Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

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

G. Folland - "Real Analysis" - Wiley

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.

[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, Curves and Surfaces. Springer, (2006).

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

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).

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

W. Rudin, Real and complex Analysis, McGraw-Hill.

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

1. Theory of distributions

Definition and convergence of distributions.
Derivative of a distribution.
Spaces of distributions.
Convolutions.
Schwartz space, tempered distributions.
Fourier transform of tempered distributions.
Sobolev spaces and pseudo-differential operators.

2. Application to Schrödinger equations

Linear Schrödinger equation, local and global smoothing effects.
Nonlinear Schrödinger equation, local theory in L^2, H^1, H^2.
Asymptotic results and singularity formation for NLS.

Bony - Cours d'analyse, Théorie des distributions et analyse de Fourier
Ponce, Linares - Introduction to nonlinear dispersive equations

Fundamental grupo and covering spaces.
Cathegories.
Anstract and geometric simplicial complexes.
Singular and simplicial homology.
Cohomology.
Duality theorems.
Data analysis.

Allen Hatcher: Algebraic topology Cambridge University press.
Vidit Nanda: Computational Algebraic Topology - Lecture notes
Lucia Caporaso : Lezioni di topologia

See the program in italian.

R. Miranda: Algebraic Curves and Riemann Surfaces

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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

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.

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.
QuantumWalks:
Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.

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

Brownian motion: definition and property of BM, continuity and non-differentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multi-dimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.

Stochastic integration: Paley-Wiener-Zygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.

Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf

An introduction to Stochastic Differential Equations (Evans)

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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)

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

- Introduction to parallel and distributed computing
- Base concepts: hardware architectures and memory hierarchy
- The C language
- Parallel programming models
- Message Passing Interface (MPI)
- OpenMP
- Introduction to general purpose programming on Graphics Processing Unit (GPU)
- CUDA and OpenCL

Introduction to Parallel Computing: From Algorithms to Programming on State-of-the-Art Platforms, Trobec, Slivnik, Bulić, Robič, Springer

Programming on Parallel Machines, Norm Matloff

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

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

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.

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

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.

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.
QuantumWalks:
Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.

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

Brownian motion: definition and property of BM, continuity and non-differentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multi-dimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.

Stochastic integration: Paley-Wiener-Zygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.

Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf

An introduction to Stochastic Differential Equations (Evans)

1. Modules

Modules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
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. Nakayama's Lemma.

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

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


4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings

5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.

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.

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

G. Strang,
Linear Algebra and Learning from Data,
Wellesley-Cambridge Press

M. Nielsen,
Neural Networks and Deep Learning (free online book)
http://neuralnetworksanddeeplearning.com

Various authors,
Distill, dedicated to clear explanations of machine learning
https://distill.pub

Sequent Calculus Proofs
***************************************
Natural Deduction

Sequent Calculus for Classical Logic (LK) and Intuitionistic Logic (LJ)

Cut elimination for LK and LJ sequent proofs

Sequent calculus for Linear Logic(LL)

Cut Elimination Theorem for LL

Focusing Theorem for LL proofs

Proof Nets
*************************************
(proof-structures, correctness, normalization, adequacy, sequentialization, focusing, complexity)

Pure Multiplicative Proof Nets

Multiplicative and Additive Proof Nets

Multiplicative and Exponenial Proof Nets

Denotational Semantics

NOTES AND SLIDES AVAILABLE ON THE COURSE WEB PAGE
https://sites.google.com/view/lm510/

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
Gioia, A. A. The theory of numbers. An introduction. Reprint of the 1970 original. Dover Publications, Inc., Mineola, NY, 2001. xii+207 pp. ISBN: 0-486-41449-3
Rosen, K. H. Elementary number theory and its applications. Fourth edition. Addison-Wesley, Reading, MA, 2000. xviii+638 pp. ISBN: 0-201-87073-8
Tattersall, J. J. Elementary number theory in nine chapters. Cambridge University Press, Cambridge, 1999. viii+407 pp. ISBN: 0-521-58531-7

The Hahn-Banach theorem; the open mapping theorem, the closed graph theorem and the Banach-Steinhaus theorem. Weak topologies, Banach-Alaoglu theorem and reflexivity. Linear continuous operators and the spectral theorem for compact, self-adjoint operators. Sobolev spaces and elliptic problems in one dimension. The spectral theorem for unitary operators.

