# Year 1 in L'Aquila - Interdisciplinary Mathematics

• ## Applied partial differential equations (6 credits)

• ECTS credits 6
• Semester 1
• University University of L'Aquila
• Objectives

Students will know basic of properties (existence, uniqueness, etc.) and techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic PDEs (conservation laws, heat, Laplace, wave equations).

• Topics

Integral curves and surfaces of vector fields. First order partial differential equations. Linear and quasi linear partial differential equations (PDEs) of first order. Method of characteristics. The initial value problem: existence and uniqueness. Development of shocks.

The Cauchy-Kovalevsky theorem. Linear partial differential operators and their characteristic curves and surfaces. Methods for finding characteristic curves and surfaces. The initial value problem for linear first order equations in two independent variables. Holmgren's uniqueness theorem. Canonical form of first order equations. Classification and canonical forms of second order equations in two independent variables. Second order equations in two or more independent variables. The principle of superposition.
The divergence theorem and the Green's identities. Equations of Mathematical Physics.

LAPLACE'S EQUATION AND HARMONIC FUNCTIONS Elementary harmonic functions. Separation of variables. Inversion with respect to circles and spheres. Boundary value problems associated with Laplace's equation. Representation theorem. Mean value property. Maximum principle. Harnack’s inequality and Liouville’s theorem. Well-posedness of the Dirichlet problem. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson's integral. Analytic functions of a complex variable and Laplace's equation in two dimensions. The Neumann problem.

GREEN'S FUNCTIONS. Solution to the Dirichlet problem for a ball in three dimensions. Further properties of harmonic functions. The Dirichlet problem in unbounded domains. Method of electrostatic images.

THE WAVE EQUATION. Cauchy problem. Energy method and uniqueness. Domain of dependence and range of influence. Conservation of energy. One-dimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multi-dimensional wave equation. Well posed problems. Fundamental solution (n=3) and strong Huygens’ principle. Kirchhoff formula. Method of descent. Poisson?s formula (n=2). Wave propagation in regions with boundaries. Uniqueness of solution of the initial-boundary value problem. Separation of variables. Reflection of waves.

THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initial-boundary value problem for the one dimensional heat equation. Method of separation of variables. The initial value problem for the one dimensional heat equation. Fundamental solution. Non homogeneous case and Duhamel’s method. Heat conduction in more than one space dimension.

• Books

E. C. Zachmanoglou and Dale W. Thoe, lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc.. 1986. ISBN 0-486-65251-3

L.C. Evans, Partial Differential Equations. American Mathematical Society. 2010. Second edition, ISBN-13: 978-0821849743

S. Salsa, Partial Differential Equations in Actions: from Modelling to Theory. Springer-Verlag Italia. 2008. ISBN 978-88-470-0751-2

W. A. Strauss, Partial Differential Equations, Student Solutions Manual: An Introduction. John Wiley & Sons, LTD. 2008. Second edition, ISBN-13: 978-0470260715

W. A. Strauss, Partial Differential Equations: an introduction. John Wiley & Sons, LTD. 2007. Second edition, ISBN-13 978-0470-05456-7

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• ## Dynamical Systems and Bifurcation Theory (6 credits)

• ECTS credits 6
• Semester 1
• University University of L'Aquila
• Prerequisites

Ordinary Differential Equations

• Topics

Linear systems of differential equations: uncoupled linear systems, diagonalization, exponentials of operators, the fundamental theorem for linear systems, planar linear systems, complex eigenvalues, multiple eigenvalues, stability theory, nonhomogeneous linear systems.

Local theory of nonlinear systems: initial value problem, hyperbolic equilibrium point, Stable Manifold Theorem. Hartman-Grobman Theorem. Stability and Liapunov functions. Saddles, nodes, foci and centers. Nonhyperbolic critical points. Center manifold theory.

Global theory of nonlinear systems: limit set, attractor, limit cycle, Poincaré map, stable manifold theorem for periodic orbits, Poincaré-Bendixson theory. Mathematical background: Fundaments of perturbation analysis. The Multiple Scale Method. Basic concepts of bifurcation analysis: Bifurcation points, Linear codimension of a bifurcation, Imperfections, Fundamental path, Center Manifold Theory.

Basic mechanisms of multiple bifurcations: divergence, Hopf, nonresonant or resonant double-Hopf, Divergence-Hopf, Double-zero bifurcation.

• Books

Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001

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• ## Functional analysis (6 credits)

• ECTS credits 6
• Semester 1
• University University of L'Aquila
• Objectives

Learn the fundamental structures of Functional Analysis.

Get familiar with the main examples of functional spaces, in particular with the theory of Hilbert spaces and Lebesgue spaces.

