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) 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).

Local existence and uniqueness theorems, existence times and extensions. Escape from compact sets. Comparison theorems. Behavior of linear systems with constant coefficients. Jordan canonical form.
Differentiable functions on a Banach space. The Implicit Function Theorem. Applications to the search of periodic solutions. Lyapunov Schmidt decomposition. Hopf theorem. Dependence on initial data. The flow box theorem. Coordinate changes generated by the flow of a vector field. The Lie exponential. Poincare normal form.

Note del docente.
Chierchia Analisi Matematica 2

Wave equation, Heath equation, Introduction to quantum mechanics

W. Craig, A Course on Partial Differential Equations - American Mathematical Society (2018)

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

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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)

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. 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 - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

The education consists of lectures and programming sessions.
The main programming language is C.

• Introduction to C
• Introduction to High Performance Computing
• Key concepts: Hardware Architecture and Memory Hierarchy
• Parallelizzation techniques
• Measuring parallel performance: theory and benchmark
• Version Control of your code: Git software
• Parallel programming with MPI: Message Passing Interface
• Parallel programming with OpenMP: Open Multiprocessing
• Parallel Input/Output
• Introduction to GPU computing and OpenCL Programming

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

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

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 Availability

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269
Taylor, J. - An introduction to error analysis - University Science Books Sausalito, California
Dubi, A. - Monte Carlo applications in systems engineering - Wiley

Introduction and generality; What is machine learning; definitions; supervised and unsupervised learning; regression and clustering;
Univariate linear regression; representation; the hypothesis function; the choice of the parameters of the hypothesis function; the cost function; the Gradient Descent algorithm; the choice of the alpha parameter;
Multivariate linear regression; vector notation of the hypothesis function and of the cost function; Gradient Descent algorithm for multivariate; matrix notation; feature scaling and normalization; polynomial regression; the Normal Equation for multivariate regression; final notes on the comparison of the Gradient Descent algorithm and the calculation of the Normal Equation;
Logistic Regression; binary classification; representation of the hypotheses; the logistics function; the decision boundary; the cost function for logistic regression; the gradient descent algorithm for logistic regression; analytic derivation of the gradient of the cost function for logistic regression; notes on the implementation in Octave of the cost function and of the gradient descent algorithno in the case of logistic regression; considerations on advanced optimization methods; multi-class classification; the one-vs-all method;
The regularization; the problem of overfitting / underfitting (ie high variance / high bias); modification of the cost function; the regularization parameter; regularization of linear regression; the algorithm of gradient descent with regularization; the regularized normal equation; logistic regression with regularization; Neural networks history; AI and connectionism; the perceiver; Rosenblatt's learning rule; learning of boolean functions; the limits of the perceiver;
Neural networks; reasons; neurons; neuroplasticity and the one-learning-algorithm hypothesis; model representation; the neuron as a logistic unit; the weight matrix; the bias; the activation function; forward propagation; vector version; the NNs as an extension of the logistic regression; calculation of the Boolean functions AND, OR, NOT, XNOR; multiclass classification with Neural Networks;
Neural Network Learning; cost function of a Multi Layer Perceptron; the Backpropagation algorithm; Intuition and formalization;
Neural Network learning; Error BackPropagation Algorithm (scalar version, vector version); Notes on implementation; rolling and unrolling of the parameters for passing the weight matrix into Octave; Gradient checking by calculating the numerical approximate gradient; initialization of weights and symmetry breaking; The ALVINN network (an autonomous driving system);
Machine Learning Diagnostic; Evaluating a Learning Algorithm; The test set error; Model selection + training, validation and test set; The concept of Bias and variance; Regularization and Bias / Variance; Choosing the regularization parameter; Putting all together: diagnostic method; Learning curves; Machine Learning system design; Debugging a learning algorithm; Diagnosing Neural Networks; Model selection; Error analysis; The importance of numerical evaluation; Error Metrics for Skewed Classes; Precision / Recall and Accuracy; Trading Off Precision and Recall; The F1 score; Data for Machine Learning; Designing a high accuracy learning system; Rationale for large data;
Support Vector Machines; SVM Cost function; SVM as Large margin Classifiers; the Kernels; choice of landmarks; choice of parameters C and sigma; Multi-class Classification with SVM Comparison between Logistic Regression and SVM and between NN vs. SVM;
Clustering; the K-means algorithm; cluster assignment step; move centroids step; optimization objective; choosing the number of clusters, the elbow method; Dimensionality Reduction; Principal Component Analysis; Motivation I: Data compression; Motivation II: data visualization - Problem Formulation; Goal of PCA; The role of Singular Value Decomposition in the PCA algorithm; Reconstruction from compressed representation; Algorithm for choosing k; Advice for Applying PCA; The most common use of PCA; Misuse of PCA;
Anomaly Detection; Problem motivation; Density estimation; Gaussian distribution; Anomaly Detection; Gaussian distribution; Parameter estimation; The Anomaly Detection Algorithm; Anomaly Detection vs. Supervised Learning; Multivariate Gaussian Distribution; Recommender Systems; Collaborative Filtering; Motivation; Problem Formulation; Content Based Recommendations; Notation; Optimization objective; Gradient descent update; Low Rank Matrix Factorization;
Learning with large datasets; Online learning; Stochastic gradient descent; Mini-batch gradient descent; Checking for convergence; Map reduce and data parallelism;
Machine Learning pipeline; the OCR systeml ceiling analysis; Laboratory: exercise related to Recommender Systems;