H. Brezis, Analisi funzionale.

W. Rudin, Functional Analysis.

Part 1: Introduction to Lebesgue's theory in R^n
Definition of L^1 functions.
Theorems on the integration of limits (monotone convergence, dominated convergence, Fatou's lemma).
Completeness of L^1 (Riesz-Fischer Theorem).
Iterated integrals and Fubini's theorem.
Measurable functions and Lebesgue measure.
Convolution and regularization.
Theorem of the change of variables in R^n.
Divergence theorem in R^n.

Part 2: Fourier in L^2
The Hilbert space L^2 (on bounded domains and on R^n).
Fourier series and transforms in L^2.

Part 3: Fundamentals of the theory of ordinary differential equations
Examples and classes of ordinary differential equations.
Local existence and uniqueness theorem (Picard-Lindelof).
Lipschitz dependence on initial data.
Maximum and global solutions; globality criteria.
Linear systems (linear structure, Wronskian); non-homogeneous systems (variations of constants; Liouville's theorem.
Linear systems with constant coefficients (exponential solution).
Jordan canonical form and qualitative analysis of solutions).
Flows. Variational equation. Parameter dependence C^k.
Introduction to qualitative analysis.
Phase space.
Use of Fourier theory in differential equations (outline).

During the lessons, typed notes will be provided.

Review of elementary group theory. Semidirect products. Permutation groups and simplicity. Group actions on a subset. Sylow theorems. Abelian finitely generated groups. Free, nilpotents and solvable groups. Group representations.

'Groups' by A. Machì, Springer

Introduction to algebraic number theory:
Rings of integers in number fields and unique factorisation of ideals.
Absolute values in number fields.

The Weil Height and the Mahler measure:
Definitions and properties.
The product Formula
Northcott’s Theorem.
Kroneker’s Theorem.

Thue equations:
Thue’s Theorem on diophantine approximation.
Siegel’s Lemma.
Thue equations have a finite number of integer solutions.

Arithmetic dynamics:
(Pre)periodic points.
The canonical height.
Rational functions.

Diophantine equations in roots of unity:
Revision about roots of unity and cyclotomic polynomials.
The Theorem of Ihare-Serre-Tate.

Equidistribution:
Definitions and examples.
Bilu’s Theorem.
Bogomolov’s Conjecture.

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.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.

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

1. Random walks and Markov Chains. Sequences of random variables. Random walks. Discrete-time Markov chains. Invariant measure, time-reversal, and reversibility.

2. Examples and classical models. Random walks on graphs. Birth and death processes. Exclusion processes. Monte Carlo method: Metropolis algorithms and Glauber dynamics for the Ising model, graph coloring, and other interacting systems.

3. Convergence to equilibrium I. Variation distance, mixing times. Ergodic theorems. Coupling techniques. Strong stationary times. Applications to the coupon collector problem and shuffling a deck of cards.

4. Convergence to equilibrium II. L2-norm convergence. Spectral gap and relaxation time estimates. Cheeger's inequality.

5. Continuous-time Markov chains. Poisson process. Q-matrices and continuous-time transition probabilities. Infinitesimal generator of a continuous-time Markov chain. Explicit construction and graphical representation.

6. Conductance and path method. Comparison method. Spectral gap for the Bernoulli-Laplace process and the exclusion process on d-dimensional torus. Convergence to equilibrium in terms of entropy and logarithmic Sobolev inequalities.

7. Other selected topics. Glauber dynamics for the Ising model: dynamic phase transition for the mean-field model and the lattice model. The cut-off phenomenon. Algorithms for "perfect simulation."

Markov Chains and Mixing Times, by D. Levine, Y. Peres and E. Wilmer, AMS 2009 - disponibile online pdf
An introduction to Markov processes, by D. Stroock, Springer 2013
Markov Chains, by J.R. Norris, Cambridge University Press 2006
Lectures on finite Markov chains, by L. Saloff-Coste, Lecture Notes in Math. 1665, pp.301-413 (1997).

