Structure 2024

Programme #Structure 2021 ~ 2024

InterMaths Erasmus Mundus Joint Master Degree

Looking for the new 2025 specialisations?

InterMaths students follow the mobility path sketched below:

01

Semester 1

A first semester common to all students on Foundations of Applied Mathematics in L’Aquila

02

Semester 2

A second semester on Numerical – Modelling Training, either in Hamburg or in Vienna

02

Year 2

A second year of Interdisciplinary training in one of the five partner universities, devoted to one of our six specialization tracks:

Modelling and Simulation of Infectious Diseases (L’Aquila)
Computational Methods in Biomedical Imaging (Hamburg)
Stochastic Modelling in Neuroscience (Nice)
Cancer Modelling and Simulation (L’Aquila)
Computational Fluid Dynamics in Industry (Vienna)
Decision Making and Applications to Logistics (Barcelona)

These specialization paths have been designed on grounds of the fields of expertise of the five reference groups, often with links with groups in the same institutions from other applied disciplines. Most importantly, they address innovative methodologies and deal with societal challenges in nowadays society, in particular in medicine and in industry.

The mobility paths are assigned to the new cohort students at the beginning of the 2-year period. We try as much as possible to satisfy the students’ preferences. The mobility scheme is sketched in the following section.

Semester 1 • L'Aquila

Foundations of Applied Mathematics

The first semester in L'Aquila (UAQ) is common to all students. It provides a sound background in applied mathematics based on advanced theoretical subjects such as functional analysis, applied partial differential equations, dynamical systems, continuum mechanics, and control systems.

Semester 2 · Hamburg

Numerical – Modelling training

The second semester at TU Hamburg provides a selected list of subjects in this framework, plus some additional subjects that complement the student’s knowledge to continue with Year 2 studies...

Semester 2 · Vienna

Numerical – Modelling training

The second semester at TU Vienna is entirely devoted to numerical methods, with particular focus on finite elements for ordinary and partial differential equations, numerical optimization, and parallel computing...

Year 2 L'Aquila

Modelling and Simulation of Infectious Diseases

Year 2 Hamburg

Computational Methods in Biomedical Imaging

Year 2 Nice

Stochastic Modelling in Neuroscience

Year 2 L'Aquila

Cancer Modelling and Simulation

Year 2 Barcelona

Decision Making and Applications to Logistics

Year 2 Vienna

Computational Fluid Dynamics in Industry

Semester #1 Cohort #2021~2024 @ UAQ
Foundations of Applied Mathematics;

Cohort

2021~2024

Semester

1

ECTS Credits

30

Campus

University of L'Aquila

The first semester at UAQ is common to all students. It provides a sound background in applied mathematics based on advanced theoretical subjects such as functional analysis, applied partial differential equations, dynamical systems, continuum mechanics, and control systems.

This semester prepares the students to perform simulations in diverse modelling frameworks, as well as to successfully tackle subjects in semester 2 such as advanced numerical calculus, optimization, and stochastic calculus. To perform this task, Semester 1 courses provide a systematic approach to the formulation of applied problems in interdisciplinary fields, and a rigorous approach to mathematical modelling. More precisely, students in this semester are provided “exact” resolution methods for (ordinary and partial) differential equations, the “infinite dimensional” approach of functional analysis (needed in approximation theory, variational calculus, and numerical analysis), a modern and “engineering oriented” approach to control, and an introduction to the mathematical theory of continuum media, a subject that is touched by several specialization tracks.

Teaching staff with longstanding experience with international joint programs in applied mathematics is in charge of this semester. The University of L’Aquila features a research group in mathematical analysis combining three generations of applied mathematicians with excellent international reputation in their field and with an outstanding research record, with main focus on partial differential equations with applications to physics, engineering, social sciences, biology and medicine.

Unit Coordinator: Bruno Rubino

Aims:

The course is intended to introduce and develop an understanding of the concepts in nonlinear dynamical systems and bifurcation theory, and an ability to analyze nonlinear dynamic models of physical systems. The emphasis is to be on understanding the underlying basis of local bifurcation analysis techniques and their applications to structural and mechanical systems.

Content:

Review of: first-order nonlinear ODE, first-order linear systems of autonomous ODE. Local theory for nonlinear dynamical systems: linearization, stable manifold theorem, stability and Liapunov functions, planar non-hyperbolic critical points, center manifold theory, normal form theory. Global theory for nonlinear systems: limit sets and attractors, limit cycles and separatrix cycles, Poincaré map. Hamiltonian systems. Poincaré-Bendixson theory. Bifurcation theory for nonlinear systems: structural stability, bifurcation at non-hyperbolic equilibrium points, Hopf bifurcations, bifurcation at non hyperbolic periodic orbits. Applications.

Pre-requisites:

Ordinary differential equations

Reading list:

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

Unit Coordinator: Donatella Donatelli

Aims:

Learning Objectives:
The aim of the course is 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 the analysis of other partial differential equations.

Learning Outcomes:
On successful completion of this course, the student should:

- understand the basic principles governing the dynamics of non-viscous fluids;
- be able to derive and deduce the consequences of the equation of conservation of mass;
- be able to apply Bernoulli's theorem and the momentum integral to simple problems including river flows;
- understand the concept of vorticity and the conditions in which it may be assumed to be zero;
- calculate velocity fields and forces on bodies for simple steady and unsteady flows derived from potentials;
- demonstrate skill in mathematical reasoning and ability to conceive proofs for fluid dynamics equations.
- demonstrate capacity for reading and understand other texts on related topics.

Content:

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical Modelling of Continuum Media (3 ECTS)

- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Fluidi di tipo Poiseulle e Couette
- Vorticity equation in 2D and 3D

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS),

- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS)

- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.

Pre-requisites:

PREREQUISITES for Mathematical Modelling of Continuum Media:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

PREREQUISITES for Mathematical fluid and biofluid dynamics, Mathematical fluid dynamics, Modelling and analysis of fluids and biofluids:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations, Sobolev spaces.

Reading list:

- Alexandre Chorin, Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer.
- Roger M. Temam, Alain M. Miranville, Mathematical Modeling in Continum Mechanics. Cambridge University Press.
- Franck Boyer, Pierre Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer-Verlag Italia.
- Andrea Bertozzi, Andrew Majda, Vorticity and Incompressible Flow. Cambridge University Press.

Unit Coordinator: Roberta Antonetti

Aims:

The aim of this course is to provide students with knowledge of Italian language at level A1 (beginner). The course will concentrate on four aspects of communication: speaking, writing, listening and reading on a basic level. Attention is given to correct pronunciation of Italian.
Furthermore, the goals of the course are to enable students to:
- develop the language proficiency required to communicate effectively in Italian at level A1;
- develop awareness of the nature of language and language learning;
- develop transferable skills;
- form a sound base of the skills, language and attitudes required for progression to work or further study, either in Italian or another subject area.
On successful completion of this module, a student should be able to:
- understand familiar words and very basic phrases concerning himself/herself, his/her family and immediate concrete surroundings when people speak slowly and clearly;
- read familiar names, words and very simple sentences, for example on notices and posters or in catalogues;
- interact in a simple way provided the other person is prepared to repeat or rephrase things at a slower rate of speech;
- ask and answer simple questions in areas of immediate need or on very familiar topics;
- use simple phrases and sentences to describe where he/she lives and people he/she knows;
- write short messages, postcards, fill in forms with personal details or a hotel registration form.

