Programme
The General Mathematics track of the Master in Mathematics is a twoyear programme taught in English, featuring core and optional courses, including project work and seminarstyle presentations.
Students can choose courses from algebra, number theory, geometry, probability, statistics and others.
New students are requested to take the Refresher Courses scheduled before the start of the first semester.
Academic Contents
Course offer for General Mathematics , Semestre 1

Details
 Number of ECTS: 4
 Course code: F1_MA_MAT_GM6
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The purpose of the course is to introduce students to various aspects of algorithmic number theory linked to publickey cryptography.

Course learning outcomes
On successful completion of the course, the student should be able to: Explain the main algorithms for primality testing, factorizing large integers, solving the discrete logarithm problem, both in the multiplicative group of finite fields, as well as in the context of elliptic curves defined over finite fields Read and understand some scientific articles published in the domain, and ask relevant questions Give a talk for peers on related topics Organize his approach to general problems in an algorithmic way 
Description
The lecture will introduce some of the most important algorithms used for computing with integers modulo m, including the Chinese Remainder Theorem, will introduce a series of theorems (Fermat’s small theorem, theorem of Lagrange, exponentiation method, etc) that allow to accelerate computations in these rings. Fields, especially finite fields, will be introduced from an algorithmic angle. Usual probable primality tests will be described. They will provide the context for the introduction of Legendre symbols, the quadratic reciprocity law, etc. Usual integer factorization methods will be introduced as well, including the method of Fermat up to the quadratic field sieve. A series of cryptographic primitives will be given both for a finite group in general, and for specific groups in particular, arising from finite fields, or from elliptic curves defined over finite fields. If time allows, some methods for the computation of the number of points of an elliptic curve over a finite field will be given. This series of lectures will be completed by some broader conferences presenting the material in a more general setting. 
Assessment
One single exam (probably in written form, depending on the number of students registered for the exam). 
Note
Note / Literature / Bibliography “How big is big? How fast is fast? A handson tutorial on mathematics of computation” by Franck Leprévost (available at the documentation center of the university and on Amazon). https://www.amazon.fr/HowBigFastHandsMathematics/dp/B087SGXLSH/ref=sr_1_1?__mk_fr_FR=%C3%85M%C3%85%C5%BD%C3%95%C3%91&crid=1ZWKY4WZEN0DO&keywords=Franck+Lepr%C3%A9vost&qid=1649316964&sprefix=franck+lepr%C3%A9vost%2Caps%2C219&sr=81

Details
 Number of ECTS: 2
 Course code: F1_SECEDUC15
 Module(s): General Mathematics 1
 Language: FR
 Mandatory: No

Lecturer
Coming soon 
Objectives
Comment lire et interpréter les différents programmes de mathématiques ? Quelle est la progression didactique à travers les 7 années de l’enseignement secondaires ? Quelles sont les similarités et les différences entre les attentes dans les différents ordres d’enseignement/sections ? Quelles sont les attentes finales et les questions d’examen qui les précisent ? Comment rendre opérationnel les notions de compétences procédurales ?

Course learning outcomes
Working through the program from a higher standpointsecondary school analysis and geometry 
Description
Le programme cadre pour l’enseignement des mathématiques dans l’enseignement secondaire classique et général :les domaines de l’analyse et de la géométrieles compétences à développerExplicitation des finalités de l’enseignement des mathématiques à l’aide de questions d’examenPréparation au concours de recrutementConceptsclé pour l’enseignement des mathématiques activités mathématiques de baseapprentissage autonomesituations d’apprentissage ouvertes et complexesrésolution de problèmes

Details
 Number of ECTS: 8
 Course code: F1_MA_MAT_GM1
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: Yes

Lecturer
Coming soon 
Objectives
Learn the concepts of commutative algebra in relation to applications in algebraic number theory, algebraic geometry and other fields of mathematics.