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.

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

1. Introduction to information theory.
Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.

2. Source coding and data compression
Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. Ziv-Lempel compression.

3. Channel coding
Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.

4. Further codes and applications
Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.

David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2004.

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.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.

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

- Fundamental notions of molecular biology (nucleic acids, protein structures, central dogma)
- Reminders of probability theory (probability distributions, Bayes' theorem, random walks)
- Sequence alignment: scoring matrices, global and local alignment methods (dynamic programming, BLAST)
- Phylogenetic trees: distance methods (ultrametric and additive trees), probabilistic methods (maximum likelihood, Bayesian methods)
- Population genetics: Hardy-Weinberg law, genetic drift, natural selection
- Boolean networks: gene regulatory networks, asyncronous Boolean networks, attractors and attractive cycles, positive and negative cycles.

- Clote P., Backofen, R., Computational Molecular Biology. An Introduction, Wiley Series in Mathematical and Computational Biology, 2000.
- Allman E. S., Rhodes J. A., Mathematical Models in Biology. An Introduction, Cambridge University Press, 2004.
- Citterich M. H., Ferr ́e F., Pavesi G., Romualdi C., Pesole G., Fon- damenti di bioinformatica, Biologia Zanichelli, 2018.
- Gillespie, J., Population Genetics: a Coincise Guide, Johns Hopkins University Press, 2004.
-R uet P., Local cycles and dynamical properties of Boolean networks, Mathematical Structures in Computer Science, Volume 26, Issue 4, May 2016, pp. 702 - 718.

Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

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

G. Folland - "Real Analysis" - Wiley

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.

[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, Curves and Surfaces. Springer, (2006).

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

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).

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

W. Rudin, Real and complex Analysis, McGraw-Hill.

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

1. Theory of distributions

Definition and convergence of distributions.
Derivative of a distribution.
Spaces of distributions.
Convolutions.
Schwartz space, tempered distributions.
Fourier transform of tempered distributions.
Sobolev spaces and pseudo-differential operators.

2. Application to Schrödinger equations

Linear Schrödinger equation, local and global smoothing effects.
Nonlinear Schrödinger equation, local theory in L^2, H^1, H^2.
Asymptotic results and singularity formation for NLS.

Bony - Cours d'analyse, Théorie des distributions et analyse de Fourier
Ponce, Linares - Introduction to nonlinear dispersive equations

Fundamental grupo and covering spaces.
Cathegories.
Anstract and geometric simplicial complexes.
Singular and simplicial homology.
Cohomology.
Duality theorems.
Data analysis.

Allen Hatcher: Algebraic topology Cambridge University press.
Vidit Nanda: Computational Algebraic Topology - Lecture notes
Lucia Caporaso : Lezioni di topologia

See the program in italian.

R. Miranda: Algebraic Curves and Riemann Surfaces

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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

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.

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.
QuantumWalks:
Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.

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

Brownian motion: definition and property of BM, continuity and non-differentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multi-dimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.

Stochastic integration: Paley-Wiener-Zygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.

Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf

An introduction to Stochastic Differential Equations (Evans)

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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)

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

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.

- Introduction to parallel and distributed computing
- Base concepts: hardware architectures and memory hierarchy
- The C language
- Parallel programming models
- Message Passing Interface (MPI)
- OpenMP
- Introduction to general purpose programming on Graphics Processing Unit (GPU)
- CUDA and OpenCL

Introduction to Parallel Computing: From Algorithms to Programming on State-of-the-Art Platforms, Trobec, Slivnik, Bulić, Robič, Springer

Programming on Parallel Machines, Norm Matloff

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

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

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.

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

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.

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.
QuantumWalks:
Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.

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

Brownian motion: definition and property of BM, continuity and non-differentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multi-dimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.

Stochastic integration: Paley-Wiener-Zygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.

Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf

An introduction to Stochastic Differential Equations (Evans)

1. Modules

Modules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
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. Nakayama's Lemma.

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

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


4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings

5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.

