MathMods :: Joint MSc

Sem3 UAQ-GSSI Social S.

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Applications  @  UAQ - GSSI  30 ECTS credits

Mathematical models in social sciences

The specialization track “Mathematical models in social science” will take place at the University of L’Aquila in collaboration with the Gran Sasso Science Institute (GSSI) in L'Aquila. The proposed curriculum covers newly developed mathematical modelling approaches, aiming at realizing practical solutions for smart and healthy communities. UAQ has longstanding collaborations with the Imperial College of London (UK) and with the University of Muenster (Germany) on many-agent modelling and mean-field limits in several branches of social sciences such as opinion formation, emergence of collective motion, and the social behaviour of largely crowded communities (the latter also in collaboration with the Universities of Catania and Ferrara, in Italy); the UAQ team has a longstanding expertise on "nonlocal aggregation models", an emerging subject in applied mathematical research with applications in particular to the modelling of criminal behaviour in urban areas. Important achievements have been also carried out at UAQ in the area of nonlinear conservation laws with applications to traffic modeling and population dynamics. The crucial contribution of GSSI concerns with classical and modern topics in fluid dynamics and real analysis providing the student with a complementary expertise in advanced mathematical theory required in some of the other courses of this semester.

This track has three compulsory courses: “Advanced analysis” (in collaboration with GSSI), providing tools to deepen the understanding of advanced nonlinear models; “Mathematical fluid dynamics” (in collaboration with GSSI), featuring advanced analytical methods on classical and modern fluid-dynamical theories; “Mathematical model for collective behavior”, presenting a selection of topics in the modeling of collective motion in social sciences and of transport modeling in real world applications, particularly in urban and behavioural sciences.

Two optional sub-paths can be undertaken to complete the semester. The first one specialises the student on "modelling in life sciences", a subject sharing several methodologies with social science modeling, which has become a classic topic in modern applied mathematics. Such sub-path consists of the "Biomathematics" and "System biology" (in collaboration with the IASI-CNR institute) courses, both presenting mathematical a modeling approach via ODEs, PDEs, and multiscale modelling, in contexts arising from modern applications such as population biology, medicine, and genetics. The second sub-path is focused on "modelling in seismology", a topic of great interest for the UAQ team after the earthquake that struck the Abruzzo Region in 2009. A selection of topics mostly related with geo-dynamics and seismic wave propagation will be presented in the course "Modelling seismic wave propagation". The course "Time series and prediction" will complete this sub-path with additional expertise in statistical methods which are essential to analyse earthquake catalogues and seismic event sequences. The INGV (Italian Institute of Geophysics and Volcanology, the reference Italian public institution for seismic events) will offer its collaboration in the offer of this course. The group of Andrea Bertozzi at UCLA (associated partner) gives the opportunity for a master thesis on mathematical models for criminal behavior. The INGV will offer collaborations on research projects and thesis with impact on the seismical activity around the Abruzzo Region.

 

Below you can find information about the subjects for this semester.

  • Advanced analysis 1 [6 credits]

    Advanced analysis 1

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      The main objectives of the course are as follows: to provide knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics; to apply this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about nonlinear equations (traffic flow, gas dynamics, fluid dynamics).

    • Topics

       

      Measure and integration theory. Functions of bounded variation. Distributions theory. Fourier transform. Sobolev spaces. Application to the study of partial differential equations: elliptic equations of second order, parabolic equations of second order, hyperbolic systems of first-order equations, nonlinear conservation laws. An outline of semigroup theory.


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  • Mathematical models for collective behaviour [6 credits]

    Mathematical models for collective behaviour

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

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

    • Topics

       

      Macroscopic traffic models: LWR model, fundamental diagrams. The Riemann problem. Point-type constraints (the "Toll gate" problem). Second order models for traffic flow: Aw-Rascle model, shocks description, instabilities near vacuum. Basics on the theory of systems of conservation laws. Junction of roads, networks. Distribution rules along the roads, optimization of the flux. Solution to the Riemann problem at a junction. Pedestrian flow: normal and panic situation. Macroscopic models, conservation of "mass", eikonal equation. The Hughes model for pedestrian flow; in one space dimension: curve of turning points, Rankine-Hugoniot conditions. 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. Introduction to the theory of flocking. Cucker-Smale model for flocking, conservation of moment and decay of the kinetic energy. Asymptotic flocking. Singular communication rate. The kinetic limit. Introduction to synchronization: Kuramoto model, basic properties.