Get familiar with the basic notions of operator theory. Be able to frame a functional equation in an abstract functional setting.

• Topics

Lebesgue Measure and Integration.

L^p Spaces.

Basic of Topological Vector Spaces, Normed and Banach Spaces, Linear Operators and linear functionals.

Hilbert Spaces.

Weak topology, Weak * topology, weak compactness.

Applications of Baire Category in Functional Analysis: Uniform Boundedness, Open Mapping, Closed Graph, Inverse Mapping.

Compact Operators.

Riesz Fredholm spectral theory.

• Books

Terence Tao, An introduction to measure theory.. American Mathematical Society, Providence, RI, ISBN: 978-0-8218-6919-2 . 2011.

Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations.. Universitext. Springer, New York,. 2011. xiv+599 pp. ISBN: 978-0-387-70913-0

Alberto Bressan, Lecture notes on functional analysis. With applications to linear partial different. Graduate Studies in Mathematics, 143. American Mathematical Society, Providence, RI,. 2013. xii+250 pp. ISBN: 978-0-8218-8771-4.

Michael Reed, Barry Simon, Methods of modern mathematical physics. I. Functional analysis. Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York,. 1980. xv+400 pp. ISBN: 0-12-585050-6.

Stein, Elias M.; Shakarchi, Rami , Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III. Princeton University Press, Princeton, NJ,. 2005. xx+402 pp. ISBN: 0-691-11386-6

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• ## High performance computing laboratory and applications to differential equations (6 credits)

• ECTS credits 6
• Semester 1
• University University of L'Aquila
• Topics

Linux/Unix OS and tools;

Basic Fortran (or C);

HPC architecture and libraries;

Application (ex ODEs, PDEs, elastodynamics).

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• ## Italian Language and Culture for foreigners (level A1) (3 credits)

• ECTS credits 3
• Semester 1
• University University of L'Aquila
• Objectives Students will reach a basic level of both written and spoken Italian (A1 level according to CEFR), and will acquire a smattering of Italian culture.
• Topics Greetings and introductions. Expressing likes and dislikes. Talking about daily activities. Understanding and using everyday expressions as well as basic phrases related to daily needs (buying something, asking for directions, ordering a meal). Interacting in a very simple way about known topics (family, nationality, home, studies). Italian gestures. Italian geography. Introduction to the most important Italian cities. Italian food.
• Books Nuovo Espresso 1, Alma Edizioni, ISBN: 9788861823181

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• ## Combinatorics and cryptography (6 credits)

• ECTS credits 6
• Semester 2
• University University of L'Aquila
• Objectives

The student will be requested to have a good preparation on the presented topics, and to be able to implement some of the algorithms in a programming language

• Topics

Abstract: Basic cryptograpy and coding theory will be developed. Some protocols and algorithms will be discussed focusing an security and data integrity.

Programme: Elementary arithmetics: Integers, divisibility, prime numbers, Euclidean division and g.c.d., Congruence classes, Chinese remainder theorem, cyclic and abelian groups, Lagrange theorem, Euler theorem, the structure of invertible classes mod p^n, Fields with p elements, polynomials, Euclidean division and g.c.d., Congruence classes of polynomials, Finite fields, primitive elements and polynomials, Legendre/Jacoby symbols and quadratic reciprocity. Cryptography: Classical cryptosystems: Shift cyphers, Vigenère Chipher, Substitution Chiper, One time pads, LFSR Data Encryption Standard: Simplified DES and differential cryptanalysis, Attacks, password encryption RSA: the algorithm, Attacks, Primality testing, the public key concept. Discrete logarithms: Bit commitment, Diffie-Helman Key exchange, ELGAMAL Hash function: SHA, birthday attacks Digital signatures: RSA signatures, Hashing and signing, DSA Error correcting codes: Binary block codes, distance and correction of errors, classical bounds, linear codes, cyclic codes, Hamming codes, BCH and Reed-Solomon codes.

• Books

[1] Wade Trappe, Lawrence C. Washington, Introduction to cryptography: with coding theory 2nd ed.. Pearson Prentice Hall. 2006.

[2] https://www.disim.univaq.it/didattica/content.php?corso=424&pid=88&did=0&lid=en

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• ## Foundations of advanced geometry (6 credits)

• ECTS credits 6
• Semester 2
• University University of L'Aquila
• Objectives

The goal of this course is to provide  the motivations, definitions and techniques for the translation of  topological problems into algebraic ones, hopefully easier to deal with. On successful completion of this module, the student should understand the fundamental concepts of algebraic geometry and should be aware of potential applications of algebraic topological invariants in other fields as theoretical physics , including the computational fluid mechanics and electrodynamics.