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.

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

G. Strang,
Linear Algebra and Learning from Data,
Wellesley-Cambridge Press

M. Nielsen,
Neural Networks and Deep Learning (free online book)
http://neuralnetworksanddeeplearning.com

Various authors,
Distill, dedicated to clear explanations of machine learning
https://distill.pub

Sequent Calculus Proofs
***************************************
Natural Deduction

Sequent Calculus for Classical Logic (LK) and Intuitionistic Logic (LJ)

Cut elimination for LK and LJ sequent proofs

Sequent calculus for Linear Logic(LL)

Cut Elimination Theorem for LL

Focusing Theorem for LL proofs

Proof Nets
*************************************
(proof-structures, correctness, normalization, adequacy, sequentialization, focusing, complexity)

Pure Multiplicative Proof Nets

Multiplicative and Additive Proof Nets

Multiplicative and Exponenial Proof Nets

Denotational Semantics

NOTES AND SLIDES AVAILABLE ON THE COURSE WEB PAGE
https://sites.google.com/view/lm510/

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
Gioia, A. A. The theory of numbers. An introduction. Reprint of the 1970 original. Dover Publications, Inc., Mineola, NY, 2001. xii+207 pp. ISBN: 0-486-41449-3
Rosen, K. H. Elementary number theory and its applications. Fourth edition. Addison-Wesley, Reading, MA, 2000. xviii+638 pp. ISBN: 0-201-87073-8
Tattersall, J. J. Elementary number theory in nine chapters. Cambridge University Press, Cambridge, 1999. viii+407 pp. ISBN: 0-521-58531-7

The Hahn-Banach theorem; the open mapping theorem, the closed graph theorem and the Banach-Steinhaus theorem. Weak topologies, Banach-Alaoglu theorem and reflexivity. Linear continuous operators and the spectral theorem for compact, self-adjoint operators. Sobolev spaces and elliptic problems in one dimension. The spectral theorem for unitary operators.

H. Brezis, Analisi funzionale.

W. Rudin, Functional Analysis.

Part 1: Introduction to Lebesgue's theory in R^n
Definition of L^1 functions.
Theorems on the integration of limits (monotone convergence, dominated convergence, Fatou's lemma).
Completeness of L^1 (Riesz-Fischer Theorem).
Iterated integrals and Fubini's theorem.
Measurable functions and Lebesgue measure.
Convolution and regularization.
Theorem of the change of variables in R^n.
Divergence theorem in R^n.

Part 2: Fourier in L^2
The Hilbert space L^2 (on bounded domains and on R^n).
Fourier series and transforms in L^2.

Part 3: Fundamentals of the theory of ordinary differential equations
Examples and classes of ordinary differential equations.
Local existence and uniqueness theorem (Picard-Lindelof).
Lipschitz dependence on initial data.
Maximum and global solutions; globality criteria.
Linear systems (linear structure, Wronskian); non-homogeneous systems (variations of constants; Liouville's theorem.
Linear systems with constant coefficients (exponential solution).
Jordan canonical form and qualitative analysis of solutions).
Flows. Variational equation. Parameter dependence C^k.
Introduction to qualitative analysis.
Phase space.
Use of Fourier theory in differential equations (outline).

During the lessons, typed notes will be provided.

Review of elementary group theory. Semidirect products. Permutation groups and simplicity. Group actions on a subset. Sylow theorems. Abelian finitely generated groups. Free, nilpotents and solvable groups. Group representations.

'Groups' by A. Machì, Springer

Introduction to algebraic number theory:
Rings of integers in number fields and unique factorisation of ideals.
Absolute values in number fields.

The Weil Height and the Mahler measure:
Definitions and properties.
The product Formula
Northcott’s Theorem.
Kroneker’s Theorem.

Thue equations:
Thue’s Theorem on diophantine approximation.
Siegel’s Lemma.
Thue equations have a finite number of integer solutions.

Arithmetic dynamics:
(Pre)periodic points.
The canonical height.
Rational functions.

Diophantine equations in roots of unity:
Revision about roots of unity and cyclotomic polynomials.
The Theorem of Ihare-Serre-Tate.

