# Year 2 in L'Aquila - Interdisciplinary Mathematics

• 2 Year
• Interdisciplinary Mathematics Pathway
• University of L'Aquila Place
• 60* ECTS Credits
• Not available if you spent your Year 1 in L'Aquila Note

#### List of course units

*Students are required to earn 60 ECTS credits, at least, during their second year by successfully attending the following compulsory course units (Semester 1 and 2 amounting to 48 ECTS credits) and picking other 12 ECTS credits (minimum) from the elective ones listed below.

• ## Advanced Analysis 1 (6 credits)

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

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.

• Objectives

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.

• Topics

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.

• Books

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.

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• ## Mathematical fluid dynamics (6 credits)

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

Basic notions of functional analysis, functions of complex values, standard properties of the  heat equation, wave equation, Laplace and Poisson's equations.

• Objectives

This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to  the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a  qualitative and quantitative analysis of  solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.

• Topics

Derivation of the governing equations: Euler and Navier-Stokes.

Eulerian and Lagrangian description of fluid motion; examples of fluid flows.

Vorticity equation in 2D and 3D.

Dimensional analysis: Reynolds number, Mach Number, Frohde number.

From compressible to incompressible models.

Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.

Existence of solutions for viscid and inviscid fluids.

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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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• ## Advanced Analysis 2 (6 credits)

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

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.

• Objectives

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.

• Topics

Abstract Measure theory.

AC and BV functions.

Fourier transforms.

Second order elliptic equations.

Variational methods.

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• ## Master's thesis (UAQ) (24 credits)

• ECTS credits 24
• Semester 2
• University University of L'Aquila
• Objectives
The topic of the thesis can be proposed to the student by the local InterMaths coordinator or by the student him/herself. In any case, the InterMaths executive committee is the responsible to approve the thesis project before its formal start. The taste and expectations of the students are respected whenever possible. The local InterMaths coordinator in the hosting institution is the responsible to provide an academic advisor to the student, although proposals from the students will always be heard in this respect.

In some cases, after the agreement with the local InterMaths coordinator, the thesis topic can be related to a problem proposed by a private company. In this case, a tutor will be designated by the company as responsible person of the work of the student, especially if he/she is eventually working in the facilities of the company; however, the academic advisor is, in any case, the responsible to ensure the progress, adequacy and scientific quality of the thesis. The necessary agreements between the university and the company will be signed in due time, according to the local rules, in order that academic credits could be legally obtained during an internship, and the students be covered by the insurance against accidents outside the university.

NOTE: Although the thesis is scheduled for the 4th semester, some preliminary work may be anticipated due to the local rules - such as preliminary local courses in the 3rd semester, ensuring that the student can follow the main courses of the 3rd semester without problems. In this point, the personalised attention to the students has to be intensified, and decisions taken case by case.

• More information Students work on their Master's Thesis over the 4th semester following their agreement with their thesis advisor.

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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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• ## Mathematical models for collective behaviour (6 credits)

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

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.

• Topics

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.

• Books

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

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• ## Biomathematics (6 credits)

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

Basic calculus and analysis (differential and integral calculus with functions of many variables).

Ordinary differential equations.

Basics in finite dimensional dynamical systems.

Elementary methods for the solution of linear partial differential equations (separation of the variables).

• Objectives

1) To learn the basics in the mathematical modelling of population dynamics.

2) To provide a mathematical description of ODE models in population dynamics and the intepretation of the qualitative behaviour of the solutions.

3) To get the basic notions in mathematical models in epidemiology and reaction kinetics.

4) To learn the mathematical modelling of population models in heterogeneous environment, described by partial differential equations.

5) To deal with advanced models in biology such as chemotaxis models and structured dynamics equations.

6) To get a sound background in reaction diffusion phenomena, Turing instability, and pattern formation.

• Topics

Continuous Population Models for Single Species. Continuous Growth Models. Delay models. Linear Analysis of Delay Population Models: Periodic Solutions.

Continuous models for Interacting Populations. PredatorPrey Models: Lotka-Volterra Systems. Realistic Predator–Prey Models. Competition Models: Principle of Competitive Exclusion. Mutualism or Symbiosis.

Reaction kinetics. Enzyme Kinetics: Basic Enzyme Reaction. Transient Time Estimates and Nondimensionalisation. Michaelis-Menten Quasi-Steady State Analysis.

Dynamics of Infectious Diseases: Epidemic Models and AIDS. Simple Epidemic Models (SIR, SI) and Practical Applications. Modelling Venereal Diseases. AIDS: Modelling the Transmission Dynamics of the Human Immunodeficiency Virus (HIV).

Time-space dependent models: PDEs in biology. Diffusion equations. Diffusion and Random walk. The gaussian distribution. Smoothing and decay properties of the diffusion operator. Nonlinear diffusion.

Reaction–diffusion models for one single species. Diffusive Malthus equation and critical patch size. Travelling waves. Fisher–Kolmogoroff equation.

Reaction–diffusion systems. Multi species waves in Predator-Prey Systems. Turing instability and spatial patterns.

Chemotaxis modelling. Diffusion vs. Chemotaxis: stability vs. instability. Diffusion vs. Chemotaxis: stability and blow–up. Chemotaxis with nonlinear diffusion. Models with maximal density.

Nonlocal interaction models in biology. Mathematical models of swarms. Approximation with interacting particle systems. Asymptotic behaviour.

Structured population dynamics. An example in ecology: competition for resources. Continuous traits. Evolutionary stable strategy in a continuous model.

• Books

James D. Murray, Mathematical Biology I: an introduction. Springer.

James D. Murray, Mathematical Biology II: Spatial models and biomedical applications . Springer.

Benoit Perthame, Transport equations in biology. Birkaeuser.

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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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• ## Kinetic and hydrodynamic models (6 credits)

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

Mathematical Analysis, Fourier transform.

• Objectives

This course provides an introduction to the classical kinetic theory of gases and the principles of kinetic modeling.

A special focus is given to the derivation of hydrodynamic equations from kinetic models by means of non-perturbative techniques and to the analysis of numerical schemes for the simulation of fluid flows.

On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.

• Topics

Boltzmann equation and the principles of kinetic description.

Kinetic models: BGK,Maxwell molecules, Vlasov equation and Fokker-Planck equation.

The closure problem and methods of reduced description: Chapman-Enskog expansion, Grad's Moment method.

Non-perturbative techniques in kinetic theory: the method of the slow invariant manifold.

Overview on Lattice Boltzmann models.

Monte Carlo simulations of lattice gas models.

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• ## Time series and prediction (6 credits)

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

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.

• Topics

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

• Books

[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.

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• ## Mathematical economics and finance (6 credits)

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

I assume familiarity with vector and topological spaces, and with the standard model of the real numbers. I assume that you know the basic facts about metric spaces, normed and seminormerd spaces, Banach and Hilbert spaces.

• Objectives

On successful completion of this course, the student should:

- Know the fundamental fixed point theorems for set-valued maps and the basic existence results for equilibrium problems and variational inequalities.

- Explain some interconnections among these various results.

- Apply this analysis to game and economic theory

• Topics

Sperner’s lemma

The Knaster-Kuratowski-Mazurkiewicz lemma

Brouwer's fixed point theorem

Variational inequalities and equilibrium problems

Generalized monotonicity and convexity

Brézis-Nirenberg-Stampacchia theorem and Fan's minimax principle

Continuity of correspondences

Browder, Kakutani and Fan-Glicksberg fixed point theorems

Gale-Nikaido-Debreu theorem

Nash equilibrium of games and abstract economies

Walrasian equilibrium of an economy

An application to traffic network

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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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