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  • Mathematical fluid dynamics [6 credits]

    Mathematical fluid dynamics

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • 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 with a particular attention to biofluid dynamics. At the end of the course the 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 the 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. Existence of solutions for viscid and inviscid fluids. Modeling for biofluids mechanics: hemodynamics, cerebrospinal fluids, animal locomotion: swimming and flying; Bioconvection for swimming microorganisms.


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Pick 1 sub-path

a) Modelling in Life Science

  • Biomathematics [6 credits]

    Biomathematics

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

      Aim of the course is to provide basic and advanced tools needed to provide a differential-equations-based model for a selected set of applied settings from population biology, epidemiology, chemistry, cell biology and similar topics. Moreover, the course aims to provide the methods to partly solve the proposed models or to detect their qualitative behaviour. The methods proposed include energy estimates, viscous approximation methods, compactness methods. Finally, the mathematical (analytical) results are complemented with suitable comments and interpretations.

    • Topics

       

      ODE models. Single species population models. ODEs with delay. Dynamical systems. Multi species models: predator-prey, competition, mutualism. Models in epidemiology. SIR and XSI models. Modelling AIDS. Modelling reaction kinetics. Michaelis-Menten approach. Singular limits. PDE models. Diffusion equations. Nonlinear diffusion, existence of solutions and asymptotic behavior. Reaction diffusion equations. Threshold phenomena and travelling waves. Reaction diffusion systems. Turing instability. Models for chemotaxis. Blow-up of solutions. Models for structured populations.


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  • Systems biology [6 credits]

    Systems biology

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      The systems biology course aims at providing mathematical models helping to understand the dynamic interactions occurring within and between cells. This course gives the mathematical tools to model and analyze gene transcription networks and enzymatic reactions. To this end, both deterministic and stochastic approaches are exploited.

    • Topics

       

      Transcription networks (TNs): graph properties and ODE approach to model gene regulation. Network motifs and their biological functions: speed up of the response time, increase robustness, produce oscillations. Enzymatic reactions: Reaction Rate Equations and Quasi Steady-State Approximations (QSSA). Master Chemical Equations: mathematical development and stochastic numerical simulations. The Gillespie algorithm.


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b) Modelling in Seismology

  • Modelling seismic wave propagation [6 credits]

    Modelling seismic wave propagation

    • ECTS credits 6
    • Semester 3
    • University University of L'Aquila
    • Objectives

       

      This is an interdisciplinary course involving geophysicists and applied mathematicians. Its main objective is to introduce some basic theoretical and numerical tools needed to understand and simulate the propagation of seismic waves.

    • Topics

       

      The course is ideally divided into three parts. In the first one we introduce general concepts of earth structure, seismic sources, and earthquake detection. In the second part we summarize the basic concepts of elasticity theory and present some mathematical model often used to describe the propagation of the seismic waves. Finally, in the last part, we introduce some numerical methods and apply our knowledge to simulate seismic waves propagation on a simple domain.


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  • Time series and prediction [6 credits]

    Time series and prediction

    • ECTS credits 6
    • Semester 3
    • 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. Estimation of ARIMA models. Building ARIMA models. Forecasting from ARIMA models

    • Books

       

      Time Series Analysis Univariate and Multivariate Methods, 2nd Edition, W. W. Wei, 2006, Addison Wesley.
      Time Series Analysis, J. Hamilton, 1994, Princeton University Press.
      Time Series Analysis and Its Applications with R Examples, Shumway, R. and Stoffer, D., 2006, Springer.
      Introduction to Time Series and Forecasting. Second Edition, P. Brockwell and R. Davis, 2002, Springer.


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University of L'Aquila, Italy (UAQ)

Department of Information Engineering, Computer Science and Mathematics, via Vetoio (Coppito), 1 – 67100 L’Aquila (Italy)

Autonomous University of Barcelona, Catalonia - Spain (UAB)

Departament de Matemàtiques, Edifici Cc - Campus UAB 08193 Bellaterra – Catalonia

Hamburg University of Technology, Germany (TUHH)

Institute of Mathematics
Schwarzenberg-Campus 3, Building E-10
D-21073 Hamburg - Germany

University of Nice - Sophia Antipolis, France (UNS)

Laboratoire J.A.Dieudonné
Parc Valrose, France-06108 NICE Cedex 2

Vienna Univ. of Technology, Austria (TUW)

Technische Universität Wien
Institute of Analysis & Scientific Computing
Wiedner Hauptstr. 8, 1040 Vienna - Austria

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