Content:

Students will be required to show knowledge and understanding of the broad topic areas listed below. These provide contexts for the acquisition of vocabulary and the study of grammar and structures.
VOCABULARY AND TOPIC AREAS
Clothes and accessories; colors; countries, nationalities and languages; culture, customs and celebrations; education e.g. learning institutions, education and training, learning tools, subjects; family and friends; feelings and emotions; food and drink; interests, sports and activities; jobs; measurements e.g. size, shape, weight; numbers (cardinal, ordinal) and money; rooms and furniture; shops and places; street directions; the human body and health e.g. parts of the body, health and illness; time expressions e.g. telling the time, dates, days of the week, months, seasons; travel and transport; weather.

FUNCTIONAL SYLLABUS
Accepting/ thanking; applying for a job; asking about personal information; asking permission; buying and asking prices; describing people and objects; giving opinions; giving instructions; greeting and introducing; inviting, refusing; making suggestions; requesting, offering; talking about likes and dislikes; talking about routines; telling the time.

GRAMMAR SYLLABUS
Sounds and spelling; nouns: gender, number; irregular plural nouns; definite article; indefinite article; adjectives, first group adjectives and second group adjectives, position of adjectives; formal address; pronouns, subject pronouns and direct and indirect object pronouns; possessive adjectives; prepositions simple and prepositions with articles; interrogatives; adverbs; present tense of regular and irregular verbs; modal verbs and the verb sapere; present perfect and some verbs with an irregular past participle.

Pre-requisites:

There are no prerequisites for attending the course.

Reading list:

"Dieci A1" - Alma edizioni, Firenze 2019.
“New Italian grammar in practice”, Alma edizioni, Firenze 2015.

Unit Coordinator: Michele Palladino

Aims:

Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form.

Content:

Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
Introduction to the theory of unbounded operators. Linear differential operators. Applications.
Introduction to infinite-dimensional differential calculus and variational methods.

Pre-requisites:

Basic calculus and analysis in several variables, linear algebra.

Reading list:

John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.

Unit Coordinator: Amadori Debora

Aims:

LEARNING OBJECTIVES.
The course aims at providing basic properties and main techniques to solve basic partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematical Modelling was verified at the time of the planning of the master program.

LEARNING OUTCOMES.
At the end of the course, the student should:

1. know basic properties (existence, uniqueness, etc.) and main techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic partial differential equations (semilinear first order PDEs, heat, Laplace, wave equations);
2. understand and be able to explain thesis and proofs in the field of basic partial differential equations;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

First order partial differential equations. Definition of characteristic vectors and characteristic surfaces. Characteristics for (semi)linear partial differential equations of first order in two independent variables. Existence and uniqueness to initial value problems for first order semilinear partial differential equations in two independent variables Duhamel’s principle for non homogeneous first order partial differential equations.

Second order partial differential equations. Classification of second order semilinear partial differential equations in two independent variables. Canonical form for second order semilinear partial differential equations in two independent variables. Classification for second order semilinear partial differential equations in many independent variables.

Heat equation. Derivation of heat equation and well–posed problems in one space dimension. Solution of Cauchy–Dirichlet problem for one dimensional heat equation by means of Fourier method of separation of variables. Energy method and uniqueness. Maximum principle. Fundamental solution. Solution of global Cauchy problem. Non homogeneous problem: Duhamel’s principle.

Laplace equation. Laplace and Poisson equation: well-posed problems; uniqueness by means of energy method. Mean value property and maximum principles. Laplace equation in a disk by means of separation of variables. Poisson’s formula. Harnack’s inequality and Liouville’s Theorem. Fundamental solution of Laplace operator. Solution of Poisson’s equation in the whole space. Green’s functions and Green’s representation formula.

Wave equation. Transversal vibrations of a string. Well–posed problems in one space dimension. D’Alembert formula. Characteristic parallelogram. Domain of dependence and range of influence. Fundamental solution for one dimensional wave equation. Duhamel’s principle for non homogeneous one dimensional wave equations. Special solutions of multi–d wave equation: planar and spherical waves. Well–posedness for initial, boundary value problems: uniqueness by means of energy estimates. Separation of variables. Domain of dependence and range of influence in several space variables. Fundamental solution for multi–dimensional wave equation. Solution of 3–d wave equation: Kirchhoff’s formula and strong Huygens’ principle. Wave equation in two dimensions: method of descent. Fundamental solution in 2–d. Duhamel’s principle for non homogeneous wave equation in 3–d: delayed potentials.

Pre-requisites:

Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.

Reading list:

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010.

- S. Salsa. Partial Differential Equations in Actions: from Modelling to Theory. Springer–Verlag Italia, 2008.

- S. Salsa, G. Verzini. Equazioni a derivate parziali: complementi ed esercizi. Springer–Verlag Italia. 2005.

- W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons Inc., 2008.

- E.C. Zachmanoglou, D.W. Thoe. lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc., 1986.

Semester #2 Cohort #2021~2024 @ TUHH
Numerical – Modelling Training;

Cohort

2021~2024

Semester

2

ECTS Credits

30

Campus

Hamburg University of Technology

Mathematical modelling and simulation heavily rely on scientific computing, seen as a scientific area encompassing numerical analysis, finite-element methods, numerical optimization, parallel computing. These keywords are the core or applied mathematics in that they train the use of advanced computing capabilities to solve complex models in a wide set of disciplines. The second semester at TU Hamburg provides a selected list of subjects in this framework, plus some additional subjects that complement the student’s knowledge to continue with Year 2 studies.

The Institute of Mathematics at TU Hamburg puts into place a group combining four chairs: Applied Analysis, Computational Mathematics, Numerical Mathematics, and Stochastics.

The computational part of this semester is taught by experts in the fields such as Sabine Le Borne (Professor in numerical mathematics with longstanding experience with computational mathematics programs) and Daniel Ruprecht (an expert in the parallelization of numerical methods).

Two additional courses are offered in this semester. A first one on Probability Theory provides a sound theoretical basis on stochastic modelling and an overview on its applications. This course is taught by Matthias Schulte, an internationally acknowledged expert in the field with wide range of expertise on probability, stochastics, large deviations, and random graphs (this course prepares for the specialization branch taught in Nice). A second complementary course on Variational Calculus provides a sound basis to the topics of Semester 3 at TUHH on biomedical imaging. This course is taught by Thomas Schmidt, an expert of calculus of variations and geometric PDEs, and by Ingenuin Gasser, who is an internationally acknowledged applied mathematician and a longstanding expert in managing international MSc programmes as well.

Unit Coordinator: Daniel Ruprecht

Aims:

Students are able to list numerical methods for the solution of ordinary differential equations and explain their core ideas, repeat convergence statements for the treated numerical methods (including the prerequisites tied to the underlying problem), explain aspects regarding the practical execution of a method, select the appropriate numerical method for concrete problems, implement the numerical algorithms efficiently and interpret the numerical results.

Content:

Numerical methods for Initial Value Problems: single step methods, multistep methods, stiff problems, differential algebraic equations (DAE) of index 1;
Numerical methods for Boundary Value Problems: multiple shooting method, difference methods

Pre-requisites:

Analysis, Linear Algebra, Basic MATLAB knowledge

Reading list:

E. Hairer, S. Noersett, G. Wanner: Solving Ordinary Differential Equations I: Nonstiff Problems
E. Hairer, G. Wanner: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems

Unit Coordinator: Thomas Schmidt

Aims:

The module introduces to variational minimization problems and/or variational methods for PDEs.

It may cover problems in a classical smooth setting as well as theory in Sobolev spaces.

Content:

A selection out of the following:

Model problems and examples (Dirichlet energy, isoperimetric and brachistochrone problems, minimal surfaces, Bolza and Weierstrass examples, …),
Existence and uniqueness of minimizers by direct methods,
Weak lower semicontinuity of (quasi)convex variational integrals,
Necessary and sufficient (PDE) conditions for minimizers,
Problems with constraints (obstacles, capacities, manifold and volume constraints, ...),
Generalized minimizers (relaxation, Young measures, ...),
Variational principles and applications,
Duality theory,
Outlook on regularity.

