields and normal extensions.

Separable, inseparable and purely inseparable extensions. Primitive element theorem.

Galois extensions. Galois group and Galois correspondence for finite extensions.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

Evolution equations of Mathematical Physics: transport, wave and heat equations. Introduction to quantum mechanics. Fourier transform

[B] P. Butta', Note del corso di Fisica Matematica
[Cr] W. Craig, A course on Partial Differential Equations
[L1] V. Lubicz, Apppunti di Meccanica Quantistica

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available from the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available from the course page

Slides of the lessons, available from the course page

Branching processes, introduction to Sigma-algebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pi-systems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. Borel-Cantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 0-1 law.
Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws.
Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers.
Doob's inequalities for martingales and sub-martingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.

D. Williams, Probability with martingales
R. Durrett, Probability: Theory and examples

Algebraic varieties in affine snd projective spaces on an algebraically closed field. Rational maps and morphisms, Segre and Veronese varieties, products, projections. Local geometry of an algebraic variety. Normal varieties and normalization. Divisors, linear systems and morphisms of projective varieties.

1) R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2) I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1977.
3) J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.
4) Notes of the course by Lucia Caporaso

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst-case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. The transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms concerning alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

The crisis of classical physics. Waves and particles. State vectors and operators. Measurements, observables and uncertainty relation. The position operator. Translations and momentum. Time evolution and the Schrödinger equation. One-dimensional problems. Parity. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.

Lecture notes available on the course website

J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Zanichelli
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics - Addison-Wesley

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak form of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

§I.Relativistic Field Theory

The Poincaré Group. Symmetries: global vs local. Noether's first and second theorems and conservation laws. The canonical stress-energy and angular momentum tensors. Improvements. Belinfante's argument and symmetric energy-momentum tensor. Local symmetries and conserved quantities.

§II.Gravity as a relativistic field theory

Particles and fields in Special Relativity. Irreps of the Poincaré group: Wigner's induced representation method. Massless particles: ISO(D-2) little group and gauge invariance. From relativistic massless spin-2 particles to full GR. Fierz-Pauli quadratic Lagrangian. Nöther method and non-linear completions. Nöther's construction of Yang-Mills Lagrangian. The transverse-traceless gravitational cubic vertex. Weinberg's Equivalence Principle from relativistic invariance of the S matrix. Spin and the sign of static forces.

§III.Elements of differential geometry

Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Levi-Civita connection. Torsion. Differential forms: definition, wedge product, interior and exterior derivatives, Hodge dual. Lie derivative of forms and Cartan's formula. Yang-Mills theory in the language of forms. Weyl tensor. Riemann and Weyl tensors in various dimensions: counting components for irreps of GL(D). Conformal transformations of the metric tensor. Conformally flat spaces. Conformally coupled scalar fields.

§IV. The Cartan-Weyl formulation of GR and Fermionic couplings

Local inertial frames. The frame field and its relation to the metric field. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Gravity as a gauge theory of the Poincaré algebra. Connection one-forms on the Poincar\'e algebra. Local Poincar\'e transformations. Torsion and curvature over the Poincar\'e algebra. First-order formulation and Cartan-Weyl's action. Relation between gauge transformations and diffeomorphisms. Spinors on curved manifolds. Minimally coupled Fermionic matter. Dirac Lagrangian.

§V. Maximally symmetric spaces

Homogeneous and isotropic spaces. Characterisation of maximally symmetric spaces: curvature constant and signature. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients.

§VI. The Schwarzschild black hole

Spherically symmetric spaces. The Schwarzschild solution. Birkhoff's theorem. Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. Free-fall towards the horizon. The tortoise coordinate. Extension of a space-time. Eddington-Finkelstein coordinates. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal's diagram and eternal black holes. (A)dS-Schwarzschild space-time.

§VII. More general black holes

Conformal diagrams. Event horizons. Reissner-Nordström and Kerr black holes. Black hole thermodynamics.

§VII. Gravitational energy

Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Einstein-Hilbert equations for asymptotically flat metrics. Candidate for gravitational energy-momentum tensor. The superpotential. ADM energy and momentum. Example: ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Generic background with Killing vectors. Quadrupole radiation.

§VIII. Asymptotic symmetries

General notion of asymptotic symmetry group. The example of Maxwell's theory in flat space. Covariant phase space formalism. Asymptotically flat spacetime and Bondi-van der Burg-Metzner-Sachs supertranslations. Applications: soft theorems and memory effects.

Note: some topics may be assigned as homework problems, as an alternative to the oral exam

-Carroll S, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Wald R, General Relativity (The University of Chicago Press, 1984)
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972)

The aim of the course is to provide the student with the general cognitive elements underlying the acquisition, control and monitoring systems of Nuclear and Subnuclear Physics experiments. The course is divided into the following topics:
-Introduction to DAQ
-Parallelism and Pipelining systems
-Derandomization
-DAQ and Trigger
-Data Transmission
-Front End Electronics
-Trigger
-Architecture Computing Systems
-Real Time Systems
-Real Time Operating Systems
-C Language
- VHDL Language
-TCP / IP Network Protocols
-DAQ Architecture
- Event Building
-VME Bus
-Run Control
-Farming
-Data Archiving

Lecture notes prepared by the teacher on the basis of the slides presented and available on the Moodle server:
https://matematicafisica.el.uniroma3.it

Parallel architectures: shared and distributed memory systems; GPGPU
Parallel programming Patterns: embarassingly parallel problems; work farms; partitioning; reduce; stencils
Performance evaluation of parallel programs: speedup, efficiency, scalability
Programming shared memory architectures with OpenMP
Programming distributed memory architectures with MPI
Programming GPUs with CUDA
Notes on innovative programming languages ​​for HPC (OpenACC, Rust + libraries, SIMD, OpenCL, ...)

Peter Pacheco, Matthew Malensek, An Introduction to Parallel Programming, 2nd ed., Morgan Kaufmann, 2021, ISBN 9780128046050

CUDA C++ programming guide

Course Slides by the Lecturer

The objective of Linguaggi di Programmazione course is to introduce main formal language theory concepts and results as well as their application for programming language classification.
Most relevant approaches for syntactic analysis of programming languages are introduced. Learning how to recognize the structure of a programming language and the implementation techniques for the abrstract machine.
Understanding the Object Oriented paradigm together with other non imperative approaches.

[1] Maurizio Gabbrielli, Simone Martini,Programming Languages - Principles and paradigms, 2/ed. McGraw-Hill, (2011).
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, 2 edizione. O’Reilly Media, (2014).
[3] David Parsons, Foundational Java Key Elements and Practical Programming. Springer- Verlag, (2012).
Course Slides provided by the lecturer.

- Affine and projective curves
- Cubics and elliptic curves
- The group law and equations of elliptic curves
- Isogenies
- Torsion points
- Elliptic curves over finite fields and Hasse's Theorem
- Brief discussion of symmetric and public key cryptosystems
- Algorithms on Elliptic Curves: Double and Add and Schoof Algorithm
- Public Key and Digital Signature algorithms on Elliptic Curves
- Weil Pairing and Identity based elliptic cryptosystems

1. Machine learning. Types of learning. Loss functions. Empirical risk minimization. Generalization and overfitting.
2. Model optimization. Convex functions. Gradient descent. Stochastic gradient descent.
3. Regression. Linear regression. Basis functions. Feature selection. Regularization.
4. Classification. Generative models. Nearest neighbor. Logistic regression. Support vector machines. Neural networks.
5. Ensemble methods. Decision trees. Boosting. Bagging.
6. Unsupervised learning. K-means clustering. Hierarchical clustering. Principal component analysis.
7. Application of the methods using the Python language. Examples using the NumPy, Pandas, SciKit-Learn, and TensorFlow libraries.

J. Watt, R. Borhani, A. Katsaggelos. Machine Learning Refined. Cambridge University Press, 2nd edition, 2020.

Random variables and their distribution, moment generating function, mean variance and covariance.
Random sampling model and statistical model.
Statistics: concept, examples, sufficient statistics.
Point estimators: definition and desired properties, moments, maximum likelihood and Bayes.
Computational methods: Newton-Raphson, EM algorithm
Improving an estimator: Rao-Blackwell, UMVU estimator, full statistic, Lehman-Scheff ́e II and Cramer-Rao
Confidence intervals: intuitive, pivotal quantity, IC for Bayes and asymptotic IC.
Hypothesis testing: likelihood ratio, pivotal quantity test (Z and T test), duality with IC, UMP, Neyman-Pearson and Karlin-Rubin tests.
Non-parametric methods: goodness-of-fit, contingency table, Kolmogorov-Smirnov and ranking tests.
Analysis of variance (ANOVA) and F.
Regression: linear, multiple linear, generalized linear and Logistic / Poisson

Statistical Inference, Casella e Berger, 2nd Edition, Duxbury Advanced Series.

Additional reference:
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi

ntroduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

The high scool mathematics: programs, key and critical points; the mathematicians' mathematics and the mathematics for the citizen
From exercise to problem solving. Didactic theories and theorists. Guido and Emma Castelnuovo. Croce, Gentile and mathematics.
Didactic methodologies. Scholastic laws.

Battaglini Frank, Di Martino, Natalini, Rosolini "Didattica della matematica", Mondadori Università

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Calculus in Banach spaces, derivation, Implicit Function Theorem;
Bifurcation theorem, Ljapunov-Schmidt method;
Energy-minimizing solutions, coercivity, lower semi-continuity;
Saddle point solutions, Mountain Pass Theorem;
Topological degree, Linking Theorems;
Ljusternik-Schnirelmann Category, existence of infinitely many solutions.

