Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.

Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.

Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo-Nirenberg-Sobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.

Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.

First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.

V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc..

C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer.

L.C. Evans, Partial Differential Equations. AMS.

M.E. Taylor, Partial Differential Equations, Nonlinear equations. Springer.

H. Brezis, Sobolev Spaces and Partial Differential Equations. Springer.

A good knowledge  of the basic arguments of a course of Functional Analysis, in particular,  a good knowledge of the theory of Lebesgue's integral and the L^p spaces.

The first module of the course, in particular a good knowledge of the theory of distributions and Sobolev spaces.

Aim of the course is the  knowledge of advanced techniques of  mathematical analysis  and in particular the basic techniques of the modern theory of  the  partial differential  equations.

Abstract Measure theory.

AC and BV functions.

Fourier transforms.

Second order elliptic equations.

Variational methods.

Basics of Algebra

The course aims to provide the arithmetical and algebraic background and the basic techniques for symmetric cryptography, public-key cryptography and error correction coding.
At the end of the course the student should be able to understand the fundamental concepts of modular arithmetic and finite fields and to be able to apply them to the study of basic cryptographic techniques and basic error correcting codes described during the course.
On successful completion of this course, the student should
1) have knowledge of the basic techniques of cryptography and error correction codes introduced;
2) understand the fundamental concepts of arithmetic and algebra and their interactions and be aware of their applications in cryptography and coding theory;
3) have knowledge of how to apply the notions of arithmetic and algebra to the study of cryptographic techniques and error correction codes;
4) understand and analyze the mathematical and application problems underlying the cryptographic schemes studied;
5) demonstrate skill in reasoning and arithmetic calculation and ability to understand the proofs of the theoretical and cryptographic results studied;
6) demonstrate ability to read and understand other scientific texts on related subjects.

Topics of the module include:
Overview of Cryptography and attack scenarios.

Elementary arithmetics: Integers, divisibility, prime numbers, Euclidean division and g.c.d., Bezout's Identity, Eucledian Algorithm, Extended Eucledian Algorithm, Congruence classes, Chinese remainder theorem, cyclic and abelian
groups, Lagrange theorem, Fermat's Little Theorem, Euler theorem, the structure of invertible classes mod N, Fields with p elements, Primitive Roots, polynomials, Euclidean division and g.c.d., Congruence classes of polynomials, Finite fields, primitive elements and polynomials.

Introduction to Probability. Probability and Ciphers, Introduction to Shannon Theory, Perfect secrecy, Shannon Theorem, one time pad, Substitution Ciphers.

Symmetric Cryptography, Feistel Networks, Substitution Permutation Networks, Advanced Encryption Standard - Rijandel.

Group generated by a round functions and Imprimitive attack.

Differential cryptanalysis, example of differential cryptanalysis on a small variant of PRESENT.

Public-key Cryptography, Discrete logarithms problem (DLP), Computational Diffie-Hellmann Problem (DHP), between DLP and DHP, Diffie-Helman Key exchange.

RSA Algorithm, Trial Division, Fermat's test, Miller Rabin Test, AKS primality test, Factoring and factoring-related problems (SQRROOT and RSA Problem), Security of RSA, Coppersmith Theorem, Hastad Attack, Wiener Attack.

Hash function, Digital signatures, RSA signatures, Hashing and signing, DSA.

Error correcting codes, Binary block codes, distance and correction of errors, singleton bound, Hamming bound, Gilbert-Varshamov bound, linear codes, Syndrome decoding, dual codes, Hamming codes, Simplex codes,cyclic codes, Reed-Solomon codes.

1) Trappe and Washington, "Introduction to Cryptography with Coding Theory", second edition, Pearson Pretince Hall, 2006;
2) Smart, "Cryptography made simple", Information Security and Cryptography, Springer, 2016;
3) Heys, "A Tutorial on Linear and Differential Cryptanalysis",
https://www.engr.mun.ca/~howard/PAPERS/ldc_tutorial.pdf

Aim of the course is to present some mathematical models currently used in the analysis of collective phenomena, such as vehicular and pedestrian traffic, and flocking phenomena. Emphasis will be given to the mathematical treatment of specific problems coming from real world applications.

Macroscopic traffic models. LWR model, its derivation. Fundamental diagrams. The Riemann problem, examples. Second order models for traffic flow: Payne-Whitham model, description, drawbacks; Aw-Rascle model, shocks description, domains of invariance, instabilities near vacuum.

Theory: systems of conservation laws, strict hyperbolicity, Rankine-Hugoniot conditions; Lax admissibility condition. The Riemann problem for systems: the linear case; GNL and LD fields; rarefactions and contact discontinuities. BV functions, examples and properties. A compactness theorem.