J. Watt, R. Borhani, A. K. Katsaggelos. Machine Learning Refined. Cambridge Univ. Press 2016

Wave equation, Heath equation, Introduction to quantum mechanics

W. Craig, A Course on Partial Differential Equations - American Mathematical Society (2018)

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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).

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

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

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

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

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

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

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

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

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

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.

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

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

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Functional analysis, H. Bre'zis

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

Highlights of Linear Algebra:
- matrix multiplication; column & row space; rank; the four fundamental subspaces;
- elimination method; decomposition in lower and upper trinagular matrices; permutations;
- orthogonal matrices;
- eigenvalues and eigenvectors for ODE;
- positive definite symmetric matrices; the energy function;
- gradient and Hessian matrix;
- singular value decomposition; Eckart-Young theorem; principal component analysis; generalized evectors;
- norms; least squares method; convexity and Newton’s method; Lagrange multipliers.

Machine Learning:
- Gradient Descend; GD with Matlab;
- Learning and Loss; Deep Neural Network;
- loss functions: Quadratic VS Cross entropy;
- Stocastics Gradient Descend (SGD) & Kaczmarcz; SGD convergence rates & ADAM
- Matlab interface for DNN; Construction of DNN;
- 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 (2019).

A) STRUCTURAL RULES INTERPRETED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE NON-DETERMINISM: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC AND PROOF SEARCH
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

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

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

Arithmetic functions and their properties:
-Definition and Dirichlet convolution.
-Number and sum of divisors function.
-Möbius function.
-Euler function.

Congruences:
-Sets of residues.
-Polynomial congruences.
-Primitive roots.

Quadratic residues:
-Legendre symbol.
-Quadratic reciprocity.
-Jacobi symbol.

Sums of squares:
-Sums of two squares.
-Number of representations.
-Sums of four squares.
-Sums of three squares.

Continued fractions and diophantine approximation:
-Simple continued fractions.
-Continued fractions and diophantine approximation.
-Infinite simple continued fractions.
-Periodic continued fractions.
-Pell's equation.
-Liouville's Theorem.

Lecture notes

Note di W. Chen
http://www.williamchen-mathematics.info/lnentfolder/lnent.html

An Introduction to the Theory of Numbers by G. H. Hardy, E. M. Wright

M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html

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

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

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

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Dynamical system and Reaction Diffusion problems; morphogenesis.

Steven H. Strogatz,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering
CRC Press, 2018

https://www.google.it/books/edition/Nonlinear_Dynamics_and_Chaos/1kpnDwAAQBAJ?hl=it&gbpv=0

A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952

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

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

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

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

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html

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

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

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

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Dynamical system and Reaction Diffusion problems; morphogenesis.

Steven H. Strogatz,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering
CRC Press, 2018

https://www.google.it/books/edition/Nonlinear_Dynamics_and_Chaos/1kpnDwAAQBAJ?hl=it&gbpv=0

A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952

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.