A. Ambrosetti, A. Malchiodi - "Nonlinear Analysis and Semilinear Elliptic Problem" - Cambridge
P. Rabinowitz - "Minimax methods in critical point theory with application to differential equations" - American Mathematical Society

Lebesgue integration in Rn, Fourier transform, theory of differential equations

Luigi Chierchia, Analisi matematica due
Reed-Simon

Measure theory, outer measures, construction of Borel measures.
Integration theory, limit theorems, convergence in mean and in measure, integration on product spaces.
Radon measures, regularity, positive linear functionals, Riesz representation theorem.
Signed measures, decomposition theorems, differentiation, BV functions, fundamental theorem of calculus.
Lp spaces, basic properties, dual spaces, density theorems.
Introduction to geometric measure theory

G. Folland - "Real Analysis" - Wiley

Basic Linear Algebra:
Hilbert Spaces, Products and Tensor Products, Matrices, Complex Spaces and Inner Products, Matrices, Graphs, and Sums Over Paths.
Boolean Functions, Quantum Bits, and Feasibility:
Feasible Boolean Functions, Quantum Representation of Boolean Arguments Quantum Feasibility.
Special Matrices:
Hadamard Matrices, Fourier Matrices, Reversible Computation and Permutation Matrices, Feasible Diagonal Matrices, Reflections. Tricks: Start Vectors, Controlling and Copying Base States, The Copy-Uncompute Trick, Superposition Tricks, Flipping a Switch, Measurement Tricks, Partial Transforms.
Algorithms:
Phil’s Algorithm: Phil Measures Up, Quantum Mazes versus Circuits versus Matrices.
Deutsch’s Algorithm: Superdense Coding and Teleportation.
The Deutsch-Jozsa Algorithm. Simon’s Algorithm. Shor’s Algorithm, Quantum Part of the Algorithm, Analysis of the Quantum Part, Continued Fractions.
FactoringIntegers: Basic Number Theory, Periods Give the Order, Factoring.
Grover’s Algorithm: The binary case, the general case, with k Unknowns, Grover Approximate Counting.

Richard J. Lipton, Kenneth W. Regan Introduction to Quantum Algorithms via Linear Algebra, Second Edition, ISBN 9780262045254, (2021), MIT Press

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.

Part I. Elements of ergodic theory.
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.

Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.

Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.

G. Gallavotti, F. Bonetto, G. Gentile
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.

I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.

[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159

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

[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.

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

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

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

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

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

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

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

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 semi-Eulerian 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 4-color 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 Max-Flow Min-Cut 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 Riemann-Roch 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.

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.

Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens

Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546

The main theorems of Functional Analysis.

H. Brezis, Functional Analysis.

Banach and Hilbert spaces, general properties, projections in Hilbert spaces, orthonormal systems.
Hahn-Banach theorem, analytic and geometric form, consequences.
First and second category spaces, Baire theorem, Banach-Steinhaus theorem, open map and closed graph, applications.
Weak, closed and convex topologies, Banach-Alaoglu Theorem, separability, reflexivity and uniform convexity.
Sobolev spaces in one dimension, embedding theorems, Poincaré inequality, application to variational problems.
Spectral theory, Fredholm alternative, spectral theorem for compact and self-adjoint operators, application to variational problems.

H. Brezis - Analisi Funzionale - Liguori (1986);
H. Brezis - Functional Analysis, Sobolev Spaces and Partial Differential Equations - Springer (2010);
W. Rudin - Functional Analysis - McGraw-Hill (1991);

Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.

“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

Definition and basic properties of the Sobolev spaces W^{1,p} (Ω). Extension operators. Sobolev inequalities. The space W^{1,p}_0 (Ω). Variational formulation of some elliptic boundary value problems. Existence of weak solutions. Regularity of weak solutions.

"Analisi funzionale", H. Brézis, Liguori Editore
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

Ring of integers in number fields.
Unique factorisation of ideals in the rings of integers.
Class group.
Group of units.
Fermat's last Theorem for regular primes
Local fields.

Notes by the lecturer.
Marcus, D. Number fields, 3rd Ed Springer-Verlag. 1977.
Samuel, P. Théorie algébrique des nombres, Hermann, Paris. 1971.
Schoof, R. Algebraic Number Theory, dispense Università di Roma Tor Vergata, 2003.
Milne, J. Algebraic Number Theory, Lecture Notes, 2017.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– The goals of statistical mechanics
– Review of thermodynamics. Convex functions and Legendre transform
– Models of statistical mechanics: canonical ensemble, grand canonical and Gibbs states.
– The Ising model and the lattice gas models. Existence of the thermodynamic limit for the free energy of the Ising or lattice gas model.
– The general structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs state.

THE ISING MODEL
– Review of known results on the Ising model in one or more dimensions.
– The solution of the one-dimensional Ising model via the transfer matrix method.
– The mean field Ising model: exact solution. Phase transition and loss of equivalence between statistical ensembles
– Ising with long-range interactions (Kac potentials) in the mean-field limit. The Maxwell construction.
– FKG and Griffiths inequalities. Existence of the infinite volume correlation functions of states with + and − conditions in the ferromagnetic Ising model.
- The geometric representation of the Ising model: high and low temperature contours.
- Existence of a phase transition in the low temperature Ising model: the Peierls's argument.
– Absence of a phase transition at high temperature and exponential decay of boundary effects boundary conditions.
– Lee-Yang theorem and analyticity of the pressure at non-zero magnetic field.
– Existence of a phase transition in the one dimensional Ising model with power law interaction |x − y|^{−p}, 1

S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017.

G. Gallavotti: Statistical Mechanics. A short treatise, ed. Springer-Verlag, 1999.

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) 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)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf on the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf on the course page

Lecture slides in pdf on the course page

Additional notes provided by the teacher

Linear dynamical systems. Forced harmonic oscillator with or without
friction. Stability theorems. Parametric resonance. Chain of
coupled harmonic oscillators: continuum limit and equations of
vibrating rope. Classic elastic diffusion. Hidden prime integrals
in the two-body problem and the harmonic oscillator problem

V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Editors
Riuniti, Rome, 1979 G. Gallavotti, Meccanica Elementare, ed. P.
Boringhieri, Turin, 1986 G. Gentile, Introduction to systems
dynamics, 1 (Ordinary differential equations, qualitative analysis and
some applications) and
2 (Lagrangian and Hamiltonian mechanics) L.D. Landau, E.M. Lifshitz,
Meccanica, Editori Riuniti, Rome, 1976

Euler angles. Euler's equations for body dynamics
rigid. Integrability of the rigid body with a point not subjected to
strength. Lagrange spinning top. Arnold–Liouville theorem. Variables
action-angle for the harmonic oscillator and for the problem of the two
bodies. Formulation in action-angle variables of the 3 problem
bodies restricted.
Calculation of the precession of Mercury's perihelion.
Notes on the KAM theory on the convergence of the theory of
classic perturbations. Notes on the statistical theory of motion:
integrable, quasi-integrable and chaotic systems. Demonstration of the
dense and uniform filling of the torus by the flow
quasi-periodic irrational. Visiting frequencies.

V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Editors
Riuniti, Rome, 1979 G. Gallavotti, Meccanica Elementare, ed. P.
Boringhieri, Turin, 1986 G. Gentile, Introduction to systems
dynamics, 1 (Ordinary differential equations, qualitative analysis and
some applications) and
2 (Lagrangian and Hamiltonian mechanics) L.D. Landau, E.M. Lifshitz,
Meccanica, Editori Riuniti, Rome, 1976

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.

Main reference:
D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Additional reference (in Italian)
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi

STOCHASTIC PROCESSES, BROWNIAN MOTION, STOCHASTIC INTEGRALS, STOCHASTIC DIFFERENTIAL EQUATIONS. ITO FORMULA. FEYNMANN-KAC FORMULAS AND APPLICATIONS. MARKOV TIMES AND PROBABILISTIC SOLUTION OF THE DIRICHLET PROBLEM.

P. Morters, Y. Peres: Bronian Motion, Cambridge 2010
T. Liggett, Continuous time Markov processes: an introduction, AMS 2010
L.C. Evans:Introduction to stochastic differential equations, AMS 2014,
J.F. Le Gall: Brownian motion, martingales, and stochastic calculus, Springer 2016

1. Optimization and combinatorial optimization problems. Enumeration of solutions.
2. Basics of algorithm analysis. Computational tractability. Asymptotic order of growth.
3. Graphs. Graph connectivity and graph traversal. Graph bipartiteness. Connectivity in directed graphs. Directed acyclic graphs and topological ordering.
4. Greedy algorithms. Interval scheduling. Optimal caching. Shortest paths in a graph. Minimum spanning trees.
5. Divide and conquer. Mergesort. Counting inversions. Closest pair of points.
6. Dynamic programming. Weighted interval scheduling. Principles of dynamic programming. Subset sums and knapsacks. All-pairs shortest paths. Shortest paths and distance vector protocols.
7. Network flow. Maximum flow and the Ford-Fulkerson algorithm. Maximum flows and minimum cuts in a network. Augmenting paths. Bipartite matching. Disjoint paths in directed and undirected graphs.
8. Computational intractability. Polynomial-time reductions. Reductions via "gadgets". Efficient certification and the definition of NP. NP-complete problems. Covering, packing, partitioning, sequencing, and numerical problems. Other examples.

Jon Kleinberg, Eva Tardos. Algorithm Design. Pearson Education, 2013.

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

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

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

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

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

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

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

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

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

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

F Cesarone (2020), Computational Finance. MATLAB oriented modeling, Routledge-Giappichelli Studies in Business and Management, ISBN 978-0-367-49303-5
https://www.giappichelli.it/computational-finance

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– The goals of statistical mechanics
– Review of thermodynamics. Convex functions and Legendre transform
– Models of statistical mechanics: canonical ensemble, grand canonical and Gibbs states.
– The Ising model and the lattice gas models. Existence of the thermodynamic limit for the free energy of the Ising or lattice gas model.
– The general structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs state.

THE ISING MODEL
– Review of known results on the Ising model in one or more dimensions.
– The solution of the one-dimensional Ising model via the transfer matrix method.
– The mean field Ising model: exact solution. Phase transition and loss of equivalence between statistical ensembles
– Ising with long-range interactions (Kac potentials) in the mean-field limit. The Maxwell construction.
– FKG and Griffiths inequalities. Existence of the infinite volume correlation functions of states with + and − conditions in the ferromagnetic Ising model.
- The geometric representation of the Ising model: high and low temperature contours.
- Existence of a phase transition in the low temperature Ising model: the Peierls's argument.
– Absence of a phase transition at high temperature and exponential decay of boundary effects boundary conditions.
– Lee-Yang theorem and analyticity of the pressure at non-zero magnetic field.
– Existence of a phase transition in the one dimensional Ising model with power law interaction |x − y|^{−p}, 1

S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017.