Course learning outcomes
The successful students possesses deepened and extended knowledge of the topics treated in Commutative Algebra. 
Description
In number theory one is naturally led to study more general numbers than just the classical integers and, thus, to introduce the concept of integral elements in number fields. The rings of integers in number fields have certain very beautiful properties (such as the unique factorisation of ideals) which characterise them as Dedekind rings. Parallelly, in geometry one studies affine varieties through their coordinate rings. It turns out that the coordinate ring of a curve is a Dedekind ring if and only if the curve is nonsingular (e.g. has no selfintersection).With this in mind, we shall work towards the concept and the characterisation of Dedekind rings. Along the way, we shall introduce and demonstrate through examples basic concepts of algebraic geometry and algebraic number theory. Moreover, we shall be naturally led to treat many concepts from commutative algebra.Depending on the previous knowledge of the audience, the lecture will cover all or parts of the following topics:(1) General concepts in the theory of commutative ringsrings, ideals and modulesNoetherian ringstensor productslocalizationcompletiondimension(2) Number ringsintegral extensionsideals and discriminantsNoether's normalisation theoremDedekind ringsunique ideal factorisation(3) Plane Curvesaffine spacecoordinate rings and Zariski topologyHilbert's Nullstellensatzresultant and intersection of curvesmorphisms of curvessingular points 
Assessment
Mode: Continuous or Combined For Master in Mathematics The final mark is the arithmetic mean of a mark from continuous assessment and a mark for the final exam.For Master in Secondary Education – Mathematics The final mark is the arithmetic mean of a mark from continuous assessment and a mark for a didactical paper. 
Note
Note / Literature / Bibliography Lecture notes, exercise sheets (available on Moodle) E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry Dino Lorenzini. An Invitation to Arithmetic Geometry, Graduate Studies in Mathematics, Volume 9, American Mathematical Society M. F. Atiyah, I. G. Macdonald. Introduction to Commutative Algebra, AddisonWesley Publishing Company.

Details
 Number of ECTS: 6
 Course code: F1_MA_MAT_FM1
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Introduction to basic concepts of modern probability theory

Course learning outcomes
On successful completion of the course, the student should be able to: Understand and use concepts of modern probability theory (e.g., filtrations, martingales, stopping times) Apply the notion of martingale to model random evolutions Know and apply classical martingale convergence theorems Describe and manipulate basic properties of Brownian motion 
Description
RadonNikodym Theorem, conditional expectations, martingales, stopping times, optional stopping theorems, Doob’s inequalities, martingale convergence theorems, martingale central limit theorem, Brownian motion. 
Assessment
Written exam 
Note
Rick Durrett: Probability Theory and Examples Yuval Peres, Peter Morters: Brownian Motion

Details
 Number of ECTS: 5
 Course code: MA_DS31
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Course learning outcomes
At the end of the course, a student should be familiar with the basic concepts of probability theory (event, random variables, distributions) and master the tools that allow him or her to make calculations (calculations of expectations, variances, distributions etc). The student should also be able to master the law of the large numbers as well as the central limit theorem. In particular, he or she must be able to calculate limits of random variables. Given an integral, the student should also be able to provide a probabilistic approximation of it and assess the probability of error. 
Description
What is an event, the probability of an event, a random variable?What are the main notations associated to probability theory?What is a law or a distribution? The main probability distributions: Bernoulli, binomial, Poisson, uniform, exponential, Cauchy, Gaussian. What random phenomena do they model? What is a density with respect to the Lebesgue measure? What is a density with respect to the counting measure? What is the distribution of a random variable?What is the expectation (or mean) of a random variable? How can we interpret it? What is the variance of a random variable?How to calculate the probability that a random variable belongs to a set from its distribution? How to calculate the expectation of a function of a given random variable from its distribution? What is a distribution function? What are its properties? What link densities and distribution functions? What are the quantiles of a realvalued random variable? What are the median and quartiles of a random variable?The main inequalities: the Markov inequality, the BienayméTchebychev inequality, the Jensen inequality.What does it mean that two random variables are independent? What is the density of a pair of independent random variables? How to generalize to n independent random variables? What is a random vector? What are the marginal distributions? How to calculate them?How to calculate the distributions of random variables (from distribution functions, change of variables)What does it mean that a sequence of random variables converges almost surely? In probability? In distribution? What are the connections between these convergence modes?What is Law of large numbers? What is the Monte Carlo method for calculating integrals?What is the central limit theorem? How can it be used to evaluate the error in the Monte Carlo method.