Pre-requisites:

A solid background in analysis and linear algebra is necessary.

Familiarity with functional analysis, Sobolev spaces, and PDEs can be advantageous.

Reading list:

H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, MOS-SIAM Series on Optimization 17, Philadelphia, 2014.
G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, An Introduction, Oxford Lecture Series in Mathematics and its Applications 15, Clarendon Press, Oxford, 1998.
B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London, 2014.
B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Springer, Berlin, 2008.
I. Ekeland, R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28, SIAM, Philadelphia, 1999.
M. Giaquinta, S. Hildbrandt, Calculus of Variations 1, The Lagrangian Formalism, Grundlehren der Mathematischen Wissenschaften 310, Springer, Berlin, 1996.
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.
F. Rindler, Calculus of Variations, Universitext, Springer, Cham, 2018.
F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications 87, Birkhäuser/Springer, Cham, 2015.
M. Struwe, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 34, Springer, Berlin, 2008.

Semester #2 Cohort #2021~2024 @ TUW
Numerical – Modelling Training;

Cohort

2021~2024

Semester

2

ECTS Credits

30

Campus

Vienna University of Technology

Mathematical modelling and simulation heavily rely on scientific computing, seen as a scientific area encompassing numerical analysis, finite-element methods, numerical optimization, parallel computing. These keywords are the core or applied mathematics in that they train the use of advanced computing capabilities to solve complex models in a wide set of disciplines. The second semester at TU Vienna is entirely devoted to numerical methods, with particular focus on finite elements for ordinary and partial differential equations, numerical optimization, and parallel computing.

The Institute for Analysis and Scientific Computing at TU Vienna features the perfect group for such a task. This institute is fully in charge of the Technical Mathematics courses at TU Vienna and features outstanding record of training new applied mathematicians in industry and academia.

Prof. Dirk Praetorius, a top researcher in the field of numerical methods with outstanding record and experience in research group leadership, will cover the computer programming part. He was recently awarded the Best Lecture Award at TU Vienna in 2019. Prof. Joachim Schoeberl, another leading figure in the field of numerics for differential equations and Head of the Scientific Computing and Modelling research unit at the Institute, is in charge of the course on numerical PDEs. This task is particularly relevant for the specializations on Computational Fluid Mechanics and Cancer Modelling. Proff. Praetorius and Schoeberl, jointly with Prof. Lothar Nannen, are also in charge of the course on numerical ordinary differential equations. Dr. Kevin Sturm, an assistant professor in the same group, will cover the Numerical Optimization course, which prepares for the specialization branch at UAB devoted also on optimization methods. Prof. Rudolf Fruehwirth will introduce parallel computing, a rapidly growing subject which is relevant to most of the specialisations.

Unit Coordinator: Markus Faustmann, Claudia Blaas-Schenner

Aims:

Scientific Programming
- formulate (certain) mathematical problems in algorithmic form,
- explain the difference between imperative and object-oriented programming,
- implement mathematical algorithms in Matlab, C, and C++,
- present and explain own solutions, and
- constructively discuss and analyze own solutions as well as those of other students.

Parallel Programming
- understand and apply the main concepts of parallel programming
- master the basic skills to write parallel programs using MPI and OpenMP
- parallelize serial programs using basic features of MPI and OpenMP
- be familiar with the components of an high-performance computing cluster
- know the principles to take advantage of shared and distributed memory systems as well as accelerators and how to exploit the capabilities of modern high-performance computing systems

Content:

Scientific Programming:
- Introduction to Matlab, C, and C++.
- Representation of integer and floating point numbers.
- Conditioning of given problems.
- Computational cost of algorithms.
- Variables and standard data types.
- Pointers.
- Loops and if-else.
- Functions and recursion.
- Call by value vs. call by reference.
- Objects and classes (resp. structures),
- Operator overloading, Inheritance.
- Templates.
- Visualization in MATLAB.
- Programming exercises.

Parallel Programming:
- Basic features of parallel programming with MPI (Message Passing Interface) and OpenMP (Open Multi-Processing) using C
- A look at CUDA to offload parts of the computation to GPUs
- Students will do the hands-on labs directly on the Vienna Scientific Cluster, the high-performance computing facility of Austrian universities, and hence will learn about and get some experience in high-performance computing.

Pre-requisites:

Basic skills in programming in C (e.g., as learnt during the lecture "Scientific Programming for Interdisciplinary Mathematics") as well as Linux command line and usage of an editor (vi or nano).

Reading list:

Scientific programming in mathematics:

lecture notes

Programming with MATLAB:

Otto and Denier, An Introduction to Programming and Numerical Methods in MATLAB

Brian Hahn, Essential MATLAB for Engineers and Scientists

Stormy Attaway, Matlab: A Practical Introduction to Programming and Problem Solving

Basics of Parallel Computing:

Rauber, Rünger: Parallel programming. Second Edition, Springer 2013.

Schmidt, Gonzalez-Dominguez, Hundt, Schlarb: Parallel Programming. Concepts and Practice. Morgan Kaufmann 2018.

Unit Coordinator: Markus Faustmann

Aims:

Being able to solve stationary partial differential equations numerically, analyse the quality of numerical solutions, select proper methods and implement them in a computer program.

Content:

- Variational formulations
- Sobolev spaces, H(div), H(curl)
- Finite element spaces (h, p, hp)
- Mixed formulations,
- Discontinuous Galerkin Methods,
- Time-dependent problems

Pre-requisites:

Applied Mathematics Foundations

Reading list:

Lecture notes,
Dietrich Braess: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2007
Cleas Johnson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, 1987, Dover 2009
Susanne Brenner & Ridgway Scott: The Mathematical Theory of Finite Elements, Springer 2008
Alexandre Ern & Jean-Luc Guermond: Theory and Practice of Finite Elements, Springer, 2010

Year #2 Cohort #2021~2024 @ UAQ
Modelling and Simulation of Infectious Diseases;

Cohort

2021~2024

Semester

3

ECTS Credits

30

Campus

University of L'Aquila

The Covid-19 emergency addressed the attention of many researchers in science and technology towards the mathematical modelling and simulation of the spread of epidemic diseases. The InterMaths Consortium features a specialization branch specifically devoted to computational methods in the modelling, analysis, simulation, and control of the spread of epidemic diseases, combining expertise from applied mathematicians, engineers, and medical scientists. The specialisation provides methodologies in multi-agent systems, statistics, network and graph modelling, and mathematical control theory to deal with models suited to predict the spread of epidemic diseases in specific situations.

The course Advanced Analysis, taught by Prof. Lattanzio, provides advanced mathematical modelling techniques that are needed in other course of the same specialization, in particular in the field of partial differential equations of nonlocal transport and diffusive type. These models are then specifically tackled in the course “Mathematical models and control of infectious diseases”, taught by staff members of the group of Mathematical Analysis at the University of L’Aquila. The course “Modelling and control of networked distributed systems”, taught by Prof. Giordano Pola, an engineer, will provide a wide range of methodologies in multi-agent modelling and network modelling that are used to predict the spread of epidemics. The course “Time Series and Prediction”, taught by a statistician, Prof. Umberto Triacca, deals with statistical methods, stationary processes, ARIMA type models. The course also provides skills to adapt these methods to predictive models in epidemics. Finally, a course taught by scientists from the Department of Biotechnological and Applied Clinical Sciences will deal with medical methodologies to deal with epidemic diseases.

ECTS Credits: 9 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). 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. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and 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. Embedding theorem for p = n. Characterization of the dual space H-1.

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

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.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Diﬀerentiation. The Radon-Nikodym Theorem, Diﬀerentiation on Euclidean space, Diﬀerentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

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.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

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

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

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

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Carmela Scalone

Aims:

Learning objectives.