G. Gallavotti: Statistical Mechanics. A short treatise, ed. Springer-Verlag, 1999.

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) 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)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf on the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf on the course page

Lecture slides in pdf on the course page

Additional notes provided by the teacher

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 non-Euclidean 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.

List of problems given in class concerning visualization and solutions with the help of software Mathematica or GeoGebra.

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; Substitution-Permutation Networks (SPN); Linear cryptanalysis for SPN: Piling-Up Lemma, linear approximation of S-boxes, linear attacks on S-boxes; 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 pre-image, resistance to the second pre-image, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second pre-image, collision on the problem of the pre-image. Iterated hash functions; the construction of Merkle-Damgard. Safe Hash Algorithm (SHA-1). Authentication Codes (MAC): nested authentication codes (HMAC).

[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.

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 semi-Eulerian 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 4-color 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 Max-Flow Min-Cut 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 Riemann-Roch 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.

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.

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

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

Taylor, J. - Introduzione all'analisi degli errori : lo studio delle incertezze nelle misure fisiche - Zanichelli Disponibile nella biblioteca Scientifica di Roma Tre

Dubi, A. - Monte Carlo applications in systems engineering - Wiley Disponibile nella biblioteca Scientifica di Roma Tre

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. Quasi-coherent 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 quasi-coherent and coherent sheaves on a scheme.

Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.

Invertible sheaves and linear systems

Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems.

Notes from Prof. Lopez, Prof. Sernesi
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).

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)
Maxwell-Boltzmann 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 fluctuation-dissipation theorem. (1, Par. 8.4).
Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2).
Johnson-Nyqvist theory of thermal noise. (1 Par. 11.3).
Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4)
Fermi-Dirac and Bose-einstein 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 Bose-Einstein condensation. (1, Par. 7.3)
Quantum theory of black-body radiation. (1, Par. 7.5)

e-Learning Moodle Platform with Supplementary Material

Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.

Topics Part A

• Coordinates and Telescopes
• Elements of Spectroscopy
• Stars and Stellar Evolution • Galaxies
• Active Galactic Nuclei


Program Part A

• Overview

• Celestial coordinates (1.3)

• Telescopes and resolving power (6.1)

• Parallax distance (3.1)

• Flux, brightness, apparent and absolute magnitudes, colors (3.2, 3.3, 3.6)

• The black body (3.4, 3.5)

• Hertzsprung-Russel diagram (8.2)

• Open and globular clusters: position, stellar populations and HR diagram (13.3)

• White dwarfs, Novae and SuperNovae (notes and partly in 15 and 16)

• The classification of galaxies (24.1)

• The rotation curve of galaxies and dark matter (25.3)

• The center of the Galaxy and its Black Hole (25.4)

• Hubble's law and expansion of the Universe (27.2)

• Probability of collision between stars and galaxies (handouts)

• Black Holes: outline of General Relativity (outline 17)

• Active Galactic Nuclei (28.1, 28.2, 28.3)


Topics Part B

• Structure and stellar evolution
• Elements of Spectroscopy
• Distances and expansion of the Universe • Galaxies
• GRB and gravitational waves

Program Part B

• Acretion disks and X-ray emission in Active Galactic Nuclei (28.2)

• Stars of Neutrons and Pulsars (16.6, 16.7)

• Gamma Ray Bursts (handouts)

• Gravitational Waves (lecture notes)

• Spectroscopy: eq. Boltzmann-excitation and Saha-ionization (8.1)

• Spectroscopy: speed, temperature and density measurements (handouts)

• Eq. of star structure, time and Kelvin-Helmholtz instability (11.1-4)

• Nuclear reactions of hydrogen (11.3)

• Jeans mass of gravitational collapse, free-fall time and Initial Mass Function (12.2, 12.3)

• The Milky Way and the local group (25.1, 25.2)

• Metallicity (25.2)

• Transit of Venus and measurement of the Earth-Sun distance (handouts)

• Distance scale (27.1)

• Hubble's law and expansion of the Universe (27.2)

• Local Group, Clusters of Galaxies, large scale structure of the Universe (27.3)

• The Big Bang and the background radiation (29.2 brief notes and lecture notes)

A copy of the lecture notes can be downloaded from the course website.

In brackets, the paragraphs from “An Introduction to Modern Astrophysics, II ed. - B.W. Carrol, D.A. Ostlie - Ed. Pearson, Addison Wesley ”(copies available in the library). The discussion in the course has been simplified compared to what is reported in the text. Alternative text in Italian: Attilio Ferrari, Stars, Galaxies, Universe - Fundamentals of Astrophysics - Ed. Springer

1. The themes of logic
2. Classical logic: propositions, proofs
3. Classical logic: connectives
4. Classical logic: types, variables, quantifiers
5. First-order Classical logic
6. Classes and sets
7. Codes, digits, Boolean algebra
8. Turing Machine
9. Axiomatization and formalization of first-order classical logic
10. Logic and other disciplines

V. Michele Abrusci, LOGICA - Lezioni di primo livello, Quarta edizione, Wolters Kluwer, 2018

Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens

Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546

1. Optimization and combinatorial optimization problems. Enumeration of solutions.
2. Basics of algorithm analysis. Computational tractability. Asymptotic order of growth.
3. Graphs. Graph connectivity and graph traversal. Graph bipartiteness. Connectivity in directed graphs. Directed acyclic graphs and topological ordering.
4. Greedy algorithms. Interval scheduling. Optimal caching. Shortest paths in a graph. Minimum spanning trees.
5. Divide and conquer. Mergesort. Counting inversions. Closest pair of points.
6. Dynamic programming. Weighted interval scheduling. Principles of dynamic programming. Subset sums and knapsacks. All-pairs shortest paths. Shortest paths and distance vector protocols.
7. Network flow. Maximum flow and the Ford-Fulkerson algorithm. Maximum flows and minimum cuts in a network. Augmenting paths. Bipartite matching. Disjoint paths in directed and undirected graphs.
8. Computational intractability. Polynomial-time reductions. Reductions via "gadgets". Efficient certification and the definition of NP. NP-complete problems. Covering, packing, partitioning, sequencing, and numerical problems. Other examples.

Jon Kleinberg, Eva Tardos. Algorithm Design. Pearson Education, 2013.

The main theorems of Functional Analysis.

H. Brezis, Functional Analysis.

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.

Main reference:
D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Additional reference (in Italian)
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi

Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.

“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

Definition and basic properties of the Sobolev spaces W^{1,p} (Ω). Extension operators. Sobolev inequalities. The space W^{1,p}_0 (Ω). Variational formulation of some elliptic boundary value problems. Existence of weak solutions. Regularity of weak solutions.

"Analisi funzionale", H. Brézis, Liguori Editore
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

• Python:https://github.com/steguar/DAIL/blob/main/Lecture_1/Lecture_1_Python_crash_ course.ipynb
• Mathematical Biology, J.D. Murray (Population models - Epidemic models)
• Population Ecologies: First Principles Deborah Goldberg and John H.Vandermeer
(Simple population models, CAP1)
• Non Linear Dynamics and Chaos, S. Strogatz (Stability of dynamical systems)
• Computational Physics, M. Newman (Euler and RK methods)
• Networks, M. Newman (ER and CM random graphs, Epidemics on Networks)
• Understanding Bioinformatics, Marketa Zvelebil & Jeremy O. Baum
• Biological sequence analysis, R. Durbin et al. (CAP 1,2,6,7)
• Bioinformatics Algorithms: an Active Learning Approach, Pavel A. Pevzner and Phillip Compeau
• Bioinformatics - an Introduction, Jeremy Ramsden
• Understanding Bioinformatics, Marketa Zvelebil, Jeremy O. Baum
• Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids
• Statistical Methods in Bioinformatics, An Introduction, Warren J. Ewens , Gregory Grant

STOCHASTIC PROCESSES, BROWNIAN MOTION, STOCHASTIC INTEGRALS, STOCHASTIC DIFFERENTIAL EQUATIONS. ITO FORMULA. FEYNMANN-KAC FORMULAS AND APPLICATIONS. MARKOV TIMES AND PROBABILISTIC SOLUTION OF THE DIRICHLET PROBLEM.

P. Morters, Y. Peres: Bronian Motion, Cambridge 2010
T. Liggett, Continuous time Markov processes: an introduction, AMS 2010
L.C. Evans:Introduction to stochastic differential equations, AMS 2014,
J.F. Le Gall: Brownian motion, martingales, and stochastic calculus, Springer 2016

Ring of integers in number fields.
Unique factorisation of ideals in the rings of integers.
Class group.
Group of units.
Fermat's last Theorem for regular primes
Local fields.

Notes by the lecturer.
Marcus, D. Number fields, 3rd Ed Springer-Verlag. 1977.
Samuel, P. Théorie algébrique des nombres, Hermann, Paris. 1971.
Schoof, R. Algebraic Number Theory, dispense Università di Roma Tor Vergata, 2003.
Milne, J. Algebraic Number Theory, Lecture Notes, 2017.

Highlights of Linear Algebra:
Matrix-matrix multiplication; column & row space; rank
The four fundamental subspaces of linear algebra
Fundamentals of Matrix factorizations:
A=LU rows & columns point of view
A=LU elimination & factorization; permutations
A=RU=VU; Orthogonal matrices
Eigensystems and Linear ODE
Intro to PSym; the energy function
Gradient and Hessian
Singular Value Decomposition
Eckart-Young; derivative of a matrix norm
Principal Component Analysis
Generalized evectors;
Norms
Least Squares
Convexity & Newton’s method
Newton & L-M method; Recap of non-linear regression
Lagrange multipliers

Machine Learning:
Gradient Descend; exact line search; GD in action; GD with Matlab
Learning & Loss; Intro to Deep Neural Network; DNN with Matlab
Loss functions: Quadratic VS Cross entropy
Stocastics Gradient Descend (SGD) & Kaczmarcz; SGD convergence rates & ADAM
Matlab interface for DNN
Construction of DNN: the key steps
Backpropagation and the Chain Rule
Machine Learning examples with Wolfram Mathematica
Convolutional NN + Mathematica examples of 1D convolution
Convolution and 2D filters + Mathematica examples of 2D convolution
Matlab Live Script, Network Designer, Pretrained Net

Symplectic geometry and Hamiltonian formalism. Other topics, depending on the interests of the audience

[Z] Zehnder - "Lectures on dynamical systems".

Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's construction.

Splitting fields and normal extensions.

Separable, inseparable and purely inseparable extensions. Primitive element theorem.

Galois extensions. Galois group and Galois correspondence for finite extensions.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Algebraic varieties in affine snd projective spaces on an algebraically closed field. Rational maps and morphisms, Segre and Veronese varieties, products, projections. Local geometry of an algebraic variety. Normal varieties and normalization. Divisors, linear systems and morphisms of projective varieties.

1) R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2) I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1977.
3) J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.
4) Notes of the course by Lucia Caporaso

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst-case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. The transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms concerning alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Calculus in Banach spaces, derivation, Implicit Function Theorem;
Bifurcation theorem, Ljapunov-Schmidt method;
Energy-minimizing solutions, coercivity, lower semi-continuity;
Saddle point solutions, Mountain Pass Theorem;
Topological degree, Linking Theorems;
Ljusternik-Schnirelmann Category, existence of infinitely many solutions.

A. Ambrosetti, A. Malchiodi - "Nonlinear Analysis and Semilinear Elliptic Problem" - Cambridge
P. Rabinowitz - "Minimax methods in critical point theory with application to differential equations" - American Mathematical Society

Lebesgue integration in Rn, Fourier transform, theory of differential equations

Luigi Chierchia, Analisi matematica due
Reed-Simon

Measure theory, outer measures, construction of Borel measures.
Integration theory, limit theorems, convergence in mean and in measure, integration on product spaces.
Radon measures, regularity, positive linear functionals, Riesz representation theorem.
Signed measures, decomposition theorems, differentiation, BV functions, fundamental theorem of calculus.
Lp spaces, basic properties, dual spaces, density theorems.
Introduction to geometric measure theory

G. Folland - "Real Analysis" - Wiley

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.

I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.

[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159

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

[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.

Evolution equations of Mathematical Physics: transport, wave and heat equations. Introduction to quantum mechanics. Fourier transform

[B] P. Butta', Note del corso di Fisica Matematica
[Cr] W. Craig, A course on Partial Differential Equations
[L1] V. Lubicz, Apppunti di Meccanica Quantistica

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available from the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available from the course page

Slides of the lessons, available from the course page

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak form of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

Branching processes, introduction to Sigma-algebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pi-systems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. Borel-Cantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 0-1 law.
Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws.
Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers.
Doob's inequalities for martingales and sub-martingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.

D. Williams, Probability with martingales
R. Durrett, Probability: Theory and examples

Part I. Elements of ergodic theory.
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.

Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.

Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.

G. Gallavotti, F. Bonetto, G. Gentile
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.

Fields extensions and their basic properties.

Algebraic closure of a field: existence and uniqueness. Kronecker's construction.

Splitting fields and normal extensions.

Separable, inseparable and purely inseparable extensions. Primitive element theorem.

Galois extensions. Galois group and Galois correspondence for finite extensions.

Prefinite groups and Krull topology. Galois correspondence for infinite extensions.

Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.

Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.

Solvable groups. Solvable and solvable by radicals extensions.

More examples and applications.

Algebra S. Bosch

Algebra S. Lang

Algebra M. Artin

Class Field Theory J. Neukirch

Evolution equations of Mathematical Physics: transport, wave and heat equations. Introduction to quantum mechanics. Fourier transform

[B] P. Butta', Note del corso di Fisica Matematica
[Cr] W. Craig, A course on Partial Differential Equations
[L1] V. Lubicz, Apppunti di Meccanica Quantistica

1. Topological classification of curves and compact surfaces. Triangulations, Euler characteristic.
2. Smooth and regular curves in Euclidean space. Immersions and imbeddings. Arc length. Curvature and the Fundamental Theorem of local geometry of plane curves.
3. Regular surfaces in R^3, local coordinates. Inverse image of a regular value. Maps and diffeomorphisms. Tangent plane and derivative of a map. Normal unit vector, orientation
and the Gauss map of a surface in R^3. Orientable surfaces, examples. The Moebius band is non-orientable.
4. Riemannian metric, examples. The shape operator is self-adjoint. Principal and asymptotic directions. The Mean and Gauss curvatures.
5. The geometry of the Gauss map. The sign of the Gauss curvature and position of the tangent plane. Theorems of Meusnieur and Olinde Rodrigues. Geometric properties of compact surfaces, Minimal surfaces and Ruled surfaces.
6. Isometries and local isometries. The Shape operator in isothermal coordinates. Proof of Gauss' Theorema Egregium. Examples, counter-examples and applications.
7. Homeworks.
8. 12 hours of lab for the visualization and computation on curves and surfaces.

Textbooks
[1] J.M. Lee, Introduction to topological manifolds. Springer, (2000). - – http://dx.doi.org/10.1007/b98853
[2] E. Sernesi, Geometria 2. Boringhieri, (1994).
[3] M. Do Carmo , Differential Geometry of Curves and Surfaces. Prentice Hall, (1976).
[4] M.Abate, F.Tovena, Curve e Superfici. Springer, (2006).

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available from the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available from the course page

Slides of the lessons, available from the course page

Branching processes, introduction to Sigma-algebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pi-systems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. Borel-Cantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 0-1 law.
Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws.
Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers.
Doob's inequalities for martingales and sub-martingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.

D. Williams, Probability with martingales
R. Durrett, Probability: Theory and examples

Algebraic varieties in affine snd projective spaces on an algebraically closed field. Rational maps and morphisms, Segre and Veronese varieties, products, projections. Local geometry of an algebraic variety. Normal varieties and normalization. Divisors, linear systems and morphisms of projective varieties.

1) R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2) I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1977.
3) J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.
4) Notes of the course by Lucia Caporaso

1) Computability, complexity and representability:

- Introduction to decision problems, algorithmic and non-algorithmic procedures, deterministic computations, discrete procedures, the notion of alphabet, of speech. Decidability and semi-decidability of a set. Deterministic, finitary and discrete computations. Formal algorithms: formal definition of algorithm, configurations of input, output, transition function. Example of formalization of an algorithm. Decidability for finished automata. Representation of the automata by matrices. Free Monoid of words. Formal semi-rings. Non-deterministic finite automata. Regular Languages. Equivalence between deterministic and non-deterministic automata.

- Turing machines: definition, decidability for Turing machine, stopping time, stopping space. Cost of computation. Complexity: worst-case and average case. Independence of decision time from a finite number of input configurations. Complexity functions, complexity classes DTIME and DSPACE (deterministic time and space). Inclusion DTIME (T (n)) ⊂ DSPACE (T (n)) ⊂ DTIME (2 ^ {cT (n)}). Pumping Lemma. Simulation of algorithms, simulation of the half tape Turing machine, simulation of a multi-tape machine. Special Turing machines. Linear Speedup theorem for Turing machines with an extended alphabet. Evaluation of acceleration coefficient in relation to alphabets. Decisions of natural number sets. Independence from representation. Considerations concerning complexity.

- Turing computability: definition of Turing computable function, characteristic functions of Turing decidable sets, the class of Turing computable functions is closed by composition, concatenation, primitive recursion and minimization. Examples of Turing computable functions. Recursive Functions: equivalence between Turing computability and recursive functions. Ackermann function ([1] chapter 1,2,3,4,5 and [4] chapter 1).

- Time-constructible functions. The notion of T-clock. Examples of some time constructible function. Closure by composition.

- Non-deterministic Turing machines: characterization through the decidability of projection sets. Definition of the class of polynomial non-deterministic functions. NP-complete problems.

2) Lambda calculus and functional programming:

- Declarative programming: a historical outline on the lambda calculus, basic definitions, the terms of the lambda calculus, the simple substitution. Relations on the lambda terms. Congruences, transition to the context. α-equivalence. alpha-equivalence passes to the context. The transitive closure of a relationship, owned by Church-Rosser. Listing of lambda-terms concerning alpha-equivalence.

- Definition of beta-reduction and beta-equivalence. Church-Rosser's theorem for beta-reduction. Normal forms for beta-reduction. Beta-reduction strategies. Normalizing strategy: left reduction (left most-outer most). Head reduction. Soluble Terms. Head Normal Forms. Solvability characterization theorem.

- Representation of the recursive functions: lambda definability theorem. Existence of the fixed point for the lambda terms. Church Fixed Point and Curry fixed point.
- Representation of other data types in the lambda-calculus: pairs, lists, trees, the solution of recursive equations on lambda-terms ([2] chapters 1, 2, 5).

[1] DEHORNOY, P., COMPLEXITÈ ET DECIDABILITÈ. SPRINGER-VERLAG, (1993).
[2] KRIVINE, J.-L., LAMBDA CALCULUS: TYPES AND MODELS. ‪ELLIS HORWOOD, (1993).
[3] SIPSER,M., INTRODUCTION TO THE THEORY OF COMPUTATION.THOMSON COURSE TECHNOLOGY, (2006).

The crisis of classical physics. Waves and particles. State vectors and operators. Measurements, observables and uncertainty relation. The position operator. Translations and momentum. Time evolution and the Schrödinger equation. One-dimensional problems. Parity. Harmonic oscillator. Symmetries and conservation laws. Time independent perturbation theory. Time dependent perturbation theory.