Details
 Number of ECTS: 3
 Course code: F1_SECEDUC16
 Module(s): General Mathematics 1
 Language: FR
 Mandatory: No

Lecturer
Coming soon 
Objectives
les questions et la taxonomie de Bloom / Anderson & Krathwohl les problèmes de compréhension et les moments clés d’une leçon/d’un apprentissage la planification d’une leçon/ d’une séquence de leçons l’évaluation formative et sommative

Course learning outcomes
Être conscient du fait que chaque choix didactique a toujours un impact sur l’enseignement et l’apprentissage des contenus qui en dépendent Être capable de formuler des questions visant différents niveaux taxonomiques Être capable d’anticiper les problèmes de compréhension éventuels des élèves Être capable de déceler les moments clés d’une leçon/d’un apprentissage Être capable de planifier une séquence de leçons comprenant une introduction, des activités, des transitions, des tempsélèves, des exemples et exercices et les questions à poser ainsi que les réponses éventuelles des élèves Être capable de formuler des questions se prêtant pour une évaluation du type formatif/sommatif Être capable de planifier une leçon de façon que l’atteinte des objectifs d’apprentissage puisse se faire de plusieurs manières différentes 
Description
Présentation de leçons et de types d’introductions Taxonomie de Bloom / Anderson & KrathwohlDiscussion d’exercices et de feuilles de travailIntroduction aux différentes formes de tempsélèvesPréparation de leçons (avec un accent sur le cycle inférieur et moyen) – Comment faire ? Good practice :Public cibleImportance du sujet pour l’élèveObjectifs à atteindreMatériel disponible/à utiliserMéthodes didactiquesMoyens techniques, Visualisation, Apps, Jeux,RéflexionIntroductions et RappelsEnchaînements et TransitionsQuestions à poser, types de questions et taxonomieMoments clésExercicesÉvaluation – pourquoi, comment, types d’évaluation, fréquenceRessourcesProblèmes et conséquences des choix didactiques opérésPréparation au stage en janvier 
Assessment
Examen final : Remise d’une préparation écrite de leçons (avec présentation) Évaluation continue à l’aide de travaux hebdomadaires (à remettre individuellement, en binôme ou en groupe – suivant la tâche imposée) 
Note
Litérature Barzel, B., Büchter, A., Leuders, T. (2007), Mathematik Methodik, Handbuch für die Sekundarstufe I und II, Cornelsen Hattie, J. (2010), Visible Learning, Routledge Leuders, T. (2020), Mathematikdidaktik (9e édition), Cornelsen Leuders, T., Prediger, S. (2016), Flexibel differenzieren und fokussiert fördern im Mathematikunterricht, Cornelsen Malle, G. (1993), Didaktische Probleme der elementaren Algebra, Vieweg Padberg, F., Wartha, S. (2017), Didaktik der Bruchrechnung (5e édition), Springer Sill, H.D. (2018), Grundkurs Mathematikdidaktik, utb. Sturm, R. (2016), Schritt für Schritt zum guten Mathematikunterricht, Klett Kallmeyer

Details
 Number of ECTS: 6
 Course code: F1_MA_MAT_MMCS22
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The aim of the course is to provide students with efficient tools to study linear operators between infinite dimensional vector spaces. In particular, this class will deal with normed vector spaces, Banach and Hilbert spaces, with bounded linear operators on normed vector spaces, fundamental principles such as Fourier analysis, Lebesgue integral, HanBanach Theorem, Uniform Boundedness Principle or Closed Graph Theorem, and with spectral theory of compact (selfadjoint) linear operators.