The aim of the course is to provide knowledge and skills in the field of numerical modelling in the epidemiological field, creating an effective bridge between the understanding of the continuous model, its qualitative properties and their counterparts in the discrete.

Learning outcomes.

At the end of the course, the student should

- have a deep knowledge and understanding of the most relevant computational techniques for the treatment of models in the epidemiological field, together with aspects related to their implementation in accurate and efficient mathematical software;
- demonstrate the ability to evaluate the most appropriate discretization in relation to the problem to be solved and ability in theoretical analysis and mathematical software design;
- demonstrate ability to read and understand other texts on related topics.

Content:

- Compartmental models in Epidemiology.
- Stochastic models.
- Epidemiological models based on pds.
- An introduction to network models for the spread of epidemics.

For each theme, the presentation will be twofold: presenting the continuous model, its qualitative properties and, jointly, presenting the most appropriate numerical approach to the approximation of solutions and the discrete conservation of the properties of the model. In addition, advanced techniques of numerical linear algebra and optimization for large linear systems arising from semi-discretization of PDEs in epidemiology will be presented.

Pre-requisites:

Basic knowledge of Numerical Analysis, differential equations, linear algebra.

Reading list:

- F. Brauer et al., Mathematical Epidemiology, Springer-Verlag (2008).
- J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley (1991).
- A. Quarteroni, Numerical Models for Differential Problems, Springer (2017).
- E. Isaacson, H.Keller, Analysis of numerical methods, Dover Publications (1994).
- D.J. Higham and P. E. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.
--Walter Gander, Martin J. Gander, Felix Kwok. Scientific Computing
An introduction using Maple and MATLAB. Springer.

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Marco Di Francesco, Antonio Esposito

Aims:

This course contributes to one of the learning objectives of the degree programmes in Mathematical Modelling and Mathematical Engineering, namely the formation of the student on advanced mathematical modelling in an interdisciplinary context, in particular in the biology/medicine area, more specifically in the field of mathematical modelling applied to the diffusion of epidemics, which is the main goal of the curriculum "Modelling and simulation of infectious diseases" of the degree program in Mathematical Modelling. Moreover, the course contributes to forming the student in the context of mathematical modelling applied to population dynamics, in agreement with the objectives of the curriculum "Mathematicla Models in Social Sciences" of the degree programme in Mathematical Modelling.

At the end of the course, the student

1) will be equipped with a sound knowledge on population dynamics modelling, particularly compartmental models which can also be applied in epidemiology.

2) will be able to formulate "ad-hoc" deterministic models, such as ordinary differential equations, partial differential equations, interacting particle systems, that describe the dynamics of an epidemics in specific situations.

3) will possess analytical techniques allowing to resolve the models studied and to determine the qualitative behaviour of the solutions to those models.

4) complement the models with auxiliary terms in order to plan specific "containment strategies" for epidemiological models.

Content:

1) Introduction to population dynamics modelling via ODEs.

2) Introduction to epidemic modelling. The Kermack-McKendrick models and its variants.

3) Modelling of vector-borne diseases. Models with delay.

4) Computing the basic reproduction number.

5) Complex epidemics modelling, multi-group modelling.
Control strategies in ODE models.

6) Age structured population models and applications in epidemiology. Class-Age structured models in epidemiology.

7) Spatial heterogeneity in population dynamics. Models with diffusion.

8) Reaction-diffusion systems. Travelling waves.
Lagrangian movements. Particle models. Nonlocal models. Discrete vs Continuum modeling using integro-differential equations.

10) SIR (and variants) models with local and nonlocal diffusion.

11) Graph modeling for population dynamics

Pre-requisites:

Dynamical systems, analytical methods for to resolutoin of ordinary differential equations.
Basics of linear partial differential equations.

Reading list:

James D. Murray; Mathematical biology I: An introduction; Springer.

James D. Murray; Mathematical biology II: Spatial Models and biomedical applications; Springer.

Fred Brauer, Pauline van den Driessche, Jianhong Wu; Mathematical Epidemiology; Lecture notes in mathematics; Springer.

Maia Martcheva - An Introduction to Mathematical Epidemiology; Texts in Applied Mathematics 61, Springer.

Lecture notes by the lecturer.

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Giordano Pola

Aims:

The aim of this course is to provide students with basics about modeling, analysis and control of networked and multi-agent systems through consensus techniques.

Content:

Introduction to networked and multi-agent systems.
Recalls of graph theory.
The agreement protocol: the linear and nonlinear cases.
Formation control.

Reading list:

Graph Theoretic Methods in Multiagent Networks, Princeton University Press, Mehran Mesbahi & Magnus Egerstedt, 2010

Year #2 Cohort #2021~2024 @ UAQ
Cancer Modelling and Simulation;

Cohort

2021~2024

Semester

3

ECTS Credits

30

Campus

University of L'Aquila

The specialization track “Cancer Modelling and Simulation” will be given at the Department of Information Engineering, Computer Science and Mathematics of UAQ in collaboration with the Department of Biotechnological and Applied Clinical Sciences at the same University. The proposed track addresses modelling and simulation of cancer genetics, evolution, and treatment using continuum bio-fluid dynamic modelling, fluid-solid interaction modelling, reaction-diffusion modelling, systems biology and control theory.

The group at L’Aquila features an important research stream on these subjects, see the contribution by M. Di Francesco in the mathematical theory of chemotaxis modelling and biological aggregation phenomena, as well as the results obtained by D. Donatelli in mathematical fluid dynamics and biofluid dynamics and cancer modelling and simulation.

The course “Advanced analysis” (taught by C. Lattanzio) provides advanced modelling tools proper for mathematical analysis. The course “Mathematical biofluid dynamics” (taught by D. Donatelli) tackles modern fluid-dynamical modelling techniques in cancer modelling, specifically cells-ECM interaction models. “Biomathematics”, taught by C. Pignotti, deals with cell population models and chemotaxis modelling, with applications to cancer modelling. The course “Systems Biology” covers deterministic and stochastic modelling and control of gene transcription networks and enzymatic reactions with application to cancer drug response. The Course “Cancer Genetics and Biology for Mathematical Modelling” is jointly taught by Dr. Alessandra Tessitore and Dr. Daria Capece (formerly at Imperial College, London), who are members of the Department of Biotechnological and Applied Clinical Sciences at UAQ.

Students of this specialization branch will have the opportunity to prepare their MSc thesis in collaboration with the staff from external institutions such as CSCAMM at the University of Maryland, WWU Muenster in Germany, KAUST, University of Oxford, Imperial College London. They may also spend the thesis semester in one of the pharmaceutical companies of the Capitank Consortium carrying out R&D projects on cancer genetics, evolution and treatment.

ECTS Credits: 9 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). 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. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and 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. Embedding theorem for p = n. Characterization of the dual space H-1.

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

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.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Diﬀerentiation. The Radon-Nikodym Theorem, Diﬀerentiation on Euclidean space, Diﬀerentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

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.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

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

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

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

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Simone Fagioli

Aims:

To learn the basics in the mathematical modelling of population dynamics.
To provide a mathematical description of ODE models in population dynamics and the interpretation of the qualitative behaviour of the solutions
To get the basic notions in mathematical models in epidemiology and reaction kinetics.
To learn the mathematical modelling of population models in heterogeneous environment, described by partial differential equations.
To deal with advanced models in biology such as chemotaxis models and structured dynamics equations.
To get a sound background in reaction diffusion phenomena, Turing instability, and pattern formation.