Equidistribution:
Definitions and examples.
Bilu’s Theorem.
Bogomolov’s Conjecture.

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.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.

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

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.

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

1. Introduction to information theory.
Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.

2. Source coding and data compression
Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. Ziv-Lempel compression.

3. Channel coding
Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.

4. Further codes and applications
Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.

David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2004.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.

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

- Fundamental notions of molecular biology (nucleic acids, protein structures, central dogma)
- Reminders of probability theory (probability distributions, Bayes' theorem, random walks)
- Sequence alignment: scoring matrices, global and local alignment methods (dynamic programming, BLAST)
- Phylogenetic trees: distance methods (ultrametric and additive trees), probabilistic methods (maximum likelihood, Bayesian methods)
- Population genetics: Hardy-Weinberg law, genetic drift, natural selection
- Boolean networks: gene regulatory networks, asyncronous Boolean networks, attractors and attractive cycles, positive and negative cycles.

- Clote P., Backofen, R., Computational Molecular Biology. An Introduction, Wiley Series in Mathematical and Computational Biology, 2000.
- Allman E. S., Rhodes J. A., Mathematical Models in Biology. An Introduction, Cambridge University Press, 2004.
- Citterich M. H., Ferr ́e F., Pavesi G., Romualdi C., Pesole G., Fon- damenti di bioinformatica, Biologia Zanichelli, 2018.
- Gillespie, J., Population Genetics: a Coincise Guide, Johns Hopkins University Press, 2004.
-R uet P., Local cycles and dynamical properties of Boolean networks, Mathematical Structures in Computer Science, Volume 26, Issue 4, May 2016, pp. 702 - 718.

Cardano equations for the solvability of third degree equations, rings and fields, the characteristic of a field, reminders of polynomial rings, field extensions, construction of some field extensions,
the subring generated by a subset, the subfield generated by a subset, algebraic and transcendental elements, algebraically closed fields.

Splitting fields. Simple extensions and maps between simple extensions, splitting fields, existence of the splitting field, uniqueness except for isomorphisms of the splitting field, multiple roots, formal derivatives, separable polynomials and perfect fields, minimal polynomials and their characterizations.

The fundamental theorem of Galois theory. Group of automorphisms of a field, normal, separable and Galois extensions, characterisations of separable extensions, fundamental theorem of Galois correspondence, examples, Galois group of polynomials, root extensions, solvable groups and the Galois theorem on solving equations, Theorem of the existence of the primitive element.

The calculation of the Galois group. Galois groups as subgroups of S_n, transitive subgroups of S_n, characterization of irreducibility in terms of transitivity, polynomials with Galois groups in A_n,
Discriminant theory, Galois groups of polynomials of degree less than or equal to 4, examples of polynomials with Galois group S_p,
Dedekind theorem (only statement). Applications of Dedekind's Theorem, how to construct a polynomial with Galois group S_n.

Cyclotomic fields. Definitions, Galois group, under maximal real fields, quadratic sub-fields, Galois groups, cyclotomic polynomials and their properties, Galois inverse theory for abelian groups.

Finite Fields. Existence and uniqueness of finite fields, Galois group of a finite field, subfields of a finite field, enumeration of irreducible polynomials on finite fields.
Construction of the algebraic closure of a finite field with p elements.

Constructions with ruler and compass. Definition of constructible plane points, constructible real numbers, characterization of constructible points in terms of fields, constructible subfields and construction of constructible numbers, cube duplication, angle trisection, circle quadrature and Gauss theorem for constructing regular polygons with ruler and compass.

J. S. Milne,Fields and Galois Theory.Course Notes, (2015).

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

G. Folland - "Real Analysis" - Wiley

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.

[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, Curves and Surfaces. Springer, (2006).

Affine Spaces
Zariski topology. Affine closed subsets and radical ideals. Theorem of the zeros of Hilbert. Correspondence between closed subsets and radical ideals. Noetherian topological spaces. Irreducible closed subsets, irreducible components. Regular functions of affine closed subsets. Regular maps, isomorphisms. Principal open subsets. Examples. Projections are open. Finite morphisms.