• Topics

General topology: topological spaces and continuous maps, induced, quotient and product topology, metric spaces, Hausdorff spaces, compact spaces, connected spaces, paths and path connected spaces

Manifolds and surfaces: the pancake problems, n-dimensional manifolds, surfaces and classification of surfaces.

Homotopy: Retracts and contractible spaces, paths and multiplication, the fundamental group, the fundamental group of the circle.

Covering spaces: the fundamental group of a covering space, the fundamental group of a orbit space, lifting theory and existence theorems, the Borsuk-Ulam theorem, the Seifert-Van Kampen theorem, the fundamental group of a surface.

Introduction to singular homology: standard and simplicial simplexes.

• Books

Czes Kosniowski, A first course in algebraic topology. Cambridge University Press. 1980.

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• ## Complex Analysis (6 credits)

• ECTS credits 6
• Semester 2
• University University of L'Aquila
• Prerequisites

Knowledge of all topics treated the Mathematical Analysis courses in the first and second year: real function of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations.

• Objectives

Knowledge of basic topics of complex analysis: elementary functions of complex variable, differentiation, integration and main theorems on analytic functions.

Ability to use such knowledge in solving problems and exercises

• Topics

Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Armonic functions.

Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liuville's theorem.

Series representation of analytic functions. Taylor's theorem. Laurent's series and classification of singularities.

Calculus of residues. The residue theorem. Application in evaluation of integrals. Rouche's theorem.

Conformal mappings. Main theorems. Fractional linear transformations.

Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.

Laplace transform and applications.

• Books

J.E. Marsden, M.J. Hoffman, Basic complex analysis. Freeman New York.

W. Rudin, Real and complex analysis. Mc Graw Hill.

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• ## Stochastic processes (6 credits)

• ECTS credits 6
• Semester 2
• University University of L'Aquila
• Objectives

The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known classes of stochastic processes (such as martingales, markov processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems with randomness evolving in time in terms of a probability model and to analyze it charcterizing some of its properties.

• Topics

Stochastic Processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples.

Filtrations, stopping times, conditional expectation.

Markov processes: definition, main properties and examples. Birth and death processes.

Poisson process with applications on queueing models.

Martingales: definition, main properties and examples. Branching processes.

Brownian motion: definition, construction and main properties.

Brownian Bridge, Geometric Brownian Motion, Ornstein-Uhlenbeck process.

Ito integral and stochastic differential equations. Applications and examples.

• Books

P. Billingsley, Probability and measure. John Wiley and Sons.

G. Grimmett, D. Stirzaker, Probability and random Processes. Oxford University Press.

B. Oksendal, Stochastic Differential Equations. Springer-Verlag.

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• ## Numerical methods for differential equations (6 credits)

• ECTS credits 6
• Semester 2
• University University of L'Aquila
• Objectives

Goals of the course:
Give the mathematical instruments to handle with optimization problems and differential
equations. The course consists of 6 credits and lasts 60 hours.

Expected learning results:
Being able to solve numerically, both for the theoretical aspects and for the implementation
issues, general problems arising in differential modeling.

• Topics

Numerical methods for the Cauchy problem. Ons step methods. Stability theory.
Stiff problems and differential-algebraic problems. Numerical methods for boundary value problems.
Numerical methods for elliptic and parabolic PDEs.

• Books

E. Hairer, S.P. Norsett and G. Wanner, Solving ordinary differential equations. I.
Nonstiff problems. Second edition. Springer Verlag.
E. Hairer and G. Wanner, Solving ordinary differential equations. I.
Stiff and differential-algebtraic problems. Second edition. Springer Verlag.
P. Henrici, Discrete variable methods in ordinary differential equations. Ed. John Wiley.
J.D. Lambert, Computational methods in ordinary differential equations. Ed. John Wiley.

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• ## Data analytics and Data driven decision (6 credits)

• ECTS credits 6
• Semester 2
• University University of L'Aquila

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• ## Italian Language and Culture for foreigners (level A2) (3 credits)

• ECTS credits 3
• Semester 2
• University University of L'Aquila
• Objectives Students will reach an elementary level of both written and spoken Italian (A2 level according to CEFR).
• Books Italian Espresso 1, Alma Edizioni, ISBN: 9788889237212. Civiltàpuntoit, di Marco Mezzadri e Linuccio Pederzani, Guerra Edizioni, ISBN: 9788855700160.

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#### Programme Structure

»Year1 L'Aquila: IntMat | SciCom
»Year2 L'Aquila: IntMat | SciCom
»Year2 BrnoMath. Engineering
»Year2 Katowice: MatMod | MatFin
»Year2 LvivApplied Maths

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