Content:

Continuous Population Models for Single Species. Continuous Growth Models. Delay models. Linear Analysis of Delay Population Models: Periodic Solutions.
Continuous models for Interacting Populations. Predator-Prey Models: Lotka-Volterra Systems. Realistic Predator–Prey Models. Competition Models: Principle of Competitive Exclusion. Mutualism or Symbiosis 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
Matematical models for tumor growth.
Structured population dynamics. An example in ecology: competition for resources. Continuous traits. Evolutionary stable strategy in a continuous model.

Pre-requisites:

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).

Reading list:

Murray - Mathematical Biology I&II

Lecturers lecture notes

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Alessandra Tessitore, Daria Capece

Aims:

Tumor initiation, progression and invasion are complex processes involving multiple and different phenomena. Mathematical models and computer simulations can help to describe, schematize and comprehend them, to provide data which could be putatively used in clinical practice to prevent and/or more specifically treat cancer. In this context, a multidisciplinary approach is fundamental to reach this goal. The main objective of this course is to approach and understand the biological processes at the base of cancer, focusing on the most significant features of oncogenesis, with the aim to provide basic information which can be applied to mathematical modelling. On completion, the student should:

know fundamentals about structure and functions of nucleic acids and proteins in eukaryotic cell;
understand the significance of gene mutations and epigenetics alterations in diseases;
identify tumor classification criteria;
understand biological and functional mechanisms at the base of cancer initiation and progression; -
know the most important databases for DNA mutation classification, microRNA and protein pathway analysis as well as on-line resources for acquiring datasets to be applied to big data analysis,
know and understand the principles at the base of personalized therapy in cancer to predict the response to therapeutic schemes.

Content:

Topics of this module (6 CFU)

Fundamentals about the structure and the role of nucleic acids and proteins (8 hrs).
Genetic and epigenetic mechanisms at the base of oncogenesis (gene mutations, DNA damage repair system failure, methylation, microRNA dysregulation) (9 hrs).
Features of cancer cells and tumor classification (15 hrs).
Biological mechanisms of angiogenesis, invasion and metastasis (8 hrs).
Molecular pathways involved in cell differentiation, proliferation and survival (7 hrs).
Big data in cancer analysis (6 hrs).
Personalized therapy in cancer (5 hrs).
In vitro and in vivo models for the study of tumor biology (2 hrs).

Reading list:

Articles/reviews about the topics of the course
Study material provided by the Professor

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Donatella Donatelli

Aims:

Learning Objectives:
The aim of the course is 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 the analysis of other partial differential equations.

Learning Outcomes:
On successful completion of this course, the student should:

- understand the basic principles governing the dynamics of non-viscous fluids;
- be able to derive and deduce the consequences of the equation of conservation of mass;
- be able to apply Bernoulli's theorem and the momentum integral to simple problems including river flows;
- understand the concept of vorticity and the conditions in which it may be assumed to be zero;
- calculate velocity fields and forces on bodies for simple steady and unsteady flows derived from potentials;
- demonstrate skill in mathematical reasoning and ability to conceive proofs for fluid dynamics equations.
- demonstrate capacity for reading and understand other texts on related topics.

Content:

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical Modelling of Continuum Media (3 ECTS)

- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Fluidi di tipo Poiseulle e Couette
- Vorticity equation in 2D and 3D

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS),

- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)

CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS)

- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.

Pre-requisites:

PREREQUISITES for Mathematical Modelling of Continuum Media:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.

PREREQUISITES for Mathematical fluid and biofluid dynamics, Mathematical fluid dynamics, Modelling and analysis of fluids and biofluids:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations, Sobolev spaces.

Reading list:

- Alexandre Chorin, Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer.
- Roger M. Temam, Alain M. Miranville, Mathematical Modeling in Continum Mechanics. Cambridge University Press.
- Franck Boyer, Pierre Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer-Verlag Italia.
- Andrea Bertozzi, Andrew Majda, Vorticity and Incompressible Flow. Cambridge University Press.

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of L'Aquila | Language: English

Unit Coordinator: Alessandro Borri

Aims:

Systems biology is an emerging research area, which aims at providing mathematical models helping to understand the dynamic interactions occurring within and among cells. This course provides the basic mathematical tools to model and analyze gene transcription as well as biochemical reaction networks: the most important network motifs are investigated exploiting both deterministic and stochastic approaches.

Content:

Review of basic concepts of biology and probability; deterministic vs. stochastic approach.
Transcription factors and transcription networks; gene expression: transcription and translation; separation of time scales and the Quasi-Steady-State Approximation (QSSA); activators and repressors; Hill and logic input functions; dynamics of simple regulation and switching production in ordinary differential equation (ODE) form; response time; mass action kinetics and conservation equation; the Michaelis-Menten kinetics; cooperativity; dependent and independent bindings; multi-dimensional input functions.
Network Motifs: real and randomized networks; the Erdos-Renyi (ER) model; graph properties; negative autoregulation (NAR) and stability analysis; three-node network motifs: feedback loops (FBLs) and feedforward loops (FFLs); coherent and incoherent FFLs; sign-sensitive delay, output pulse generation, speed up of response time, biphasic response curves in I1-FFL; review of the other types of FFL; examples: Arabinose system, Galactose system, flagella motor genes in E. Coli.
Introduction to Numerical Simulation in Systems Biology; MATLAB: introduction and basic notions; integration of non-stiff differential equations via ode45; randomized computation, Monte Carlo simulation, construction of randomized networks (E-R models) and digraphs.
General Transcription and Interaction Networks: Single-Input Module (SIM), Multi-Output Feedforward Loop, LIFO and FIFO temporal programs, bi-fans and Densely Overlapping Regulons (DORs), interlocked FFLs; positive autoregulation (PAR) and response slow-down; two-node positive feedback: double-positive (lock-on) and double-negative (toggle-switch) circuits; three-node motives; protein-protein interaction (PPI) networks and hybrid motifs; network motifs in real-world networks: food webs, social networks, electrical circuits, neuronal networks. Examples: sporulation of bacterium Bacillus subtilis, phosphorylation, response to pain stimuli in humans.
Biological Oscillations: the role of negative feedback and delay; time-separation, feedback strength and cooperativity; noise-induced and delay oscillations; bistability and hysteresis; robustness in biological circuits; models at a higher level of organization: tissues and organs; the glucose-insulin feedback loop.
The stochastic approach in Systems Biology: motivation, Chemical Reaction Networks, Continuous-Time Markov Chains; the Chemical Master Equation (CME) and its properties, characterization of the CME stationary distribution, the macroscopic equation, one-step processes; stochastic simulation in Systems Biology, inverse sampling method, Gillespie Stochastic Simulation Algorithm (SSA); examples.
Mesoscopic models and noise: extrinsic vs intrinsic noise, bursty reactions, the Chemical Langevin Equation, the Wiener Process, the Euler-Maruyama integration of Stochastic Differential Equations (SDEs).
Biological and biomedical examples: metabolism, tumors, epidemics, pharmacokinetics and pharmacodynamics (PKPD), environmental pollution, ecology.

Pre-requisites:

Basic notions of mathematical analysis, dynamical systems and probability.

Reading list:

- Uri Alon, An introduction to systems biology: design principles of
biological circuits, CRC press, 2019.
- E. Klipp, W. Liebermeister, C. Wierling, and A. Kowald, Systems biology: a textbook, John Wiley & Sons, 2016.
- J. D. Murray, Mathematical biology, 3rd edition. New York: Springer, 2001.

Year #2 Cohort #2021~2024 @ TUHH
Computational Methods in Biomedical Imaging;

Cohort

2021~2024

Semester

3

ECTS Credits

30

Campus

Hamburg University of Technology

The specialisation track "Computational methods in biomedical imaging" will be mainly offered by the School of Electrical Engineering, Computer Science and Mathematics at TUHH. It will also involve teaching staff members from the University of Hamburg (UHH) and researchers from Universitätsklinikum Hamburg-Eppendorf (UKE). Courses in this track address both classical and modern mathematical modelling and simulation techniques in biomedical imaging. In total, they convey an understanding of the complete range of engineering, programming and mathematical aspects of biomedical imaging.