Varieties
Projective spaces and their Zariski topology. Quasi-projective varieties. Rational and regular maps. Projective hypersurfaces. Birational equivalence. Principal open subsets and affine closed subsets. Affine varieties. Dimension of quasi-projective varieties. Finite and generically finite morphisms. Characterizations of birational equivalence. Characterization of generically finite morphisms. Costructible sets, Chevalley's theorem. Every variety is birational to a hypersurface.

Local geometry
Local ring of a variety in a point. Zariski cotangent space. Zariski tangent space. Singular and non singular points.

L. Caporaso
Introduction to algebraic geometry
Notes of the course

I. Shafarevich
Basic Algebraic geometry
Springer-Verlag, Berlin, 1994

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).

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

W. Rudin, Real and complex Analysis, McGraw-Hill.

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

1. Theory of distributions

Definition and convergence of distributions.
Derivative of a distribution.
Spaces of distributions.
Convolutions.
Schwartz space, tempered distributions.
Fourier transform of tempered distributions.
Sobolev spaces and pseudo-differential operators.

2. Application to Schrödinger equations

Linear Schrödinger equation, local and global smoothing effects.
Nonlinear Schrödinger equation, local theory in L^2, H^1, H^2.
Asymptotic results and singularity formation for NLS.

Bony - Cours d'analyse, Théorie des distributions et analyse de Fourier
Ponce, Linares - Introduction to nonlinear dispersive equations

Fundamental grupo and covering spaces.
Cathegories.
Anstract and geometric simplicial complexes.
Singular and simplicial homology.
Cohomology.
Duality theorems.
Data analysis.

Allen Hatcher: Algebraic topology Cambridge University press.
Vidit Nanda: Computational Algebraic Topology - Lecture notes
Lucia Caporaso : Lezioni di topologia

See the program in italian.

R. Miranda: Algebraic Curves and Riemann Surfaces

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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

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.

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.
QuantumWalks:
Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.

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

Brownian motion: definition and property of BM, continuity and non-differentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multi-dimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.

Stochastic integration: Paley-Wiener-Zygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.

Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf

An introduction to Stochastic Differential Equations (Evans)

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 at the address: http://www.mat.uniroma3.it/users/ferretti/corso.pdf

Roberto Ferretti, "Esercizi d'esame di Analisi Numerica", available at the address: http://www.mat.uniroma3.it/users/ferretti/Esercizi.pdf

Slides of the lessons, available from the course page: http://www.mat.uniroma3.it/users/ferretti/bacheca.html

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)

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

- Introduction to parallel and distributed computing
- Base concepts: hardware architectures and memory hierarchy
- The C language
- Parallel programming models
- Message Passing Interface (MPI)
- OpenMP
- Introduction to general purpose programming on Graphics Processing Unit (GPU)
- CUDA and OpenCL

Introduction to Parallel Computing: From Algorithms to Programming on State-of-the-Art Platforms, Trobec, Slivnik, Bulić, Robič, Springer

Programming on Parallel Machines, Norm Matloff

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

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

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.

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

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.

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.
QuantumWalks:
Classical Random Walks, Random Walks and Matrices, Defining Quantum Walks, Interference and Diffusion.

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

Brownian motion: definition and property of BM, continuity and non-differentiability of the trajectories. Markov property, strong Markov property and reflection principle. Multi-dimensional BM, harmonic functions and Dirichlet problem. Skorokhod embedding and Donsker invariance principle.

Stochastic integration: Paley-Wiener-Zygmund integral. Stochastic integral, Ito's formula and applications. Stochastic differential equations, Theorem of existence and uniqueness of solutions of SDEs. Exercises.

Brownian Motion (Moerters and Peres): http://www.mi.uni-koeln.de/~moerters/book/book.pdf

An introduction to Stochastic Differential Equations (Evans)

1. Modules

Modules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
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. Nakayama's Lemma.

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

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


4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings

5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.

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.