The course aim is to provide the student with the general cognitive elements underlying the acquisition, control and monitoring systems of Nuclear and Subnuclear Physics experiments. The course is divided into the following topics: -Introduction to DAQ-Parallelism and Pipelining systems -Derandomization-DAQ and Trigger-Data Transmission -Front End Electronics-Trigger-Architecture Computing Systems-Real Time Systems-Real Time Operating Systems -C Language-TCP / IP Network Protocols-DAQ-Architecture Building -VME Bus-Run Control-Farming-Data Archiving During the course, laboratory exercises will take place with the execution of simple examples of: - reading and data transfer systems through pipe mechanisms with concurrent processes; - signal-based trigger simulation programs; - Run Control program for activation and termination of processes; - configuration and reading of data from board on VME bus.

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

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.

Topics Part A

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


Program Part A

• Overview

• Celestial coordinates (1.3)

• Telescopes and resolving power (6.1)

• Parallax distance (3.1)

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

• The black body (3.4, 3.5)

• Hertzsprung-Russel diagram (8.2)

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

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

• The classification of galaxies (24.1)

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

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

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

• Probability of collision between stars and galaxies (handouts)

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

• Active Galactic Nuclei (28.1, 28.2, 28.3)


Topics Part B

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

Program Part B

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

• Stars of Neutrons and Pulsars (16.6, 16.7)

• Gamma Ray Bursts (handouts)

• Gravitational Waves (lecture notes)

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

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

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

• Nuclear reactions of hydrogen (11.3)

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

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

• Metallicity (25.2)

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

• Distance scale (27.1)

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

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

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

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

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

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

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

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

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

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

Algebraic varieties in affine and projective spaces.
Veronese and Segre mappings. Products and projections.
Local geometry
Bertini's theorems.

I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
L. Caporaso Introduzione alla geometria algebrica Versione preliminare.

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).

Local existence and uniqueness theorems, existence times and extensions. Escape from compact sets. Comparison theorems. Behavior of linear systems with constant coefficients. Jordan canonical form.
Differentiable functions on a Banach space. The Implicit Function Theorem. Applications to the search of periodic solutions. Lyapunov Schmidt decomposition. Hopf theorem. Dependence on initial data. The flow box theorem. Coordinate changes generated by the flow of a vector field. The Lie exponential. Poincare normal form.

Note del docente.
Chierchia Analisi Matematica 2

Wave equation, Heath equation, Introduction to quantum mechanics

W. Craig, A Course on Partial Differential Equations - American Mathematical Society (2018)

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

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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)

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. 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 - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

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.

The education consists of lectures and programming sessions.
The main programming language is C.

• Introduction to C
• Introduction to High Performance Computing
• Key concepts: Hardware Architecture and Memory Hierarchy
• Parallelizzation techniques
• Measuring parallel performance: theory and benchmark
• Version Control of your code: Git software
• Parallel programming with MPI: Message Passing Interface
• Parallel programming with OpenMP: Open Multiprocessing
• Parallel Input/Output
• Introduction to GPU computing and OpenCL Programming

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

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

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 Availability

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269
Taylor, J. - An introduction to error analysis - University Science Books Sausalito, California
Dubi, A. - Monte Carlo applications in systems engineering - Wiley