Wave front tracking algorithm: approximate rarefactions, possible types of interactions. Bounds on number of waves and on total variation. Compactness of approximate solutions. The initial-boundary value problem on the half line: boundary Riemann problem, interactions with the boundary, control of the total variation by means of a Lyapunov-type functional. The Toll gate problem.

Networks, basic definitions, conservation of the flux. Examples. Distributions along the roads, maximization of the flux. Riemann problem on a junction composed by 2 incoming roads and 2 outgoing roads. The case of 2 incoming roads and 1 outgoing road: the "right of way" rule. Junction between one incoming and one outgoing road, different fluxes.

Pedestrian flow: normal and panic situation. Macroscopic models for evacuation, conservation of "mass", eikonal equation. The Hughes model for pedestrian flow. The eikonal equation: non uniqueness, viscosity solutions, relation with vanishing viscosity approximation. The Hughes model in one space dimension. Curve of turning points, Rankine-Hugoniot conditions. The case of constant initial density and of symmetric initial data; conservation of the left and right mass; an example with mass exchange across the turning point. Macroscopic models for pedestrian flow that include: knowledge of a preferred path, discomfort from walking along walls, tendency of avoiding high densities of pedestrian in a neighborhood (nonlocal term of convolution type), angle of vision, obstacle in the domain. Linearized stability around a smooth solution.

Introduction to the theory of flocking. Examples: Krause model for opinion dynamics, Cucker-Smale model, model for attraction-repulsion phenomena. The Cucker-Smale flocking model: basic properties, estimates on the kinetic energy. A "flocking theorem": proof by bootstrapping method (Ha and Tadmor). Some drawbacks of the model. Introduction to the kinetic limit for flocking: the N-particle distribution function, Liouville equation, marginal distribution, continuity equation. The formal mean-field limit: a Vlasov-type equation.

M.D. Rosini, Macroscopic models for vehicular flows and crowd dynamics: theory and applications. Springer. 2013. http://link.springer.com/book/10.1007/978-3-319-00155-5/page/1

M. Garavello, B. Piccoli, Traffic flow on networks. Conservation laws models. AIMS Series on Applied Mathematics. 2006. http://www.aimsciences.org/books/am/AMVol1.html

Probability theory and Real Analysis

Students should:
1. Develop the skills to model simple real problems and propose a solution;
2. Solve theoretical problems, using the appropriate mathematical tools;
3. Read the related texts and gain access to more advanced courses;
4. Get a first flavour of the relevant research problems.

1. Discrete time processes: Markov chains in finite and countable space, limiting distribution;
2. Continuous time processes: density and distribution of into-event time for Poisson process, applications and extensions: e.g. birth-and-death processes, queues, epidemics;
3. Renewal processes: ordinary renewal process, renewal theorem, equilibrium
renewal process, application to queues;
4. Wiener processes and basic stochastic calculus: basic definitions and properties, It\^o's formula, Stochastic Differential Equations.

1. Markov Chains, J.R. Norris, Cambridge University Press;
2. Introduction to Stochastic Processes, G. Lawler, Chapman & Hall;
3. Basic Stochastic Processes, A Course Through Exercises, Z. Brzezniak and T. Zastawniak, Springer;
4. Probability and Random Processes, G. Grimmett and D. Stirzaker, 3rd Edition, Oxford University Press;
5. A first look at Rigorous Probability Theory, J. Rosenthal, World Scientific.

The course is an introduction to Time Series Analysis and Forecasting. The level is the first-year graduate in Mathematics with a prerequisite knowledge of basic inferential statistical methods.

The aim of the course is to present important concepts of time series analysis (stationarity of stochastic processes, ARIMA models, forecasting etc.). At the end of the course, the student should be able to select an appropriate ARIMA model for a given time series.

Stochastic processes (some basic concepts)

Stationary stochastic processes

Autocovariance and autocorrelation functions

Ergodicity of a stationary stochastic process

Estimation of moment functions of a stationary process

ARIMA models

Estimatiom of ARIMA models

Building ARIMA models

Forecasting from ARIMA models

[1]Time Series Analysis Univariate and Multivariate Methods, 2nd Edition, W. W. Wei, 2006, Addison Wesley.

[2] Time Series Analysis, J. Hamilton, 1994, Princeton University Press.

[3] Time Series Analysis and Its Applications with R Examples, Shumway, R. and Stoffer, D., 2006, Springer.

[4]Introduction to Time Series and Forecasting. Second Edition, P. Brockwell and R. Davis, 2002, Springer.