Course learning outcomes
Students will acquire a solid understanding of functional analysis, its fundamental results and basic techniques. In particular, students will understand applications to measure theory, Fourier theory, and the spectral theorem for (unbounded) operators. Students will know the relevance of a theorem, its underlying motivation and a precise idea of its proof. Hopefully, students will demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis. 
Description
Functional analysis aims to study infinitedimensional spaces of functions and features of linear operators on these spaces. Though, from the point of view of Mathematics, functional analysis has its own interest, it plays a crucial role in many related areas, in particular in Physics, Engineering or Finance. Roughly speaking, most “reallife problems” involve nonlinear partial differential equations with infinitedimensional solution spaces of functions (or distributions). This course provides Master students with the basic tools and the fundamental results to develop skills to solve such problems. 
Assessment
There will be an exam after the first half of the semester, and a final exam at the end of semester. Attendance is mandatory. Homework assignments will be posted as recommended problems, but not graded. Selected problems will be discussed in class. If you miss a lecture, you are responsible for obtaining lecture notes and for determining if any announcements were made. 
Note
Note / Literature / Bibliography Introductory Functional Analysis with Applications, by Erwin Kreyszig.Functional Analysis, Spectral Theory and Applications, by Manfred Einsiedler and Thomas Ward.An Introduction to Fourier Analysis, by Russell L. Herman

Details
 Number of ECTS: 8
 Course code: F1_MA_MAT_GM2
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: Yes

Lecturer
Coming soon 
Course learning outcomes
On successful completion of the course, the student should be able to: compute with tensors, such as metrics and curvature, in coordinates or coordinatefree demonstrate detailed knowledge of the exponential map, including criteria for completeness, conjugate points, and Jacobi fields understand multiple interpretations of curvature, for example, as the obstruction to a local parallel framing, or how it influences the spreading of geodesics via the Jacobi equation 
Description
This foundational Masterlevel course is centered around the concepts of connection on a manifold and curvature. These are investigated in further depth in the setting of Riemannian or Lorentzian metrics, where the geometry of geodesics and its relation to curvature are studied. 
Assessment
Oral or written exam 
Note
Literatur M. P. do Carmo, Riemannian Geometry, Birkhäuser (1992) B. O’Neill, SemiRiemannian Geometry with applications to Relativity, Elsevier (1983) John M. Lee, Introduction to Smooth Manifolds (2nd edition), Springer (2012)

Details
 Number of ECTS: 6
 Course code: F1_MA_MAT_FM2
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Introduction to basic concepts of Stochastic Analysis.

Description
Continuous martingales, stochastic integration, quadratic variation, Itô calculus, theorem of Girsanov, stochastic differential equations, Markov property of solutions, connection of stochastic differential equations and partial differential equations, martingale representation theorems, chaotic expansions, FeynmanKac formula. 
Assessment
Written exam 
Note
I. Karatzas, S. Shreve: Brownian motion and stochastic calculus. 2nd edition. Springer, 1991 B. Oksendal: Stochastic differential equations. Springer, 2003 D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. Springer Grundl., 1999

Details
 Number of ECTS: 7
 Course code: F1_MA_MAT_GM3
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The goal of the course it to get acquainted with Partial differential equations (PDE) as a powerful tool for modeling problems in science, providing functional analytic techniques in order to deal with PDE.

Course learning outcomes
On successful completion of the course the student should be able to: Apply methods of Fourier Analysis to the discussion of constant coefficient differential equations Work freely with the classical formulas in dealing with boundary value problems for the Laplace equation Prove acquaintance with the basic properties of harmonic functions (maximum principle, mean value property) and solutions of the wave equation (Huygens property) Solve Cauchy problems for the heat and the wave equations Give a pedagogic talk for peers on a related topic 
Description
Fourier transform, the classical equations, spectral theory of unbounded operators, distributions, fundamental solutions. 
Assessment
Written exam 
Note
Literatur Rudin: Functional analysis Jost: Postmodern analysis Folland: Introduction to partial differential equations. ReedSimon: Methods of mathematical physics IIV

Details
 Number of ECTS: 6
 Course code: F1_MA_MAT_FM3
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Introductory course to basic concepts of Mathematical Finance, also suitable for students who are not going to choose their specialization in Finance. The goal is to deepen the knowledge of modern probability theory by studying applications of general interest in an actual field of applied mathematics.