Introduction and generality; What is machine learning; definitions; supervised and unsupervised learning; regression and clustering;
Univariate linear regression; representation; the hypothesis function; the choice of the parameters of the hypothesis function; the cost function; the Gradient Descent algorithm; the choice of the alpha parameter;
Multivariate linear regression; vector notation of the hypothesis function and of the cost function; Gradient Descent algorithm for multivariate; matrix notation; feature scaling and normalization; polynomial regression; the Normal Equation for multivariate regression; final notes on the comparison of the Gradient Descent algorithm and the calculation of the Normal Equation;
Logistic Regression; binary classification; representation of the hypotheses; the logistics function; the decision boundary; the cost function for logistic regression; the gradient descent algorithm for logistic regression; analytic derivation of the gradient of the cost function for logistic regression; notes on the implementation in Octave of the cost function and of the gradient descent algorithno in the case of logistic regression; considerations on advanced optimization methods; multi-class classification; the one-vs-all method;
The regularization; the problem of overfitting / underfitting (ie high variance / high bias); modification of the cost function; the regularization parameter; regularization of linear regression; the algorithm of gradient descent with regularization; the regularized normal equation; logistic regression with regularization; Neural networks history; AI and connectionism; the perceiver; Rosenblatt's learning rule; learning of boolean functions; the limits of the perceiver;
Neural networks; reasons; neurons; neuroplasticity and the one-learning-algorithm hypothesis; model representation; the neuron as a logistic unit; the weight matrix; the bias; the activation function; forward propagation; vector version; the NNs as an extension of the logistic regression; calculation of the Boolean functions AND, OR, NOT, XNOR; multiclass classification with Neural Networks;
Neural Network Learning; cost function of a Multi Layer Perceptron; the Backpropagation algorithm; Intuition and formalization;
Neural Network learning; Error BackPropagation Algorithm (scalar version, vector version); Notes on implementation; rolling and unrolling of the parameters for passing the weight matrix into Octave; Gradient checking by calculating the numerical approximate gradient; initialization of weights and symmetry breaking; The ALVINN network (an autonomous driving system);
Machine Learning Diagnostic; Evaluating a Learning Algorithm; The test set error; Model selection + training, validation and test set; The concept of Bias and variance; Regularization and Bias / Variance; Choosing the regularization parameter; Putting all together: diagnostic method; Learning curves; Machine Learning system design; Debugging a learning algorithm; Diagnosing Neural Networks; Model selection; Error analysis; The importance of numerical evaluation; Error Metrics for Skewed Classes; Precision / Recall and Accuracy; Trading Off Precision and Recall; The F1 score; Data for Machine Learning; Designing a high accuracy learning system; Rationale for large data;
Support Vector Machines; SVM Cost function; SVM as Large margin Classifiers; the Kernels; choice of landmarks; choice of parameters C and sigma; Multi-class Classification with SVM Comparison between Logistic Regression and SVM and between NN vs. SVM;
Clustering; the K-means algorithm; cluster assignment step; move centroids step; optimization objective; choosing the number of clusters, the elbow method; Dimensionality Reduction; Principal Component Analysis; Motivation I: Data compression; Motivation II: data visualization - Problem Formulation; Goal of PCA; The role of Singular Value Decomposition in the PCA algorithm; Reconstruction from compressed representation; Algorithm for choosing k; Advice for Applying PCA; The most common use of PCA; Misuse of PCA;
Anomaly Detection; Problem motivation; Density estimation; Gaussian distribution; Anomaly Detection; Gaussian distribution; Parameter estimation; The Anomaly Detection Algorithm; Anomaly Detection vs. Supervised Learning; Multivariate Gaussian Distribution; Recommender Systems; Collaborative Filtering; Motivation; Problem Formulation; Content Based Recommendations; Notation; Optimization objective; Gradient descent update; Low Rank Matrix Factorization;
Learning with large datasets; Online learning; Stochastic gradient descent; Mini-batch gradient descent; Checking for convergence; Map reduce and data parallelism;
Machine Learning pipeline; the OCR systeml ceiling analysis; Laboratory: exercise related to Recommender Systems;

J. Watt, R. Borhani, A. K. Katsaggelos. Machine Learning Refined. Cambridge Univ. Press 2016

Wave equation, Heath equation, Introduction to quantum mechanics

W. Craig, A Course on Partial Differential Equations - American Mathematical Society (2018)

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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).

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

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

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

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

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

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

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

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

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

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.

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Functional analysis, H. Bre'zis

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

A) STRUCTURAL RULES INTERPRETED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE NON-DETERMINISM: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC AND PROOF SEARCH
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

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

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

Arithmetic functions and their properties:
-Definition and Dirichlet convolution.
-Number and sum of divisors function.
-Möbius function.
-Euler function.

Congruences:
-Sets of residues.
-Polynomial congruences.
-Primitive roots.

Quadratic residues:
-Legendre symbol.
-Quadratic reciprocity.
-Jacobi symbol.

Sums of squares:
-Sums of two squares.
-Number of representations.
-Sums of four squares.
-Sums of three squares.

Continued fractions and diophantine approximation:
-Simple continued fractions.
-Continued fractions and diophantine approximation.
-Infinite simple continued fractions.
-Periodic continued fractions.
-Pell's equation.
-Liouville's Theorem.