The course “Computer Tomography“ is taught by Armin Iske, who is affiliated with UHH and associated with UKE. It covers the mathematical instruments involved in the process of transforming data into images with a particular focus on CT and MRI, such as Radon transform, back projection, algebraic reconstruction and kernel-based reconstruction. This is complemented by the course “Medical Imaging“, which is taught by Tobias Knopp, who has a double appointment at TUHH and UKE. This course is geared towards both the underlying physics of imaging as well as the respective numerical algorithms and their efficient implementation. Furthermore, the course “Mathematical Image Processing“, taught by M. Lindner from the Institute of Mathematics at TUHH, deals with concepts of image processing and their mathematical background such as denoising, edge detection, deconvolution, inpainting, segmentation and registration. On the other hand, the course “Intelligent Systems in Medicine“, taught by A. Schlaefer from the Institute of Medical Technology, provides another perspective by addressing topics such as kinematics, tracking systems, navigation and image guidance in clinical contexts. The application-oriented course “Case studies in medical imaging“, supervised by Anusch Taraz (the local coordinator at TUHH) and Ingenuin Gasser from UHH, brings to bear the newly achieved techniques by applying them in specific, realistic situations.

Overall, the specialisation track “Computational methods in biomedical imaging“ is founded on the close interaction between TUHH and UHH together with UKE, which is strengthened even further by the Hamburg research center on medical technology (FMTHH), where many scientists (including all of those mentioned above) run joint research projects in the areas of biomobility, networked implants and imaging. Students in this track will benefit thoroughly from this active research environment. Moreover, they can make use of the ample opportunities to write their thesis in cooperation with one of the companies working on medical technology, such as Jung Diagnostics and Basler.

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: Hamburg University of Technology | Language: English

Unit Coordinator: Dr. Jens-Peter M. Zemke

Content:

Basics: analogy; layout of neural nets, universal approximation, NP-completeness
Feedforward nets: backpropagation, variants of Stochastistic Gradients
Deep Learning: problems and solution strategies
Deep Belief Networks: energy based models, Contrastive Divergence
CNN: idea, layout, FFT and Winograds algorithms, implementation details
RNN: idea, dynamical systems, training, LSTM
ResNN: idea, relation to neural ODEs
Standard libraries: Tensorflow, Keras, PyTorch
Recent trends

Reading list:

Skript
Online-Werke:

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: Hamburg University of Technology | Language: English

Unit Coordinator: Marko Lindner

Aims:

Students are able to characterize and compare diffusion equations, explain elementary methods of image processing, explain methods of image segmentation and registration, sketch and interrelate basic concepts of functional analysis.

Content:

Basic methods of image processing
Smoothing filters
The diffusion / heat equation
Variational formulations in image processing
Edge detection
De-convolution
Inpainting
Image segmentation
Image registration

Pre-requisites:

Analysis: partial derivatives, gradient, directional derivative
Linear Algebra: eigenvalues, least squares solution of a linear system

Reading list:

Will be announced in the lecture

Year #2 Cohort #2021~2024 @ TUW
Computational Fluid Dynamics and Semiconductor Modelling;

Cohort

2021~2024

Semester

3

ECTS Credits

30

Campus

Vienna University of Technology

The specialisation track “Computational fluid dynamics in industry” will be offered at the Institute for Analysis and Scientific Computing at TUW in collaboration with the Institute of Fluid Mechanics and Heat Transfer and the Institute for Microelectronics at the same University. The reference group includes world-leading experts in PDE modelling in fluid dynamics, reaction-diffusion systems and semiconductor devices such as Anton Arnold and Ansgar Juengel.

The core course “Computational fluid dynamics” (CFD) is taught by Prof. Manual Garcia Villalba Navaridas. It covers state-of-the-art numerical methods for the (in)compressible Navier-Stokes equations along with the treatment of complex geometries and turbulence modelling. Prof. Manual Garcia Villalba Navaridas is also in charge of the course CFD-codes and turbulent flows jointly with Prof. Herbert Steinrueck.

The course “Continuum and kinetic modelling with PDEs” is taught by Prof. Anton Arnold, local InterMaths coordinator. It covers a wide range of application of classical and modern PDE models to fluid dynamics. The course “Continuum models in semiconductor theory” provides an introduction to semiconductor physics and devices and an additional part on theory, modelling and simulation of MEMS & NEMS. The course Numerical simulations and scientific computing taught by Josef Weinbub provides advanced methodologies in numerical simulations needed in this track.

Students in this specialization branch will have the chance to spend their thesis period in private industries in the semiconductor devices sector such as Infineon or in the software company CERBSim. Moreover, they will have the chance to collaborate with researchers in mathematical modelling from IST Austria.

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: Vienna University of Technology | Language: English

Unit Coordinator: Herbert Steinrück

Pre-requisites:

Partial differential equations,
Fluid mechanics,
Numerical methods for fluid mechanics

Reading list:

Lecture notes,
P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press, Cambridge (2002)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press (1961) P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press (2002)

ECTS Credits: 7 | Semester: 1 | Year: 2 | Campus: Vienna University of Technology | Language: English

Unit Coordinator: Ulrich Schmid

Aims:

Introduction to semiconductor physics and devices 360.241

Students will be able to have a basic understanding of the underlying physics involved in the operation of semiconductor devices and will have a consistent base level of comfort and familiarity with the topic before they move to more advanced courses. The primary focus of the course will be on the development of physics and concepts, however a basic introduction to the application of simulation and numerical calculation within the field of semiconductor devices will also be partially covered.

Theory, modelling and simulation of MEMS & NEMS:

Model the mechanical behavior of MEMS/NEMS by continuum mechanics.
State and test the underlying assumptions of the theory of linear elasticity.
Describe the interaction between MEMS/NEMS and fluids.
Model the piezoelectric effect.
Calculate the eigenmodes and eigenfrequencies of selected structures.
Derive the differences between macroscopic and microscopic systems
Predict the dynamics of resonators.
Derive the basic theory of the method of finite elements.
Explain the working principle of reference oscillators, mass und fluid sensors.
Discuss novel concepts like phononic crystals and quantum MEMS/NEMS.
Understand technical subject-specific technical terminology and critically evaluate relevant scientific publications.
Use open source software for eigenmode analysis.

Content:

Introduction to semiconductor physics and devices:

In this course we will learn how the quantum mechanics of solids and transport has been harnessed to build the Digital Age. We will explore the physics of semiconductors and semiconductor devices and the role they play in modern technology. We will also lay the ground-work for more advanced discussions of how computational techniques and simulation can be used to push knowledge in this field forward to new horizons.

* Basic Structure of Crystals and Solids, Quantum Mechanics of Solids, Physics of Semiconductors, Transport in Semiconductors, PN junction, PN diode, PN Junctions and Modern Technology (LEDs, Photovoltaics, etc.), MOS transistor, MOS capacitor, MOSFETs and Modern Technology (VLSI, IGFET Sensors, etc.), Metal/semiconductor interfaces, Heterostructures and Modern Technology (Schottky Diodes, 2D-FETs, etc.)

Theory, modelling and simulation of MEMS & NEMS:

The design of micro- and nanoelectromechanical systems (MEMS/NEMS) is a highly interdisciplinary field which reflects in the variety of topics of this course. Starting from an introduction to continuum mechanics and piezoelectricity we investigate different aspects of the mechanics of basic MEMS/NEMS structures like membranes and beams. By understanding the interaction of MEMS/NEMS with their environment, we are able to understand the outstanding performance of MEMS/NEMS sensors for mass and fluid sensing. Another important aspect for the modelling of MEMS/NEMS is the representation of MEMS/NEMS with discrete lumped element models and we discuss the most important discrete models. For quantitative predictions often numerical methods need to be employed for which the FEM is the most known. We discuss the fundamental theory of the FEM and its limitations. Using the above theory, we study example applications like reference oscillators or fluid sensors. Additionally, we take a look at novel concepts like phononic crystals or quantum MEMS/NEMS.