Lecture notes available on the course website

J.J. Sakurai, Jim Napolitano - Meccanica Quantistica Moderna - Zanichelli
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics - Addison-Wesley

The purpose of this course is to give a brief introduction to the Finite Elements Method (FEM), a gold standard for the numerical solution of PDEs systems. Oddly enough, the widespread use of the FEM is not accompanied by an adequate knowledge of the mathematical framework underlying the method.
This course, starting from the weak formulation of balance equations, will give an overview of the techniques used to reduce a differential problem into an algebraic one. During the course, some selected problems in mechanics and physics will be solved, covering the three main types of equations: elliptic, parabolic and hyperbolic.
The course will cover the following topics: - Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
Moreover, the students will be introduced to the use COMSOL Multiphysics, a scientific software for numerical simulations based on the Finite Element Method.

1) Integral Form at a Glance,
note a cura del docente

2) When functions have no value(s): Delta functions and distributions
Steven G. Johnson, MIT course 18.303 notes, 2011

3) Understanding and Implementing the Finite Elements Method
Mark S. Gockenbach, SIAM, 2006 Cap. 1 Some model PDE’s
Cap. 2 The weak form of a BVP Cap. 3 The Galerkin method
Cap. 4 Piecewise polynomials and the finite element method (sections 4.1, 4.2) Cap. 5 Convergence of the finite element method (sections 5.1 ~ 5.4)

§I.Relativistic Field Theory

The Poincaré Group. Symmetries: global vs local. Noether's first and second theorems and conservation laws. The canonical stress-energy and angular momentum tensors. Improvements. Belinfante's argument and symmetric energy-momentum tensor. Local symmetries and conserved quantities.

§II.Gravity as a relativistic field theory

Particles and fields in Special Relativity. Irreps of the Poincaré group: Wigner's induced representation method. Massless particles: ISO(D-2) little group and gauge invariance. From relativistic massless spin-2 particles to full GR. Fierz-Pauli quadratic Lagrangian. Nöther method and non-linear completions. Nöther's construction of Yang-Mills Lagrangian. The transverse-traceless gravitational cubic vertex. Weinberg's Equivalence Principle from relativistic invariance of the S matrix. Spin and the sign of static forces.

§III.Elements of differential geometry

Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Levi-Civita connection. Torsion. Differential forms: definition, wedge product, interior and exterior derivatives, Hodge dual. Lie derivative of forms and Cartan's formula. Yang-Mills theory in the language of forms. Weyl tensor. Riemann and Weyl tensors in various dimensions: counting components for irreps of GL(D). Conformal transformations of the metric tensor. Conformally flat spaces. Conformally coupled scalar fields.

§IV. The Cartan-Weyl formulation of GR and Fermionic couplings

Local inertial frames. The frame field and its relation to the metric field. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Gravity as a gauge theory of the Poincaré algebra. Connection one-forms on the Poincar\'e algebra. Local Poincar\'e transformations. Torsion and curvature over the Poincar\'e algebra. First-order formulation and Cartan-Weyl's action. Relation between gauge transformations and diffeomorphisms. Spinors on curved manifolds. Minimally coupled Fermionic matter. Dirac Lagrangian.

§V. Maximally symmetric spaces

Homogeneous and isotropic spaces. Characterisation of maximally symmetric spaces: curvature constant and signature. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients.

§VI. The Schwarzschild black hole

Spherically symmetric spaces. The Schwarzschild solution. Birkhoff's theorem. Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. Free-fall towards the horizon. The tortoise coordinate. Extension of a space-time. Eddington-Finkelstein coordinates. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal's diagram and eternal black holes. (A)dS-Schwarzschild space-time.

§VII. More general black holes

Conformal diagrams. Event horizons. Reissner-Nordström and Kerr black holes. Black hole thermodynamics.

§VII. Gravitational energy

Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Einstein-Hilbert equations for asymptotically flat metrics. Candidate for gravitational energy-momentum tensor. The superpotential. ADM energy and momentum. Example: ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Generic background with Killing vectors. Quadrupole radiation.

§VIII. Asymptotic symmetries

General notion of asymptotic symmetry group. The example of Maxwell's theory in flat space. Covariant phase space formalism. Asymptotically flat spacetime and Bondi-van der Burg-Metzner-Sachs supertranslations. Applications: soft theorems and memory effects.

Note: some topics may be assigned as homework problems, as an alternative to the oral exam

-Carroll S, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Wald R, General Relativity (The University of Chicago Press, 1984)
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972)

The aim of the course is to provide the student with the general cognitive elements underlying the acquisition, control and monitoring systems of Nuclear and Subnuclear Physics experiments. The course is divided into the following topics:
-Introduction to DAQ
-Parallelism and Pipelining systems
-Derandomization
-DAQ and Trigger
-Data Transmission
-Front End Electronics
-Trigger
-Architecture Computing Systems
-Real Time Systems
-Real Time Operating Systems
-C Language
- VHDL Language
-TCP / IP Network Protocols
-DAQ Architecture
- Event Building
-VME Bus
-Run Control
-Farming
-Data Archiving

Lecture notes prepared by the teacher on the basis of the slides presented and available on the Moodle server:
https://matematicafisica.el.uniroma3.it

Parallel architectures: shared and distributed memory systems; GPGPU
Parallel programming Patterns: embarassingly parallel problems; work farms; partitioning; reduce; stencils
Performance evaluation of parallel programs: speedup, efficiency, scalability
Programming shared memory architectures with OpenMP
Programming distributed memory architectures with MPI
Programming GPUs with CUDA
Notes on innovative programming languages ​​for HPC (OpenACC, Rust + libraries, SIMD, OpenCL, ...)

Peter Pacheco, Matthew Malensek, An Introduction to Parallel Programming, 2nd ed., Morgan Kaufmann, 2021, ISBN 9780128046050

CUDA C++ programming guide

Course Slides by the Lecturer

The objective of Linguaggi di Programmazione course is to introduce main formal language theory concepts and results as well as their application for programming language classification.
Most relevant approaches for syntactic analysis of programming languages are introduced. Learning how to recognize the structure of a programming language and the implementation techniques for the abrstract machine.
Understanding the Object Oriented paradigm together with other non imperative approaches.

[1] Maurizio Gabbrielli, Simone Martini,Programming Languages - Principles and paradigms, 2/ed. McGraw-Hill, (2011).
[2] Dean Wampler, Alex Payne, Programming Scala: Scalability = Functional Programming + Objects, 2 edizione. O’Reilly Media, (2014).
[3] David Parsons, Foundational Java Key Elements and Practical Programming. Springer- Verlag, (2012).
Course Slides provided by the lecturer.

- Affine and projective curves
- Cubics and elliptic curves
- The group law and equations of elliptic curves
- Isogenies
- Torsion points
- Elliptic curves over finite fields and Hasse's Theorem
- Brief discussion of symmetric and public key cryptosystems
- Algorithms on Elliptic Curves: Double and Add and Schoof Algorithm
- Public Key and Digital Signature algorithms on Elliptic Curves
- Weil Pairing and Identity based elliptic cryptosystems

1. Machine learning. Types of learning. Loss functions. Empirical risk minimization. Generalization and overfitting.
2. Model optimization. Convex functions. Gradient descent. Stochastic gradient descent.
3. Regression. Linear regression. Basis functions. Feature selection. Regularization.
4. Classification. Generative models. Nearest neighbor. Logistic regression. Support vector machines. Neural networks.
5. Ensemble methods. Decision trees. Boosting. Bagging.
6. Unsupervised learning. K-means clustering. Hierarchical clustering. Principal component analysis.
7. Application of the methods using the Python language. Examples using the NumPy, Pandas, SciKit-Learn, and TensorFlow libraries.

J. Watt, R. Borhani, A. Katsaggelos. Machine Learning Refined. Cambridge University Press, 2nd edition, 2020.

Random variables and their distribution, moment generating function, mean variance and covariance.
Random sampling model and statistical model.
Statistics: concept, examples, sufficient statistics.
Point estimators: definition and desired properties, moments, maximum likelihood and Bayes.
Computational methods: Newton-Raphson, EM algorithm
Improving an estimator: Rao-Blackwell, UMVU estimator, full statistic, Lehman-Scheff ́e II and Cramer-Rao
Confidence intervals: intuitive, pivotal quantity, IC for Bayes and asymptotic IC.
Hypothesis testing: likelihood ratio, pivotal quantity test (Z and T test), duality with IC, UMP, Neyman-Pearson and Karlin-Rubin tests.
Non-parametric methods: goodness-of-fit, contingency table, Kolmogorov-Smirnov and ranking tests.
Analysis of variance (ANOVA) and F.
Regression: linear, multiple linear, generalized linear and Logistic / Poisson

Statistical Inference, Casella e Berger, 2nd Edition, Duxbury Advanced Series.

Additional reference:
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi

ntroduction to cryptography. Classic ciphers. Introduction to cryptanalysis.
Introduction to public-key cryptography.
The RSA cryptosystem.
Primality tests.
Factorization algorithms.
Some attacks on the RSA.
The discrete logarithm problem. Diffie-Hellman key exchange.
Elgamal cryptosystem.
Massey-Omura cryptosystem.
Digital signatures.
Overview of some cryptographic protocols.

Baldoni, Ciliberto, Piacentini: Aritmetica, crittografia e codici
D. Stinson: Cryptography - theory and practice

The high scool mathematics: programs, key and critical points; the mathematicians' mathematics and the mathematics for the citizen
From exercise to problem solving. Didactic theories and theorists. Guido and Emma Castelnuovo. Croce, Gentile and mathematics.
Didactic methodologies. Scholastic laws.

Battaglini Frank, Di Martino, Natalini, Rosolini "Didattica della matematica", Mondadori Università

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Calculus in Banach spaces, derivation, Implicit Function Theorem;
Bifurcation theorem, Ljapunov-Schmidt method;
Energy-minimizing solutions, coercivity, lower semi-continuity;
Saddle point solutions, Mountain Pass Theorem;
Topological degree, Linking Theorems;
Ljusternik-Schnirelmann Category, existence of infinitely many solutions.