Lecture notes

Note di W. Chen
http://www.williamchen-mathematics.info/lnentfolder/lnent.html

An Introduction to the Theory of Numbers by G. H. Hardy, E. M. Wright

M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html

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

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

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

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Dynamical system and Reaction Diffusion problems; morphogenesis.

Steven H. Strogatz,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering
CRC Press, 2018

https://www.google.it/books/edition/Nonlinear_Dynamics_and_Chaos/1kpnDwAAQBAJ?hl=it&gbpv=0

A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952

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

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

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

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

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html

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

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

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

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Dynamical system and Reaction Diffusion problems; morphogenesis.

Steven H. Strogatz,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering
CRC Press, 2018

https://www.google.it/books/edition/Nonlinear_Dynamics_and_Chaos/1kpnDwAAQBAJ?hl=it&gbpv=0

A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952

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.

The course aim is to provide the student with the general cognitive elements underlying the acquisition, control and monitoring systems of Nuclear and Subnuclear Physics experiments. The course is divided into the following topics: -Introduction to DAQ-Parallelism and Pipelining systems -Derandomization-DAQ and Trigger-Data Transmission -Front End Electronics-Trigger-Architecture Computing Systems-Real Time Systems-Real Time Operating Systems -C Language-TCP / IP Network Protocols-DAQ-Architecture Building -VME Bus-Run Control-Farming-Data Archiving During the course, laboratory exercises will take place with the execution of simple examples of: - reading and data transfer systems through pipe mechanisms with concurrent processes; - signal-based trigger simulation programs; - Run Control program for activation and termination of processes; - configuration and reading of data from board on VME bus.

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

Topics Part A

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


Program Part A

• Overview

• Celestial coordinates (1.3)

• Telescopes and resolving power (6.1)

• Parallax distance (3.1)

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

• The black body (3.4, 3.5)

• Hertzsprung-Russel diagram (8.2)

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

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

• The classification of galaxies (24.1)

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

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

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

• Probability of collision between stars and galaxies (handouts)

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

• Active Galactic Nuclei (28.1, 28.2, 28.3)


Topics Part B

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

Program Part B

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

• Stars of Neutrons and Pulsars (16.6, 16.7)

• Gamma Ray Bursts (handouts)

• Gravitational Waves (lecture notes)

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

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

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

• Nuclear reactions of hydrogen (11.3)

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

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

• Metallicity (25.2)

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

• Distance scale (27.1)

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

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

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

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

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

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

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

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

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

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) 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).

Local existence and uniqueness theorems, existence times and extensions. Escape from compact sets. Comparison theorems. Behavior of linear systems with constant coefficients. Jordan canonical form.
Differentiable functions on a Banach space. The Implicit Function Theorem. Applications to the search of periodic solutions. Lyapunov Schmidt decomposition. Hopf theorem. Dependence on initial data. The flow box theorem. Coordinate changes generated by the flow of a vector field. The Lie exponential. Poincare normal form.

Note del docente.
Chierchia Analisi Matematica 2

Wave equation, Heath equation, Introduction to quantum mechanics

W. Craig, A Course on Partial Differential Equations - American Mathematical Society (2018)

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

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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)

Quantum mechanics: The crisis of classical physics. Waves and particles. State vectors and operators. Measurements and observables. The position operator. Translations and momentum. Time evolution and the schrodinger equation. Parity. One-dimensional problems. 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 - Seconda Edizione [Zanichelli, Bologna, 2014]
An english version of the book is also available:
Sakurai J.J., Modern Quantum Mechanics (Revised Edition) [Addison-Wesley, 1994]

The education consists of lectures and programming sessions.
The main programming language is C.

• Introduction to C
• Introduction to High Performance Computing
• Key concepts: Hardware Architecture and Memory Hierarchy
• Parallelizzation techniques
• Measuring parallel performance: theory and benchmark
• Version Control of your code: Git software
• Parallel programming with MPI: Message Passing Interface
• Parallel programming with OpenMP: Open Multiprocessing
• Parallel Input/Output
• Introduction to GPU computing and OpenCL Programming

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

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

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 Availability

Further applications

Weinzierl, S. - Introduction to Monte Carlo methods arXiv:hep-ph/0006269
Taylor, J. - An introduction to error analysis - University Science Books Sausalito, California
Dubi, A. - Monte Carlo applications in systems engineering - Wiley