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

G. Strang,
Linear Algebra and Learning from Data,
Wellesley-Cambridge Press

M. Nielsen,
Neural Networks and Deep Learning (free online book)
http://neuralnetworksanddeeplearning.com

Various authors,
Distill, dedicated to clear explanations of machine learning
https://distill.pub

Sequent Calculus Proofs
***************************************
Natural Deduction

Sequent Calculus for Classical Logic (LK) and Intuitionistic Logic (LJ)

Cut elimination for LK and LJ sequent proofs

Sequent calculus for Linear Logic(LL)

Cut Elimination Theorem for LL

Focusing Theorem for LL proofs

Proof Nets
*************************************
(proof-structures, correctness, normalization, adequacy, sequentialization, focusing, complexity)

Pure Multiplicative Proof Nets

Multiplicative and Additive Proof Nets

Multiplicative and Exponenial Proof Nets

Denotational Semantics

NOTES AND SLIDES AVAILABLE ON THE COURSE WEB PAGE
https://sites.google.com/view/lm510/

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
Gioia, A. A. The theory of numbers. An introduction. Reprint of the 1970 original. Dover Publications, Inc., Mineola, NY, 2001. xii+207 pp. ISBN: 0-486-41449-3
Rosen, K. H. Elementary number theory and its applications. Fourth edition. Addison-Wesley, Reading, MA, 2000. xviii+638 pp. ISBN: 0-201-87073-8
Tattersall, J. J. Elementary number theory in nine chapters. Cambridge University Press, Cambridge, 1999. viii+407 pp. ISBN: 0-521-58531-7

The Hahn-Banach theorem; the open mapping theorem, the closed graph theorem and the Banach-Steinhaus theorem. Weak topologies, Banach-Alaoglu theorem and reflexivity. Linear continuous operators and the spectral theorem for compact, self-adjoint operators. Sobolev spaces and elliptic problems in one dimension. The spectral theorem for unitary operators.

H. Brezis, Analisi funzionale.

W. Rudin, Functional Analysis.

Part 1: Introduction to Lebesgue's theory in R^n
Definition of L^1 functions.
Theorems on the integration of limits (monotone convergence, dominated convergence, Fatou's lemma).
Completeness of L^1 (Riesz-Fischer Theorem).
Iterated integrals and Fubini's theorem.
Measurable functions and Lebesgue measure.
Convolution and regularization.
Theorem of the change of variables in R^n.
Divergence theorem in R^n.

Part 2: Fourier in L^2
The Hilbert space L^2 (on bounded domains and on R^n).
Fourier series and transforms in L^2.

Part 3: Fundamentals of the theory of ordinary differential equations
Examples and classes of ordinary differential equations.
Local existence and uniqueness theorem (Picard-Lindelof).
Lipschitz dependence on initial data.
Maximum and global solutions; globality criteria.
Linear systems (linear structure, Wronskian); non-homogeneous systems (variations of constants; Liouville's theorem.
Linear systems with constant coefficients (exponential solution).
Jordan canonical form and qualitative analysis of solutions).
Flows. Variational equation. Parameter dependence C^k.
Introduction to qualitative analysis.
Phase space.
Use of Fourier theory in differential equations (outline).

During the lessons, typed notes will be provided.

Review of elementary group theory. Semidirect products. Permutation groups and simplicity. Group actions on a subset. Sylow theorems. Abelian finitely generated groups. Free, nilpotents and solvable groups. Group representations.

'Groups' by A. Machì, Springer

Introduction to algebraic number theory:
Rings of integers in number fields and unique factorisation of ideals.
Absolute values in number fields.

The Weil Height and the Mahler measure:
Definitions and properties.
The product Formula
Northcott’s Theorem.
Kroneker’s Theorem.

Thue equations:
Thue’s Theorem on diophantine approximation.
Siegel’s Lemma.
Thue equations have a finite number of integer solutions.

Arithmetic dynamics:
(Pre)periodic points.
The canonical height.
Rational functions.

Diophantine equations in roots of unity:
Revision about roots of unity and cyclotomic polynomials.
The Theorem of Ihare-Serre-Tate.

Equidistribution:
Definitions and examples.
Bilu’s Theorem.
Bogomolov’s Conjecture.

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.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.