A. Ambrosetti, A. Malchiodi - "Nonlinear Analysis and Semilinear Elliptic Problem" - Cambridge
P. Rabinowitz - "Minimax methods in critical point theory with application to differential equations" - American Mathematical Society

Lebesgue integration in Rn, Fourier transform, theory of differential equations

Luigi Chierchia, Analisi matematica due
Reed-Simon

Measure theory, outer measures, construction of Borel measures.
Integration theory, limit theorems, convergence in mean and in measure, integration on product spaces.
Radon measures, regularity, positive linear functionals, Riesz representation theorem.
Signed measures, decomposition theorems, differentiation, BV functions, fundamental theorem of calculus.
Lp spaces, basic properties, dual spaces, density theorems.
Introduction to geometric measure theory

G. Folland - "Real Analysis" - Wiley

Basic Linear Algebra:
Hilbert Spaces, Products and Tensor Products, Matrices, Complex Spaces and Inner Products, Matrices, Graphs, and Sums Over Paths.
Boolean Functions, Quantum Bits, and Feasibility:
Feasible Boolean Functions, Quantum Representation of Boolean Arguments Quantum Feasibility.
Special Matrices:
Hadamard Matrices, Fourier Matrices, Reversible Computation and Permutation Matrices, Feasible Diagonal Matrices, Reflections. Tricks: Start Vectors, Controlling and Copying Base States, The Copy-Uncompute Trick, Superposition Tricks, Flipping a Switch, Measurement Tricks, Partial Transforms.
Algorithms:
Phil’s Algorithm: Phil Measures Up, Quantum Mazes versus Circuits versus Matrices.
Deutsch’s Algorithm: Superdense Coding and Teleportation.
The Deutsch-Jozsa Algorithm. Simon’s Algorithm. Shor’s Algorithm, Quantum Part of the Algorithm, Analysis of the Quantum Part, Continued Fractions.
FactoringIntegers: Basic Number Theory, Periods Give the Order, Factoring.
Grover’s Algorithm: The binary case, the general case, with k Unknowns, Grover Approximate Counting.

Richard J. Lipton, Kenneth W. Regan Introduction to Quantum Algorithms via Linear Algebra, Second Edition, ISBN 9780262045254, (2021), MIT Press

We will treat some aspects of the relation between Riemannian geometry and topology of manifolds. In particular, the aim of this course is to prove Gauss-Bonnet theorem for surfaces and Hopf-Rinow theorem which holds in any dimension. Both results will be proved using geometric properties of geodesics. These are the curves which, at least locally, minimize the distance on a Riemannian manifold. Time permitting, we will give an introduction to abstract Riemannian geometry in arbitrary dimension.

Differential Geometry of Curves & Surfaces, by Manfredo Do Carmo. Second edition.
Curves and Surfaces, by Marco Abate and Francesca Tovena.

Part I. Elements of ergodic theory.
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.

Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.

Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.

G. Gallavotti, F. Bonetto, G. Gentile
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.

I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.

[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159

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

[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.

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

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

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

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

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

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

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

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 semi-Eulerian 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 4-color 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 Max-Flow Min-Cut 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 Riemann-Roch 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.

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.

The main theorems of Functional Analysis.

H. Brezis, Functional Analysis.

Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens

Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546

Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.

“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

Definition and basic properties of the Sobolev spaces W^{1,p} (Ω). Extension operators. Sobolev inequalities. The space W^{1,p}_0 (Ω). Variational formulation of some elliptic boundary value problems. Existence of weak solutions. Regularity of weak solutions.

"Analisi funzionale", H. Brézis, Liguori Editore
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

Ring of integers in number fields.
Unique factorisation of ideals in the rings of integers.
Class group.
Group of units.
Fermat's last Theorem for regular primes
Local fields.

Notes by the lecturer.
Marcus, D. Number fields, 3rd Ed Springer-Verlag. 1977.
Samuel, P. Théorie algébrique des nombres, Hermann, Paris. 1971.
Schoof, R. Algebraic Number Theory, dispense Università di Roma Tor Vergata, 2003.
Milne, J. Algebraic Number Theory, Lecture Notes, 2017.

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– The goals of statistical mechanics
– Review of thermodynamics. Convex functions and Legendre transform
– Models of statistical mechanics: canonical ensemble, grand canonical and Gibbs states.
– The Ising model and the lattice gas models. Existence of the thermodynamic limit for the free energy of the Ising or lattice gas model.
– The general structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs state.

THE ISING MODEL
– Review of known results on the Ising model in one or more dimensions.
– The solution of the one-dimensional Ising model via the transfer matrix method.
– The mean field Ising model: exact solution. Phase transition and loss of equivalence between statistical ensembles
– Ising with long-range interactions (Kac potentials) in the mean-field limit. The Maxwell construction.
– FKG and Griffiths inequalities. Existence of the infinite volume correlation functions of states with + and − conditions in the ferromagnetic Ising model.
- The geometric representation of the Ising model: high and low temperature contours.
- Existence of a phase transition in the low temperature Ising model: the Peierls's argument.
– Absence of a phase transition at high temperature and exponential decay of boundary effects boundary conditions.
– Lee-Yang theorem and analyticity of the pressure at non-zero magnetic field.
– Existence of a phase transition in the one dimensional Ising model with power law interaction |x − y|^{−p}, 1

S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017.

G. Gallavotti: Statistical Mechanics. A short treatise, ed. Springer-Verlag, 1999.

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) 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)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf on the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf on the course page

Lecture slides in pdf on the course page

Additional notes provided by the teacher

STOCHASTIC PROCESSES, BROWNIAN MOTION, STOCHASTIC INTEGRALS, STOCHASTIC DIFFERENTIAL EQUATIONS. ITO FORMULA. FEYNMANN-KAC FORMULAS AND APPLICATIONS. MARKOV TIMES AND PROBABILISTIC SOLUTION OF THE DIRICHLET PROBLEM.

P. Morters, Y. Peres: Bronian Motion, Cambridge 2010
T. Liggett, Continuous time Markov processes: an introduction, AMS 2010
L.C. Evans:Introduction to stochastic differential equations, AMS 2014,
J.F. Le Gall: Brownian motion, martingales, and stochastic calculus, Springer 2016

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.

Main reference:
D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Additional reference (in Italian)
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi

1. Optimization and combinatorial optimization problems. Enumeration of solutions.
2. Basics of algorithm analysis. Computational tractability. Asymptotic order of growth.
3. Graphs. Graph connectivity and graph traversal. Graph bipartiteness. Connectivity in directed graphs. Directed acyclic graphs and topological ordering.
4. Greedy algorithms. Interval scheduling. Optimal caching. Shortest paths in a graph. Minimum spanning trees.
5. Divide and conquer. Mergesort. Counting inversions. Closest pair of points.
6. Dynamic programming. Weighted interval scheduling. Principles of dynamic programming. Subset sums and knapsacks. All-pairs shortest paths. Shortest paths and distance vector protocols.
7. Network flow. Maximum flow and the Ford-Fulkerson algorithm. Maximum flows and minimum cuts in a network. Augmenting paths. Bipartite matching. Disjoint paths in directed and undirected graphs.
8. Computational intractability. Polynomial-time reductions. Reductions via "gadgets". Efficient certification and the definition of NP. NP-complete problems. Covering, packing, partitioning, sequencing, and numerical problems. Other examples.

Jon Kleinberg, Eva Tardos. Algorithm Design. Pearson Education, 2013.

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

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

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

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

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

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

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

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

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

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

F Cesarone (2020), Computational Finance. MATLAB oriented modeling, Routledge-Giappichelli Studies in Business and Management, ISBN 978-0-367-49303-5
https://www.giappichelli.it/computational-finance

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– The goals of statistical mechanics
– Review of thermodynamics. Convex functions and Legendre transform
– Models of statistical mechanics: canonical ensemble, grand canonical and Gibbs states.
– The Ising model and the lattice gas models. Existence of the thermodynamic limit for the free energy of the Ising or lattice gas model.
– The general structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs state.

THE ISING MODEL
– Review of known results on the Ising model in one or more dimensions.
– The solution of the one-dimensional Ising model via the transfer matrix method.
– The mean field Ising model: exact solution. Phase transition and loss of equivalence between statistical ensembles
– Ising with long-range interactions (Kac potentials) in the mean-field limit. The Maxwell construction.
– FKG and Griffiths inequalities. Existence of the infinite volume correlation functions of states with + and − conditions in the ferromagnetic Ising model.
- The geometric representation of the Ising model: high and low temperature contours.
- Existence of a phase transition in the low temperature Ising model: the Peierls's argument.
– Absence of a phase transition at high temperature and exponential decay of boundary effects boundary conditions.
– Lee-Yang theorem and analyticity of the pressure at non-zero magnetic field.
– Existence of a phase transition in the one dimensional Ising model with power law interaction |x − y|^{−p}, 1

S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, 2017.

G. Gallavotti: Statistical Mechanics. A short treatise, ed. Springer-Verlag, 1999.

Ordinary Differential Equations
Finite difference approximation for ordinary differential equations: Euler's method. Consistency, stability, absolute stability. Second order Runge-Kutta methods.
Single step implicit methods: backward Euler and Crank-Nicolson methods. Convergence of single step methods. Multi-step methods: general structure, complexity, absolute stability. Stability and consistency of multi-step methods. Adams methods, BDF methods, Predictor-Corrector methods. (Reference: Chapter 7 of curse notes "Appunti del corso di Analisi Numerica")

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) 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)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf on the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", in pdf on the course page

Lecture slides in pdf on the course page

Additional notes provided by the teacher

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 non-Euclidean 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.

List of problems given in class concerning visualization and solutions with the help of software Mathematica or GeoGebra.

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; Substitution-Permutation Networks (SPN); Linear cryptanalysis for SPN: Piling-Up Lemma, linear approximation of S-boxes, linear attacks on S-boxes; 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 pre-image, resistance to the second pre-image, collision resistance. The random oracle model: ideal hash functions, properties of independence. Randomized algorithms, collision on the problem of the second pre-image, collision on the problem of the pre-image. Iterated hash functions; the construction of Merkle-Damgard. Safe Hash Algorithm (SHA-1). Authentication Codes (MAC): nested authentication codes (HMAC).

[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.

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 semi-Eulerian 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 4-color 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 Max-Flow Min-Cut 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 Riemann-Roch 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.

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.