Introduction and generality; What is machine learning; definitions; supervised and unsupervised learning; regression and clustering;
Univariate linear regression; representation; the hypothesis function; the choice of the parameters of the hypothesis function; the cost function; the Gradient Descent algorithm; the choice of the alpha parameter;
Multivariate linear regression; vector notation of the hypothesis function and of the cost function; Gradient Descent algorithm for multivariate; matrix notation; feature scaling and normalization; polynomial regression; the Normal Equation for multivariate regression; final notes on the comparison of the Gradient Descent algorithm and the calculation of the Normal Equation;
Logistic Regression; binary classification; representation of the hypotheses; the logistics function; the decision boundary; the cost function for logistic regression; the gradient descent algorithm for logistic regression; analytic derivation of the gradient of the cost function for logistic regression; notes on the implementation in Octave of the cost function and of the gradient descent algorithno in the case of logistic regression; considerations on advanced optimization methods; multi-class classification; the one-vs-all method;
The regularization; the problem of overfitting / underfitting (ie high variance / high bias); modification of the cost function; the regularization parameter; regularization of linear regression; the algorithm of gradient descent with regularization; the regularized normal equation; logistic regression with regularization; Neural networks history; AI and connectionism; the perceiver; Rosenblatt's learning rule; learning of boolean functions; the limits of the perceiver;
Neural networks; reasons; neurons; neuroplasticity and the one-learning-algorithm hypothesis; model representation; the neuron as a logistic unit; the weight matrix; the bias; the activation function; forward propagation; vector version; the NNs as an extension of the logistic regression; calculation of the Boolean functions AND, OR, NOT, XNOR; multiclass classification with Neural Networks;
Neural Network Learning; cost function of a Multi Layer Perceptron; the Backpropagation algorithm; Intuition and formalization;
Neural Network learning; Error BackPropagation Algorithm (scalar version, vector version); Notes on implementation; rolling and unrolling of the parameters for passing the weight matrix into Octave; Gradient checking by calculating the numerical approximate gradient; initialization of weights and symmetry breaking; The ALVINN network (an autonomous driving system);
Machine Learning Diagnostic; Evaluating a Learning Algorithm; The test set error; Model selection + training, validation and test set; The concept of Bias and variance; Regularization and Bias / Variance; Choosing the regularization parameter; Putting all together: diagnostic method; Learning curves; Machine Learning system design; Debugging a learning algorithm; Diagnosing Neural Networks; Model selection; Error analysis; The importance of numerical evaluation; Error Metrics for Skewed Classes; Precision / Recall and Accuracy; Trading Off Precision and Recall; The F1 score; Data for Machine Learning; Designing a high accuracy learning system; Rationale for large data;
Support Vector Machines; SVM Cost function; SVM as Large margin Classifiers; the Kernels; choice of landmarks; choice of parameters C and sigma; Multi-class Classification with SVM Comparison between Logistic Regression and SVM and between NN vs. SVM;
Clustering; the K-means algorithm; cluster assignment step; move centroids step; optimization objective; choosing the number of clusters, the elbow method; Dimensionality Reduction; Principal Component Analysis; Motivation I: Data compression; Motivation II: data visualization - Problem Formulation; Goal of PCA; The role of Singular Value Decomposition in the PCA algorithm; Reconstruction from compressed representation; Algorithm for choosing k; Advice for Applying PCA; The most common use of PCA; Misuse of PCA;
Anomaly Detection; Problem motivation; Density estimation; Gaussian distribution; Anomaly Detection; Gaussian distribution; Parameter estimation; The Anomaly Detection Algorithm; Anomaly Detection vs. Supervised Learning; Multivariate Gaussian Distribution; Recommender Systems; Collaborative Filtering; Motivation; Problem Formulation; Content Based Recommendations; Notation; Optimization objective; Gradient descent update; Low Rank Matrix Factorization;
Learning with large datasets; Online learning; Stochastic gradient descent; Mini-batch gradient descent; Checking for convergence; Map reduce and data parallelism;
Machine Learning pipeline; the OCR systeml ceiling analysis; Laboratory: exercise related to Recommender Systems;

J. Watt, R. Borhani, A. K. Katsaggelos. Machine Learning Refined. Cambridge Univ. Press 2016

Wave equation, Heath equation, Introduction to quantum mechanics

W. Craig, A Course on Partial Differential Equations - American Mathematical Society (2018)

Introduction to statistics: random sampling of finite and infinite populations. Definition of the statistical model and the concept of statistics. Example of statistics. Properties of statistics: sufficient, minimal and complete statistics.