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

1. Random walks and Markov Chains. Sequences of random variables. Random walks. Discrete-time Markov chains. Invariant measure, time-reversal, and reversibility.

2. Examples and classical models. Random walks on graphs. Birth and death processes. Exclusion processes. Monte Carlo method: Metropolis algorithms and Glauber dynamics for the Ising model, graph coloring, and other interacting systems.

3. Convergence to equilibrium I. Variation distance, mixing times. Ergodic theorems. Coupling techniques. Strong stationary times. Applications to the coupon collector problem and shuffling a deck of cards.

4. Convergence to equilibrium II. L2-norm convergence. Spectral gap and relaxation time estimates. Cheeger's inequality.

5. Continuous-time Markov chains. Poisson process. Q-matrices and continuous-time transition probabilities. Infinitesimal generator of a continuous-time Markov chain. Explicit construction and graphical representation.

6. Conductance and path method. Comparison method. Spectral gap for the Bernoulli-Laplace process and the exclusion process on d-dimensional torus. Convergence to equilibrium in terms of entropy and logarithmic Sobolev inequalities.

7. Other selected topics. Glauber dynamics for the Ising model: dynamic phase transition for the mean-field model and the lattice model. The cut-off phenomenon. Algorithms for "perfect simulation."

Markov Chains and Mixing Times, by D. Levine, Y. Peres and E. Wilmer, AMS 2009 - disponibile online pdf
An introduction to Markov processes, by D. Stroock, Springer 2013
Markov Chains, by J.R. Norris, Cambridge University Press 2006
Lectures on finite Markov chains, by L. Saloff-Coste, Lecture Notes in Math. 1665, pp.301-413 (1997).

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.

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

1. Introduction to information theory.
Reliable transmission of information. Shannon's information content. Measures of information. Entropy, mutual information, informational divergence. Data compression. Error correction. Data processing theorems. Fundamental inequalities. Information diagrams. Informational divergence and maximum likelihood.

2. Source coding and data compression
Typical sequences. Typicality in probability. Asymptotic equipartitioning property. Block codes and variable length codes. Coding rate. Source coding theorem. Lossless data compression. Huffman code. Universal codes. Ziv-Lempel compression.

3. Channel coding
Channel capacity. Discrete memoryless channels. Information transmitted over a channel. Decoding criteria. Noisy channel coding theorem.

4. Further codes and applications
Hamming space. Linear codes. Generating matrix and check matrix. Cyclic codes. Hash codes.

David J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge University Press, 2004.

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.

Notes on graph theory: graph, directed graph, tree, free and rooted tree, connection, strong connection, acyclicality; isomorphisms between graphs, planarity, Kuratowski's theorem, Euler's formula; coloring of graphs, Eulerian paths, Hamiltonian circuits.
Review of algorithm theory and computational complexity: complexity classes, the class of NP, NP-complete, NP-hard problems. Problems of decision, search, enumeration and optimization; problems of nonlinear programming, convex programming, linear programming and integer linear programming; combinatorial optimization problems. Recalls on the elements of combinatorics.
Optimization problems on graphs: visit of graphs, verification of fundamental properties, connection, presence of cycles, strongly connected components, topological ordering of a graph. Minimum spanning tree (Prim and Kruskal algorithms). Paths of minimum cost (Dijkstra and Bellman-Ford algorithms, dynamic programming technique, Floyd-Warshall algorithm, computation of the transitive closure of a graph). Networks and calculation of the maximum flow on a network, maximum flow / minimum cut theorem, Ford-Fulkerson algorithm, Edmonds-Karp algorithm, preflow algorithms, "push-relabel" algorithms.
Partitioning problems of graphs, trees and paths into connected components, objective functions and algorithmic techniques.
Stable marriage problem, generalizations and applications of the problem, Gale and Shapley algorithm.
Huffman codes.
Programming laboratory for the implementation of algorithms in Python language and with the help of Mathematica software.

T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, "Introduction to algorithms"

Lecture notes and other teaching material in Italian provided by the teacher and made available on the course website (http://www.mat.uniroma3.it/users/liverani/IN440) and on the Microsoft Teams platform

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.