Course learning outcomes
On successful completion of the course, the student should be able to: Derive and apply formulas for option pricing and hedging strategies Carry out calculations based on arbitrage arguments Calculate the price of European call and put options using the Cox, Ross and Rubinstein model Apply the techniques of Snell envelopes to evaluate American options Derive the classical BlackScholes formulas as limiting case of a sequence of CRR markets 
Description
Discrete financial markets, the notion of arbitrage, discrete martingale theory, martingale transforms, complete markets, the fundamental theorem of asset pricing, European and American options, hedging strategies, optimal stopping, Snell envelopes, the model of Cox, Ross and Rubinstein. 
Assessment
written exam 
Note
D. Lamberton, B. Lapeyre: Introduction au calcul stochastique appliqué à la finance. Ellipses, 1997 S. E. Shreve: Stochastic calculus for finance. I: The binomial asset pricing model. Springer Finance, 2004 H. Föllmer, A. Schied: Stochastic finance. An introduction in discrete time. 2nd ed., de Gruyter, 2004 F. Delbaen, W. Schachermayer: The mathematics of arbitrage. Springer Finance, 2006

Details
 Number of ECTS: 1
 Course code: F1_MA_MAT_MMCS24
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Introduction Getting started Basics of Python Array computations with numpy Array computations with numpy (cont.) Plotting with matplotlib Tabular data manipulation with pandas Tabular data manipulation with pandas (cont.) Writing good quality and robust Python code

Course learning outcomes
You should feel comfortable writing basic scientific programs in Python, and be able to participate fully in future courses that require an element of programming. 
Description
This course covers the basics of scientific programming with Python. It is aimed at people who have done some programming before, perhaps on an undergraduate course, but need a refresher before starting their Masters or Doctoral degrees at the University.https://jhale.github.io/scientificpython/ 
Assessment
A coursework will be distributed at the end of the class. To pass the course and receive the ECTS credits you must complete the coursework.

Details
 Number of ECTS: 6
 Course code: F1_MA_MAT_MMCS25
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The objective of the course is to provide the main tools in numerical analysis to manage rigorously the computer solution of practical problems in physics and engineering.

Course learning outcomes
At the end of the course, a student will be able to understand the main methods, the rigorous mathematical analysis and algorithms developed in numerical analysis to master their concrete implementation on a computer to formulate some basic physics or engineering applications in view of their treatment by a numerical method and its computer solution. 
Description
The main results about the theory of numerical methods will be explained. During the course, the student will implement algorithms in Python to propose some concrete, robust and efficient scientific computing solutions to concrete problems. The course content is the following:Notion of error in numerical analysisPolynomial interpolation and approximationNumerical derivation and integrationNumerical solution of Ordinary Differential EquationsDirect and iterative methods for solving linear systemsNonlinear equations in one and several variablesProject presentation + Complementary topics (optimization, eigenvalues problems) 
Assessment
The evaluation will be based on continuous assessment (weekly programming exercises – 50 % of the grade) and a project presentation with a report (50 % of the grade). 
Note
X. Antoine, Numerical Analysis, course at the University of Luxembourg Ascher, Uri M and Greif, Chen, A First Course in Numerical Methods, SIAM, 2011. Heath, Michael T, Scientific computing: an introductory survey, revised second edition SIAM, 2018.

Details
 Number of ECTS: 1
 Course code: F1_MA_MAT_GM5
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon

Details
 Number of ECTS: 3
 Course code: MA_DS8
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
On successful completion of the course the students should be able to understand the relevance of the topics covered in the course for their applications, master the proofs of the main results of the course, solve problems using the toolkit developed in the course be autonomous in learning in the field of Graph Theory.