Point estimators: method of moments, maximum likelihood estimators and Bayes estimators. EM algorithm.

How to evaluate estimators: bias, consistency and mean square error. UMVU estimators and efficient estimators.

Confidence interval: the concept of pivotals, asymptotic methods and the delta method.

Hypothesis testing: definitions, likelihood ratio test and duality with confidence interval. Uniformly most powerful tests.

Non-parametric methods: Goodness-of-fit test for discrete and continuum variables, contingency tables and Kolmogorov-Smirnov test.

Other topics: analysis of variance (ANOVA), linear regression, generalized linear regression and logistic regression.

Statistical Inference
Casella e Berger
Duxbury
2nd edition.

The goal of the course is to discuss modern methods from probability theory and their use to solve fundamental problems from other areas, such as computer science (randomized algorithms and random networks), combinatorics and data science. In particular, in the course we see several applications where the problem to be solved is in fact *not random*, but we choose to resort to a probabilistic solution in an opportunistic way to solve them.

Some of the topics seen in the course are the following:
* Randomized algorithms
* Can we really use perfect randomness in computer science?
* Probabilistic method and its application to computer science, combinatorics and some games
* Concentration of random variables and martingales, and their applications to routing and dimension reduction
* Branching processes and spread of infections
* Percolation, Erdos-Renyi random graphs and random networks
* Random walks on graphs and application to clustering data

"Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis", Mitzenmacher and Upfal, Cambridge University Press
"The probabilistic method", Alon and Spencer, John Wiley & Sons

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).

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

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

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

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

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

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

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

Logic and Arithmetic: incompleteness

Part 1: Decidability and fundamental results of recursion theory. Primitive recursive functions and elementary functions: definitions and examples, elementary coding of the finite sequences of natural numbers, an alternative definition of the set of elementary functions. Ackermann's function and the (partial) recursive functions. Arithmetical hierarchy and representation (in N) of recursive functions. Arithmetization of syntax: coding of terms and formulas, satisfiability in N of Delta formulas is elementary, coding of sequence and derivations. The fundamental theorems of recursion theory. Decidability, semi-decidability, undecidability.

Part 2: Peano arithmetic. Peano's axioms and first order Peano’s axioms. The models of (first order) Peano arithmetic. The representable functions in (first order) Peano arithmetic. Incompleteness and undecidability: Church’s undecidability theorem, fixed point, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem, final remarks on incompleteness, hints on incompleteness and second order logic.

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

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.

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

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

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the
norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in
dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in
dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Omega)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Omega)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Functional analysis, H. Bre'zis

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

Highlights of Linear Algebra:
- matrix multiplication; column & row space; rank; the four fundamental subspaces;
- elimination method; decomposition in lower and upper trinagular matrices; permutations;
- orthogonal matrices;
- eigenvalues and eigenvectors for ODE;
- positive definite symmetric matrices; the energy function;
- gradient and Hessian matrix;
- singular value decomposition; Eckart-Young theorem; principal component analysis; generalized evectors;
- norms; least squares method; convexity and Newton’s method; Lagrange multipliers.

Machine Learning:
- Gradient Descend; GD with Matlab;
- Learning and Loss; Deep Neural Network;
- loss functions: Quadratic VS Cross entropy;
- Stocastics Gradient Descend (SGD) & Kaczmarcz; SGD convergence rates & ADAM
- Matlab interface for DNN; Construction of DNN;
- 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 (2019).

A) STRUCTURAL RULES INTERPRETED AS LOGICAL RULES: SEQUENT CALCULUS AND DERIVABILITY IN LINEAR LOGIC
B) POSITIVE AND NEGATIVE NON-DETERMINISM: FOCUSED SEQUENT CALCULUS FOR LINEAR LOGIC AND PROOF SEARCH
C) IMPLICIT COMPLEXITY AND LINEAR LOGIC
D) GEOMETRY OF PROOFS: PROOF NETS IN LINEAR LOGIC
E) INVARIANTS AND DEVELOPMENT OF INTERACTION OF PROOFS: COHERENT SPACES, GEOMETRY OF INTERACTION

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

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

Arithmetic functions and their properties:
-Definition and Dirichlet convolution.
-Number and sum of divisors function.
-Möbius function.
-Euler function.