Presentation of the problems that can be treated through integrals on large number of dimensions

Basics

Probability and Random variables

Measurement, uncertainty and its propagation

Curve-fitting, least-squares, optimization

Classical numerical integration, speed of convergence

Integration MC (Mean, variance)

Sampling Strategies

Applications

Propagation of uncertainties

Generation according to a distribution

Real World Applications

Cosmic Rays Shower

System Availabilty

Further applications

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

Taylor, J. - Introduzione all'analisi degli errori : lo studio delle incertezze nelle misure fisiche - Zanichelli Disponibile nella biblioteca Scientifica di Roma Tre

Dubi, A. - Monte Carlo applications in systems engineering - Wiley Disponibile nella biblioteca Scientifica di Roma Tre

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. Quasi-coherent 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 quasi-coherent and coherent sheaves on a scheme.

Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on the projective space. Eulero-Poincaré characteristic.

Invertible sheaves and linear systems

Glueing of sheaves. Invertible sheaves and their description. The Picard group.
Morphisms in a projective space. Linear systems.

Notes from Prof. Lopez, Prof. Sernesi
R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).

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)
Maxwell-Boltzmann 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 fluctuation-dissipation theorem. (1, Par. 8.4).
Einstein and Langevin theories of the Brownian motion. (Par. 1 par. 11.1,11.2).
Johnson-Nyqvist theory of thermal noise. (1 Par. 11.3).
Quantum statistical mechanics and the density matrix. (1, Par. 6.2,6.3,6.4)
Fermi-Dirac and Bose-einstein 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 Bose-Einstein condensation. (1, Par. 7.3)
Quantum theory of black-body radiation. (1, Par. 7.5)

e-Learning Moodle Platform with Supplementary Material

Suggested textbook:
1) C. Di Castro and R. Raimondi, Statistical Mechanics and Applications in Condensed Matter, Cambridge University Press, 2015.

Topics Part A

• Coordinates and Telescopes
• Elements of Spectroscopy
• Stars and Stellar Evolution • Galaxies
• Active Galactic Nuclei


Program Part A

• Overview

• Celestial coordinates (1.3)

• Telescopes and resolving power (6.1)

• Parallax distance (3.1)

• Flux, brightness, apparent and absolute magnitudes, colors (3.2, 3.3, 3.6)

• The black body (3.4, 3.5)

• Hertzsprung-Russel diagram (8.2)

• Open and globular clusters: position, stellar populations and HR diagram (13.3)

• White dwarfs, Novae and SuperNovae (notes and partly in 15 and 16)

• The classification of galaxies (24.1)

• The rotation curve of galaxies and dark matter (25.3)

• The center of the Galaxy and its Black Hole (25.4)

• Hubble's law and expansion of the Universe (27.2)

• Probability of collision between stars and galaxies (handouts)

• Black Holes: outline of General Relativity (outline 17)

• Active Galactic Nuclei (28.1, 28.2, 28.3)


Topics Part B

• Structure and stellar evolution
• Elements of Spectroscopy
• Distances and expansion of the Universe • Galaxies
• GRB and gravitational waves

Program Part B

• Acretion disks and X-ray emission in Active Galactic Nuclei (28.2)

• Stars of Neutrons and Pulsars (16.6, 16.7)

• Gamma Ray Bursts (handouts)

• Gravitational Waves (lecture notes)

• Spectroscopy: eq. Boltzmann-excitation and Saha-ionization (8.1)

• Spectroscopy: speed, temperature and density measurements (handouts)

• Eq. of star structure, time and Kelvin-Helmholtz instability (11.1-4)

• Nuclear reactions of hydrogen (11.3)

• Jeans mass of gravitational collapse, free-fall time and Initial Mass Function (12.2, 12.3)

• The Milky Way and the local group (25.1, 25.2)

• Metallicity (25.2)

• Transit of Venus and measurement of the Earth-Sun distance (handouts)

• Distance scale (27.1)

• Hubble's law and expansion of the Universe (27.2)

• Local Group, Clusters of Galaxies, large scale structure of the Universe (27.3)

• The Big Bang and the background radiation (29.2 brief notes and lecture notes)

A copy of the lecture notes can be downloaded from the course website.

In brackets, the paragraphs from “An Introduction to Modern Astrophysics, II ed. - B.W. Carrol, D.A. Ostlie - Ed. Pearson, Addison Wesley ”(copies available in the library). The discussion in the course has been simplified compared to what is reported in the text. Alternative text in Italian: Attilio Ferrari, Stars, Galaxies, Universe - Fundamentals of Astrophysics - Ed. Springer

1. The themes of logic
2. Classical logic: propositions, proofs
3. Classical logic: connectives
4. Classical logic: types, variables, quantifiers
5. First-order Classical logic
6. Classes and sets
7. Codes, digits, Boolean algebra
8. Turing Machine
9. Axiomatization and formalization of first-order classical logic
10. Logic and other disciplines

V. Michele Abrusci, LOGICA - Lezioni di primo livello, Quarta edizione, Wolters Kluwer, 2018

Division, Factorization, Some Elementary Properties of Primes, Some Results and Problems Concerning Primes.
ARITHMETIC FUNCTIONS: The Divisor Function. The Moebius Function. The Euler Function. Dirichlet Convolution
CONGRUENCES: Sets of Residues,Some Interesting Congruences, Some Linear Congruences, Some Polynomial Congruences, Primitive Roots, the Theorem of Gauss.
QUADRATIC RESIDUES: The Legendre Symbol. Quadratic Reciprocity. The Jacobi Symbol.The Distribution of Quadratic Residues.
SUMS OF INTEGER SQUARES: Sums of Two Squares. Number of Representations. Sums of Four Squares. Sums of Three Squares.
ELEMENTARY PRIME NUMBER THEORY: Euclid's Theorem Revisited. The Von Mangoldt Function. Tchebycheff's Theorem. Some Results of Mertens

Chen, W; ELEMENTARY NUMBER THEORY. https://rutherglen.science.mq.edu.au/wchen/lnentfolder/lnent.html
Chowdhury, F.; Chowdhury, M. R. Essentials of Number Theory. Pi Publications, Dhaka, Bangladesh, 2005. ISBN 984-32-2836-7
Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers. Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp. ISBN: 0-19-853170-2; 0-19-853171-0
Davenport, H. Aritmetica superiore. Un'introduzione alla teoria dei numeri. Editore: Zanichelli, 1994. 199 pp. ISBN: 8808091546

1. Optimization and combinatorial optimization problems. Enumeration of solutions.
2. Basics of algorithm analysis. Computational tractability. Asymptotic order of growth.
3. Graphs. Graph connectivity and graph traversal. Graph bipartiteness. Connectivity in directed graphs. Directed acyclic graphs and topological ordering.
4. Greedy algorithms. Interval scheduling. Optimal caching. Shortest paths in a graph. Minimum spanning trees.
5. Divide and conquer. Mergesort. Counting inversions. Closest pair of points.
6. Dynamic programming. Weighted interval scheduling. Principles of dynamic programming. Subset sums and knapsacks. All-pairs shortest paths. Shortest paths and distance vector protocols.
7. Network flow. Maximum flow and the Ford-Fulkerson algorithm. Maximum flows and minimum cuts in a network. Augmenting paths. Bipartite matching. Disjoint paths in directed and undirected graphs.
8. Computational intractability. Polynomial-time reductions. Reductions via "gadgets". Efficient certification and the definition of NP. NP-complete problems. Covering, packing, partitioning, sequencing, and numerical problems. Other examples.

Jon Kleinberg, Eva Tardos. Algorithm Design. Pearson Education, 2013.

The main theorems of Functional Analysis.

H. Brezis, Functional Analysis.

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.

Main reference:
D. Levine, Y. Peres, E. Wilmer, Markov chains and mixing times.. AMS bookstore, (2009).

Additional reference (in Italian)
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi

Preliminaries: definition of hyper-surface, integration on hyper-surfaces, the divergence theorem; the Laplace equation: the mean value inequalities, the minimum and maximum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence's theorems, interior estimates on the derivatives, the Perron method for the Dirichlet problem.

“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger, Classics in Mathematics, Springer-Verlag
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

Definition and basic properties of the Sobolev spaces W^{1,p} (Ω). Extension operators. Sobolev inequalities. The space W^{1,p}_0 (Ω). Variational formulation of some elliptic boundary value problems. Existence of weak solutions. Regularity of weak solutions.

"Analisi funzionale", H. Brézis, Liguori Editore
"Partial differential equations. Second edition", Lawrence C. Evans, Graduate Studies in Mathematics 19, American Mathematical Society

• Python:https://github.com/steguar/DAIL/blob/main/Lecture_1/Lecture_1_Python_crash_ course.ipynb
• Mathematical Biology, J.D. Murray (Population models - Epidemic models)
• Population Ecologies: First Principles Deborah Goldberg and John H.Vandermeer
(Simple population models, CAP1)
• Non Linear Dynamics and Chaos, S. Strogatz (Stability of dynamical systems)
• Computational Physics, M. Newman (Euler and RK methods)
• Networks, M. Newman (ER and CM random graphs, Epidemics on Networks)
• Understanding Bioinformatics, Marketa Zvelebil & Jeremy O. Baum
• Biological sequence analysis, R. Durbin et al. (CAP 1,2,6,7)
• Bioinformatics Algorithms: an Active Learning Approach, Pavel A. Pevzner and Phillip Compeau
• Bioinformatics - an Introduction, Jeremy Ramsden
• Understanding Bioinformatics, Marketa Zvelebil, Jeremy O. Baum
• Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids
• Statistical Methods in Bioinformatics, An Introduction, Warren J. Ewens , Gregory Grant

STOCHASTIC PROCESSES, BROWNIAN MOTION, STOCHASTIC INTEGRALS, STOCHASTIC DIFFERENTIAL EQUATIONS. ITO FORMULA. FEYNMANN-KAC FORMULAS AND APPLICATIONS. MARKOV TIMES AND PROBABILISTIC SOLUTION OF THE DIRICHLET PROBLEM.