Pre-requisites:

Some basic exposure to the concepts and equations in the physics of electromagnetism;
(Vector) calculus,
Differential equations especially.

Reading list:

Lecture notes

Year #2 Cohort #2021~2024 @ UAB
Decision Making and Applications to Logistics;

Cohort

2021~2024

Semester

3

ECTS Credits

30

Campus

Autonomous University of Barcelona

Semester 3 at UAB will be supervised by the Department of Mathematics, with the collaboration of the School of Engineering. It will focus on Decision Making, a broad term that encompasses many scientific techniques and connects to every application field, and in particular to its relation to logistics and supply chain management. In fact, one can say that the very construction of a mathematical model is often done with the purpose to do some action, hence to decide something, and not only to observe results.

Students in this track will learn the concepts and most practical modelling and simulation tools applied to logistics and supply chain, together with state-of-the-art instruction in algorithms for hard optimisation problems and modern computational statistics, as supporting courses.

The two logistics courses will be lectured by staff of the School of Engineering, prof. Miquel A. Piera and prof. Juan M. Ramos, co-founders of the company Aslogic, devoted to aeronautic logistics, and Dr. Roman Buil, from Accenture. The course in Optimisation is taught by Prof. Lluís Alsedà, director of the Centre de Recerca Matemàtica (CRM), one of our research partners. The subject Computational Statistics and Bayesian Networks is managed by Prof. Pere Puig (who is also the coordinator of the Bachelor’s degree in Computational Mathematics at UAB) and counts on the participation of Dr. Juan R. González from the Barcelona Institute of Global Health. The course Case studies of optimisation problems in industry will be led by prof. Xavier Mora, member of the Research Group in Mathematical Models and Applications, and it will count on collaborations from our partners in industry as Aslogic, Eurecat, AIS, and research centres as CRM, the IIIA, and the IERMB.

Master’s theses can be carried out along with an internship in industry. Students following this track will have employment opportunities in the logistic sector due to the knowledge acquired in diverse decision making situations. They will also have a broad spectrum of opportunities in many other sectors.

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: Autonomous University of Barcelona | Language: English

Unit Coordinator: Lluis Alseda Soler, Martin Hernan Campos Heredia, Judit Chamorro Servent, Susana Serna

Aims:

Analysis of case studies, and practice of team working dynamics and client-consultant relationship.

The cases will focus primarily on optimisation and logistics real problems, but may have a scope beyond the pure "decision making" setting.

The course will be organised and directed by a professor of the department of Mathematics and will include minicourses and presentations given by industrial collaborators and other departments' teaching staff.

Content:

Mathematical modelling, i.e. solving real-world problems by means of mathematics.

Pre-requisites:

Students must have mathematical and computational skills at the level of a science degree.

Reading list:

Ch. Rousseau, Y. Saint-Aubin, 2008. Mathematics and Technology. Springer.
P. Pevzner, R. Shamir, 2011. Bioinformatics for Biologists. Cambridge Univ. Press

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: Autonomous University of Barcelona | Language: English

Unit Coordinator: Albert Ruiz Cirera, Judit Chamorro Servent

Aims:

The objective of this course is to study and practise the paradigmatic problems (Routing, Scheduling, Location, Packing), relevant in logistics, that lead to discrete and combinatorial optimisation models, intrinsically difficult in practice due to its huge input size.

The modern metaheuristics for approximating the solutions of such problems, like Evolutionary Algorithms, Tabu Search, Particle Swarm, or Ant Colony will be introduced

Content:

Combinatorial Algorithms for graphs and routing: Dijstra and A* algorithms.
Optimisation on graphs.
Deterministic optimization for nonlinear problems (constrained and non-constrained).
Genetic Algorithms.
Simulated Annealing.
Ant colony optimisation algorithms.
Particle swarm optimization.
Neural Networks in optimization.
Scheduling.
Machine learning trough neural networks.

Pre-requisites:

Mathematical knowledge at the level of Science degree. Programming skills

Reading list:

Judea Pearl, A* Algorithms and such: Heuristics: Intelligent Search Strategies for Computer Problem Solving, Addison-Wesley, 1984.
Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, Numerical Mathematics, , Texts in Applied Mathematicsm 37, Springer, 1991.
Sean Luke, Essentials of Metaheuristics, 2009: http://cs.gmu.edu/∼sean/book/metaheuristics/
David Beasley, David R. Bully and Ralph R. Martinz, An Overview of Genetic Algorithms (Part 1: Fundamentals and Part 2: Research Topics)
S. Kirkpatrick, C. D. Gelatt Jr. and M. P. Vecchi, Optimization by Simulated Annealing, Science, May 1983, Vol. 220, no. 4598, pp. 671-680.
William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes in C.
The Art of Scientific Computing (second edition)}, Cambridge University Press.
Marco Dorigoa and Christian Blum, Ant colony optimizationtheory: A survey, Theoretical Computer Science 344 (2005) 243 - 278.
Ronald L. Graham, Combinatorial Scheduling Theory
R. Gary Parker, Deterministic Scheduling Theory, Chapman Hall.
Peter Brucker, Scheduling Algorithms, Fourth Edition, Springer
R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Khan, Optimization and approximation in deterministic sequencing and scheduling: a survey
Peter Brucker, Scheduling Algorithms, Springer-Verlag, 2007, Berlin Heidelberg New York (ISBN 978-3-540-69515-8).
Jean-Yves Potvin, Kate A. Smith, Artificial Neural Networks for Combinatorial Optimization
Kate Smith, Neural Networks for Combinatorial Optimization: A Review of More Than a Decade of Research
Kate Smith, Marimuthu Palaniswami and Mohan Krishnamoorthy. Neural Techniques for Combinatorial Optimization with Applications
Lecture notes provided by the teacher.

ECTS Credits: 3 | Semester: 1 | Year: 2 | Campus: Autonomous University of Barcelona | Language: English

Unit Coordinator: Ramon Anton Piera Eroles

Aims:

To introduce students to the knowledge, processes, skills, tools and techniques suitable for project management, such that the application of them satisfy the requirements specified for project development, and may have a significant impact on its success. Specifically: learning the terminology and basic concepts of project management area and understanding the relationship between logistics and supply chain management and project management.

Content:

Introduction to Project Management
System Development Cycle
Feasibility Study
Project Planning
Graphs-based Programming Methods
Cost Analysis
Risk Management
Project Control

Reading list:

Heagney, Joseph. Fundamentals of Project Management, 5th edition. 2016.
Martinelli, Russ, et al. Program Management for Improved Business Results, 2nd edition, 2014.
Nicholas, John M. Project management for business and technology: principles and practice , 2nd edition. Prentice Hall, 2001.
Nicholas, John M., Steyn, H. Project management for business and technology: principles and practice, 3rd edition. Elsevier, 2008.
Nicholas, John M. and Steyn, H. Project management for engineering, business, and technology, 4th edition. Routledge, 2012.
A Guide to the project management body of knowledge: (PMBOK® Guide), 6th edition. Project Management Institute, 2017.
Lewis, James P. Fundamentals of project management: developing core competencies to help outperform the competition. Amacom, 2002.