Congruences:
-Sets of residues.
-Polynomial congruences.
-Primitive roots.

Quadratic residues:
-Legendre symbol.
-Quadratic reciprocity.
-Jacobi symbol.

Sums of squares:
-Sums of two squares.
-Number of representations.
-Sums of four squares.
-Sums of three squares.

Continued fractions and diophantine approximation:
-Simple continued fractions.
-Continued fractions and diophantine approximation.
-Infinite simple continued fractions.
-Periodic continued fractions.
-Pell's equation.
-Liouville's Theorem.

Lecture notes

Note di W. Chen
http://www.williamchen-mathematics.info/lnentfolder/lnent.html

An Introduction to the Theory of Numbers by G. H. Hardy, E. M. Wright

M. Fontana, Appunti del corso TN1 (Argomenti della teoria classica dei numeri), http://www.mat.uniroma3.it/users/fontana/didattica/fontana_didattica.html#dispense

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html

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

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

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

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Dynamical system and Reaction Diffusion problems; morphogenesis.

Steven H. Strogatz,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering
CRC Press, 2018

https://www.google.it/books/edition/Nonlinear_Dynamics_and_Chaos/1kpnDwAAQBAJ?hl=it&gbpv=0

A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952

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

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

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

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

INTRODUCTION TO STATISTICAL MECHANICS AND GIBBS STATES
– Review of equilibrium thermodynamics. Convex functions and Legendre transform.
– Models of statistical mechanics: microcanonical, canonical and grandcanonical ensembles. Gibbs states.
– Models of lattice gases and Ising spins. The theorem of existence of thermodynamic limit for Ising models. Equivalence of the ensembles.
– The structure of Gibbs states. Extremal states and mixtures. The notion of phase transition: loss of analyticity and non-uniqueness of the Gibbs states

THE ISING MODEL
– Known results on the ferromagnetic Ising model in dimensions one or more
– GKS and FKG inequalities. Existence of the infinite volume Gibbs states with + or - boundary conditions
– The one-dimensional Ising model: exact solution via the transfer matrix formalism. Absence of a phase transition and exponential decay of correlations.
– The mean field Ising model (Curie-Weiss model): exact solution. Phase transition and loss of equivalence between canonical free energy and grandcanonical pressure. Connection between the mean field model and the model in dimension d with weak, long-ranged, interactions (Kac interactions): the theorem of Lebowitz-Penrose
- Geometric representation of the 2D Ising model: high and low temperature contours. Existence of a phase transition in the 2D nearest neighbor Ising model: the Peierls argument. Analyticity of the pressure at high temperatures.
– The Lee-Yang theorem
– Existence of a phase transition in the long range 1D Ising model with interaction |x-y|^{-p}, 1

S. Friedli and Y. Velenik: Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge: Cambridge University Press, 2017.
Disponibile online in preprint version su https://www.unige.ch/math/folks/velenik/smbook/index.html

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

Partial Differential Equations
Finite difference approximation for partial differential equations. Semi-discrete approximations and convergence. The Lax-Richtmeyer theorem. Transport equation: the method of characteristics. The "Upwind" (semi-discrete and fully-discrete) scheme, consistency and stability. Heat equation: Fourier approximation. Finite difference scheme, consistency and stability. Poisson equation: Fourier approximation. Finite difference scheme, convergence. (Reference: notes by R. LeVeque, "Finite Difference methods for differential equations", selected chapters 1, 2, 3, 12, 13)

Roberto Ferretti, "Appunti del corso di Analisi Numerica", in pdf at http://www.mat.uniroma3.it/users/ferretti/corso.pdf

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

Lecture slides in pdf at http://www.mat.uniroma3.it/users/ferretti/bacheca.html

Additional notes given by the teacher

Dynamical system and Reaction Diffusion problems; morphogenesis.

Steven H. Strogatz,
Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering
CRC Press, 2018

https://www.google.it/books/edition/Nonlinear_Dynamics_and_Chaos/1kpnDwAAQBAJ?hl=it&gbpv=0

A. Turing, The Chemical Basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, vol. 237, no. 641, 1952

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