Description
Through a presentation of selected topics, the course aims to be an introduction to graph theory, its applications and its algorithmic aspects. It is designed as a selfcontained course and focused on problems pertaining to Data Science. Possible topics for the course include, but are not limited toGraphs and digraphs, degree and the degree sequence algorithmConnectedness, distance, shortest paths and connected components algorithmsGraph matching problems and algorithmsElements of algebraic graph theory and PageRank algorithmGraph traversal algorithmsTrees and applicationsMinimum spanning tree algorithmsNetwork flow, min cut – max flow theorem and Ford–Fulkerson algorithmCentrality and betweness measuresCluster analysisRandom Graphs 
Assessment
First session Written exam and homework, and possibly algorithm implementation project during the semester. Retake exam Writen exam 
Note
Note / Literature / Bibliography R. Diestel, Graph Theory, Springer, 2017 D. Jungnickel, Graphs, Networks and algorithms, Springer 2017

Details
 Number of ECTS: 4
 Course code: F1_MA_MAT_GM7
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Course learning outcomes
On successful completion of the project the student should be able to: analyse complex tasks propose solution strategies, break up a longer project into subsequent steps, apply a variety of methods in one project, present a task and its solution in a scientific way. 
Description
The student project consists of project work that is carried out under the supervision of a professor or a postdoc. The work is either individual or group work. Group work needs the explicit approval of the Study Director.At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS). The student(s) need to notify the Study Director of the project and the tasks at the latest on 15 October.The project outcome is a pdf document written by the student. Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December. 
Assessment
The students are marked for their project work.In the case of a retake exam, a new project has to be done. The retake can be done with other supervisors. 
Note
Note / Literature / Bibliography Depends on the project.

Details
 Number of ECTS: 1
 Course code: F1_MA_MAT_GM8
 Module(s): General Mathematics 1
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Course learning outcomes
On successful completion of the Student Seminar 1, the students should be able to: Acquire good insight into a field by means of individual work Deliver a mathematical lecture on a topic of their choice Share their knowledge with others 
Description
Every participant chooses a supervisor among the academic staff of the Department of Mathematics or among the instructors of the Master in Mathematics and, jointly with the supervisor, a topic for a talk.The audience of the talk consists at least of the participants of the Student Seminar 1, the supervisor and one other academic staff member of the Department of Mathematics (e.g. the supervisor of another talk). The duration of the talk is 40 minutes (time for questions included). 
Assessment
The evaluation is based on the quality of the talk delivered by the student 
Note
Note / Literature / Bibliography Depends on the subject and is communicated by the chosen supervisor
Coming soon
Course offer for General Mathematics (Académique), Semestre 3

Details
 Number of ECTS: 2
 Course code: F1_SECEDUC18
 Module(s): General Mathematics 3
 Language: FR
 Mandatory: No

Lecturer
Coming soon 
Objectives
Comment lire et interpréter les différents programmes de mathématiques ? Quelle est la progression didactique à travers les 7 années de l’enseignement secondaires ? Quelles sont les similarités et les différences entre les attentes dans les différents ordres d’enseignement/sections ? Quelles sont les attentes finales et les questions d’examen qui les précisent ? Comment rendre opérationnel les notions de compétences procédurales ?

Course learning outcomes
Working through the program from a higher standpointsecondary school analysis and geometry 
Description
Le programme cadre pour l’enseignement des mathématiques dans l’enseignement secondaire classique et général les domaines de l’algèbre et des probabilitésles compétences à développerExplicitation des finalités de l’enseignement des mathématiques à l’aide de questions d’examenPréparation au concours de recrutementConceptsclé pour l’enseignement des mathématiques :activités mathématiques de baseapprentissage autonomesituations d’apprentissage ouvertes et complexesrésolution de problèmes

Details
 Number of ECTS: 3
 Course code: F1_SECEDUC19
 Module(s): General Mathematics 3
 Language: FR
 Mandatory: No

Lecturer
Coming soon 
Objectives
Savoir concevoir des situations d’apprentissage Savoir évaluer les apprentissages