Year #2 Cohort #2021~2024 @ UniCA
Stochastic Modelling in Neuroscience;

Cohort

2021~2024

Semester

3

ECTS Credits

30

Campus

University of Côte d'Azur

The specialisation track “Stochastic modelling in neuroscience” will be offered at the Université de la Côte d’Azur in Nice. The teaching staff are affiliated to the CRNS Laboratoire J. A. Dieudonne, in particular to the Probability and Statistics Team, featuring experts in the field at outstanding level, such as Francois Delarue (AMS Doob Prize 2020), Patricia Reynaud-Bouret (Director of the “NeuroMod” Institute), and Cédric Bernardin (a leading international figure in the field of stochastic calculus and applications). The former two are acknowledged researchers in the applications of probability and stochastic analysis to neuroscience. Part of the courses will be taught by staff at the “NeuroMod” Institute for Modelling in Neuroscience and Cognition.

The course “Stochastic calculus with applications to neuroscience”, taught by Cédric Bernardin, will provide stochastic calculus tools to be applied to the dynamics of biological neural networks. The course “Probabilistic numerical methods” will address Monte Carlo and quasi Monte Carlo numerical methods. It will be taught by Etienne Tanré, who runs several projects in this field. The course “Stochastic control and interacting systems” addresses a very innovative field at the interface of optimal control, stochastic calculus, and mathematical analysis, which has relevant contributions to mathematical modelling in neuroscience. It will be taught by one of the most important experts in the field such as Francois Delarue. Two courses will be taken from the existing Modelling for Neural and Cognitive Systems MSc program: “Stochastic models in neurocognition and their statistical inference”, taught by the Director of the NeuroMod Institute Patricia Reynaud-Bouret, and “ Behavioural and cognitive neuroscience”, taught by Alice Guyon, research director at CNRS at the Institute of Molecular and Cellular Pharmacology. The latter is the application-driven course of this track, it will be exclusively focused on neuroscience modelling and it will analyse the neurobiological basis for higher mental functions through several examples.

This specialisation track will give the opportunity to spend the thesis semester in one of the institutes involved with the teaching or in one of the industrial partners, such as Fotonower, a company devoted to artificial intelligence and image processing.

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of Côte d'Azur | Language: English

Unit Coordinator: Alice Guyon, Ingrid Bethus

Aims:

Neuronal and cognitive systems cannot be modeled without knowledge of the basics of Neurosciences, from the molecular to the integrated level, involved in cognition and behaviors.

The first part of the program focuses on elementary neurophysiology and neuroanatomy. What are the different subparts constituting the nervous system and what are their main roles? How are neurons constituted? How do they generate activity and communicate with other neurons?

The second part of the program explores, with an integrative perspective, the neurobiological basis for higher mental functions through several examples. Sensorimotor functions are at the root of all the other processes. So the study of feeding behaviors is a good way to learn about the bio-logic of elementary behaviors, starting from the physiology of the autonomic nervous system and ending with neuroethological issues. Learning and memory are the basic processes of higher mental functions and also hot topics with applications in many domains.

In addition to all these fundamentals, the course also explains the materials and methods used in cognitive neurosciences to obtain data at the different levels of organization of nervous, cognitive and behavioral systems. This course is taught by a teaching staff member of the Master Programme Mod4NeuCog at UCA.

Content:

Neuronal and cognitive systems
Neurophysiology and neuroanatomy
Neurobiological basis for higher mental functions

ECTS Credits: 6 | Semester: 1 | Year: 2 | Campus: University of Côte d'Azur | Language: English

Unit Coordinator: François Delarue

Aims:

The course has two purposes. The first one is to provide the basic knowledge in stochastic control, control for discrete and continuous processes, dynamic programming principle, dynamic programming equation, Hamilton-Jacobi-Bellman equation.

The second part of the course will address interacting particle systems, as some of them are now currently used in the modelling of large neural networks. Applications to self-organisation and phase transition in neuroscience will be considered and, in connection with the first part of course, some learning methods will be discussed as well.

Content:

Stochastic control
Dynamic programming principle
HJB equation, interacting particle system
Mean field models
Learning methods

Pre-requisites:

Probability with measure theory, optimization, stochastic calculus

Programme #Structure 2021 ~ 2024

InterMaths Erasmus Mundus Joint Master Degree

Semester 1

Semester 2

Year 2

Semester 1 • L'Aquila

Foundations of Applied Mathematics

Semester 2 · Hamburg

Numerical – Modelling training

Semester 2 · Vienna

Numerical – Modelling training

Year 2 L'Aquila

Modelling and Simulation of Infectious Diseases

Year 2 Hamburg

Computational Methods in Biomedical Imaging

Year 2 Nice

Stochastic Modelling in Neuroscience

Year 2 L'Aquila

Cancer Modelling and Simulation

Year 2 Barcelona

Decision Making and Applications to Logistics

Year 2 Vienna

Computational Fluid Dynamics in Industry

Semester #1 Cohort #2021~2024 @ UAQ Foundations of Applied Mathematics;

Cohort

Semester

ECTS Credits

Campus

Semester #2 Cohort #2021~2024 @ TUHH Numerical – Modelling Training;

Cohort

Semester

ECTS Credits

Campus

Semester #2 Cohort #2021~2024 @ TUW Numerical – Modelling Training;

Cohort

Semester

ECTS Credits

Campus

Year #2 Cohort #2021~2024 @ UAQ Modelling and Simulation of Infectious Diseases;

Cohort

Semester

ECTS Credits

Campus

Year #2 Cohort #2021~2024 @ UAQ Cancer Modelling and Simulation;

Cohort

Semester

ECTS Credits

Campus

Year #2 Cohort #2021~2024 @ TUHH Computational Methods in Biomedical Imaging;

Cohort

Semester

ECTS Credits

Campus

Year #2 Cohort #2021~2024 @ TUW Computational Fluid Dynamics and Semiconductor Modelling;

Cohort

Semester

ECTS Credits

Campus

Year #2 Cohort #2021~2024 @ UAB Decision Making and Applications to Logistics;

Cohort

Semester

ECTS Credits

Campus

Year #2 Cohort #2021~2024 @ UniCA Stochastic Modelling in Neuroscience;

Cohort

Semester

ECTS Credits

Campus

#Consortium InterMaths EMJM;

University of L'Aquila, coordinating institutionUAQ • Università degli Studi dell'Aquila

Get in touch

Hamburg University of TechnologyTUHH • Technische Universität Hamburg

Vienna University of TechnologyTUW • Technische Universität Wien

Autonomous University of BarcelonaUAB • Universitat Autònoma de Barcelona

University of Côte d'AzurUniCA • Université Côte d'Azur

Semester #1 Cohort #2021~2024 @ UAQ
Foundations of Applied Mathematics;

Semester #2 Cohort #2021~2024 @ TUHH
Numerical – Modelling Training;

Semester #2 Cohort #2021~2024 @ TUW
Numerical – Modelling Training;

Year #2 Cohort #2021~2024 @ UAQ
Modelling and Simulation of Infectious Diseases;

Year #2 Cohort #2021~2024 @ UAQ
Cancer Modelling and Simulation;

Year #2 Cohort #2021~2024 @ TUHH
Computational Methods in Biomedical Imaging;

Year #2 Cohort #2021~2024 @ TUW
Computational Fluid Dynamics and Semiconductor Modelling;

Year #2 Cohort #2021~2024 @ UAB
Decision Making and Applications to Logistics;

Year #2 Cohort #2021~2024 @ UniCA
Stochastic Modelling in Neuroscience;

University of L'Aquila, coordinating institution
UAQ • Università degli Studi dell'Aquila

Hamburg University of Technology
TUHH • Technische Universität Hamburg

Vienna University of Technology
TUW • Technische Universität Wien

Autonomous University of Barcelona
UAB • Universitat Autònoma de Barcelona

University of Côte d'Azur
UniCA • Université Côte d'Azur