P. Morters, Y. Peres: Bronian Motion, Cambridge 2010
T. Liggett, Continuous time Markov processes: an introduction, AMS 2010
L.C. Evans:Introduction to stochastic differential equations, AMS 2014,
J.F. Le Gall: Brownian motion, martingales, and stochastic calculus, Springer 2016

Ring of integers in number fields.
Unique factorisation of ideals in the rings of integers.
Class group.
Group of units.
Fermat's last Theorem for regular primes
Local fields.

Notes by the lecturer.
Marcus, D. Number fields, 3rd Ed Springer-Verlag. 1977.
Samuel, P. Théorie algébrique des nombres, Hermann, Paris. 1971.
Schoof, R. Algebraic Number Theory, dispense Università di Roma Tor Vergata, 2003.
Milne, J. Algebraic Number Theory, Lecture Notes, 2017.

Highlights of Linear Algebra:
Matrix-matrix multiplication; column & row space; rank
The four fundamental subspaces of linear algebra
Fundamentals of Matrix factorizations:
A=LU rows & columns point of view
A=LU elimination & factorization; permutations
A=RU=VU; Orthogonal matrices
Eigensystems and Linear ODE
Intro to PSym; the energy function
Gradient and Hessian
Singular Value Decomposition
Eckart-Young; derivative of a matrix norm
Principal Component Analysis
Generalized evectors;
Norms
Least Squares
Convexity & Newton’s method
Newton & L-M method; Recap of non-linear regression
Lagrange multipliers

Machine Learning:
Gradient Descend; exact line search; GD in action; GD with Matlab
Learning & Loss; Intro to Deep Neural Network; DNN with Matlab
Loss functions: Quadratic VS Cross entropy
Stocastics Gradient Descend (SGD) & Kaczmarcz; SGD convergence rates & ADAM
Matlab interface for DNN
Construction of DNN: the key steps
Backpropagation and the Chain Rule
Machine Learning examples with Wolfram Mathematica
Convolutional NN + Mathematica examples of 1D convolution
Convolution and 2D filters + Mathematica examples of 2D convolution
Matlab Live Script, Network Designer, Pretrained Net

Symplectic geometry and Hamiltonian formalism. Other topics, depending on the interests of the audience

[Z] Zehnder - "Lectures on dynamical systems".

1. Atomic theory and structure of the atom. Atoms, molecules, moles; atomic weight and molecular weight. Rutherford model, Bohr model, quantum theory, quantum numbers and energy levels; polyelectronic atoms, periodic system.

2. Chemical bond. Ionic bond. Covalent bond: σ bond and π bond. Polyatomic molecules. Molecular structure. Hybridization and resonance. Molecular orbital. Metallic bond. Intermolecular forces.

3. Nomenclature. Oxides, hydroxides, acids, salts, ions.

4. Chemical reactions. Balancing chemical reactions.

5. States of aggregation. Gaseous state and gas laws.

6. Thermodynamics. Matter, energy, heat. First and second principle. Enthalpy, entropy, free energy.

7. Solid state: ionic, molecular, metallic, covalent solids. Conductors, semiconductors, insulators.

8. Liquid and amorphous. State changes and state diagrams.

9. Solutions. Concentration of solutions. Colligative properties. Electrolyte solutions.

10. Chemical kinetics. Speed ​​of chemical reactions. Speed ​​constant. Influence of temperature on velocity: Arrhenius equation. Catalysts.

11. Chemical equilibrium. Equilibrium constant and speed constants. Constant of equilibrium and free energy. Equilibria in the gas and heterogeneous phase. Le Chatelier's principle. Van’t Hoff equation.

12. Equilibrium in solution. Acid-base equilibria: Acids and bases, pH, dissociation constants, polyprotic acids, hydrolysis, buffers; acid-base titrations, indicators.

13. Precipitation equilibria: solubility and solubility product, common ion effect.

14. Electrochemistry. Batteries, electrode potentials, Nernst equation.

15. Laboratory. Acid-base titrations and pH measurements.

Numerical exercises will be carried out on the topics covered during the lessons.
There will be two laboratory exercises that will take place at CeDiC (Via della Vasca Navale 79).

Shriver & Atkins' Inorganic Chemistry 7th Edition
Periodica Table
Slides online

Introduction to set theory: aggregates and sets, necessity of a theory, ordinals and cardinals, antinomies and paradoxes, main characteristics of axiomatic set theory. Zermelo’s axiomatic set theory and Zermelo-Fraenkel’s axiomatic set theory: preliminaries and conventions, Zermelo’s axioms, the replacement axiom and Zermelo-Fraenkel’s theory, extensions of the language by definition. Ordinals: orders, well-orders and well-foundedness, well-foundedness and induction principle, the ordinal numbers, well-orders and ordinals, ordinal induction (proofs and definitions), diagonal argument and limit ordinals, infinity axiom and ordinal arithmetic, hints on the use of ordinals in proof-theory. Axiom of choice: equivalent formulations (and proof of the equivalence), infinite sets and axiom of choice. Cardinals: equipotent sets and infinite sets, the cardinal numbers, cardinal arithmetic.

V. Michele Abrusci e Lorenzo Tortora de Falco, Logica. Vol. 2 Incompletezza, teoria assiomatica degli insiemi, Springer, 2018

Evolution equations of Mathematical Physics: transport, wave and heat equations. Introduction to quantum mechanics. Fourier transform

[B] P. Butta', Note del corso di Fisica Matematica
[Cr] W. Craig, A course on Partial Differential Equations
[L1] V. Lubicz, Apppunti di Meccanica Quantistica

Linear Systems
Direst methods: Gaussian elimination. Pivoting strategies. Gaussian elimination as a factorization. Doolittle and Choleshy factorizations. Iterative methods: Jacobi, Gauss-Seidel, SOR, Richardson, and related convergence results. Comparison of direct vs iterative solvers. Stability of algorithms for the solution of linear systems.

Iterative Methods for Scalar Nonlinear Equations
The intermediate zero theorem. The algorithms of bisection, Newton, secants, chords, and related convergence results. (Reference: Chapter 1 excluding Section 1.2.3, and Appendices A.1, A.2)

Approximation of Functions
General approximation strategies. Interpolating polynomial in Lagrange and Newton form. Representation of the interpolation error. Convergence of the interpolating polynomial for analytic functions. Refinement strategies in interpolation: Chebyshev nodes, composite approximations. Error estimates. Hermite polynomial, construction and representation of the error. Least Squares approximations. (Reference: Chapter 5 excluding Section 5.2, and Appendix A.4)

Numerical Integration
General principles of numerical integration. Polya's Theorem on the convergence of interpolatory quadrature formulae. Closed and open Newton-Cotes formulae. Stability results and error estimation. Generalized Newton-Cotes formulae and their convergence. Gaussian quadratures and their convergence. (Reference: Chapter 6)

Laboratory Activity
C language coding of some of the major algorithms, and in particular: Gaussian elimination, iterative methods for linear systems and scalar equations, Lagrange/Newton interpolation with a refinement strategy.

N.B.: References are provided with respect to the course notes.

Roberto Ferretti, "Appunti del corso di Analisi Numerica", available from the course page

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available from the course page

Slides of the lessons, available from the course page

Random variables and their distribution, moment generating function, mean variance and covariance.
Random sampling model and statistical model.
Statistics: concept, examples, sufficient statistics.
Point estimators: definition and desired properties, moments, maximum likelihood and Bayes.
Computational methods: Newton-Raphson, EM algorithm
Improving an estimator: Rao-Blackwell, UMVU estimator, full statistic, Lehman-Scheff ́e II and Cramer-Rao
Confidence intervals: intuitive, pivotal quantity, IC for Bayes and asymptotic IC.
Hypothesis testing: likelihood ratio, pivotal quantity test (Z and T test), duality with IC, UMP, Neyman-Pearson and Karlin-Rubin tests.
Non-parametric methods: goodness-of-fit, contingency table, Kolmogorov-Smirnov and ranking tests.
Analysis of variance (ANOVA) and F.
Regression: linear, multiple linear, generalized linear and Logistic / Poisson

Statistical Inference, Casella e Berger, 2nd Edition, Duxbury Advanced Series.

Additional reference:
Luca Leuzzi, Enzo Marinari, Giorgio Parisi
CALCOLO DELLE PROBABILITÀ: un trattatello per principianti volenterosi

This course is based on the use of case studies, intersting examples of science communication that will be presented and analysed during the lessons.

On the examples of these case studies, communication laboratories and practical activities will be organized. Students will work in team, guided by researchers and professional communicators, to plan and produce specific communication tools (articles, websites, blogs, audio/video etc).

The course will also take in account the technological aspects related to communication, introducing and examining selected open source software.

The program

The course is 52 hours long, including 40 ore of lessons and 12 hours of lab activities. 12 hours are in common with the “Communcating Science” PhD course.

Introduction to science communication
• The postulates of communication: from body language to the communication plan
• About science communication: why should we communicate science?
• Different types of communication, including in the academic & research world
• Planning an event for the public: the 5 steps strategy
• Visual communication and science

Speaking to the public about science
• Introduction to verbal communication: from public talks to press conferences
• The basics of public speaking in science
• Slides, audio/video and multimedia tools

Writing about science
• Introducing science journalism
• Differences between a scientific article, a press release and outreach articles
• Writing for video: the storyboard

Visual communication of science
• How to communicate science with images
• How to plan and produce an image

Communicating science on web
• How is science communicated on the web
• Science and web 2.0
• How to plan and produce a website

Organization of a public event
• The communication plan of a public event
• Organizing an astronomical observation event

"The hands-on guide for science communicators: a step.by-step approach to public outreach" di Lars Lindberg Christensen
https://play.google.com/store/books/details?id=GI_fpb4xFX4C&rdid=book-GI_fpb4xFX4C&rdot=1&source=gbs_vpt_read&pcampaignid=books_booksearch_viewport

Relations and functions
The natural numbers
The ring of integers and polynomials
Residue classes
Groups

G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)

I. N. Herstein, Algebra, Editori Riuniti, (2003)

Relations and functions
The natural numbers
The ring of integers and polynomials
Residue classes
Groups