Course learning outcomes
Être capable de planifier une séquence de leçons comprenant une introduction, des activités, des transitions, des tempsélèves, des exemples et exercices et les questions à poser ainsi que les réponses éventuelles des élèves Être capable d’anticiper les problèmes de compréhension éventuels des élèves Être capable de déceler les moments clés d’une leçon/d’un apprentissage Être capable de planifier une leçon permettant d’atteindre les objectifs d’apprentissage de plusieurs manières différentes Être capable de formuler des questions se prêtant pour une évaluation du type formatif/sommatif Être capable de concevoir une épreuve sommative adaptée aux objectifs d’apprentissage visés qui respecte le principe de la structure et de la répartition des points d’un devoir en classe imposé par le programme du niveau d’enseignement en question Connaître les critères de correction des devoirs en classe en mathématiques Être capable d’évaluer les tests et examens de mathématiques Être capable d’utiliser les résultats de l’évaluation de manière diagnostique et remédiative 
Description
Préparation détaillée de leçons de mathématiques (avec un accent sur le cycle supérieur et moyen)Planification des unités et séquences de mathématiques avec choix de méthodes et de formes sociales appropriéesIntroduction aux différents types d’évaluationConception d’épreuves et de devoirs en classe en mathématiquesTravail avec des grilles de compétences/indicateursGestion de l’erreur dans le processus d'apprentissage en mathématiquesMesures de différenciationPréparation au stage en janvier 
Assessment
Évaluation continue à l’aide de travaux hebdomadaires (à remettre individuellement, en binôme ou en groupe – suivant la tâche imposée) Examen final : Remise d’une préparation écrite de leçons suivie d’un examen oral 
Note
Litérature Antibi, A. (2003), La constante macabre, Math’Adore Antibi, A. (2021), La folie de l’évaluation, Math’Adore Astolfi, J.P. (2015), L’erreur, un outil pour enseigner, ESF Editeur Barzel, B., Büchter, A., Leuders, T. (2007), Mathematik Methodik, Handbuch für die Sekundarstufe I und II, Cornelsen Büchter, A., Leuders, T. (2005), Mathematikaufgaben selbst entwickeln, Lernen fördern – Leistung überprüfen, Cornelsen Hattie, J. (2010), Visible Learning, Routledge Leuders, T. (2020), Mathematikdidaktik (9. Auflage), Cornelsen Leuders, T., Prediger, S. (2016), Flexibel differenzieren und fokussiert fördern im Mathematikunterricht, Cornelsen Sill, H.D. (2018), Grundkurs Mathematikdidaktik, utb. Sturm, R. (2016), Schritt für Schritt zum guten Mathematikunterricht, Klett Kallmeyer

Details
 Number of ECTS: 6
 Course code: MAMATH173
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The purpose of this course is to give an introduction into the theory of finite dimensional Lie groups and Lie algebras, assuming some basic knowledge of differentiable manifolds.

Course learning outcomes
On successful completion of the course, the student should be able to: Expound the mathematical foundation behind symmetries of solid bodies, dynamics of mechanical systems, and geometric structures in nature. Explain the deep interrelations between Lie groups and Lie algebras, as well as the technical tools behinds these interrelations. Simplify mathematical problems admitting symmetry Lie groups actions to problems admitting symmetry actions of their Lie algebras. Master applications to the theory of manifolds and representation theory, which in turn have applications in physics, engineering and mechanics. 
Description
The Lie algebra of a Lie group, the exponential map, the adjoint representation, actions of Lie groups and Lie algebras on manifolds, the universal enveloping algebra, basics of the representation theory. 
Assessment
Written examination 
Note
Literature "Lie groups and Lie algebras" by Eckhard Meinrenken, 83 pages (free to download) “Prerequisites from Differential Geometry” by Sergei Merkulov (free to download)

Details
 Number of ECTS: 6
 Course code: MAMATH174
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Beyond learning exciting material, the course is designed to explore and experience the process of mathematical research.

Course learning outcomes
The course requires minimal prerequisites (some linear algebra, Euclidean geometry and basic topology) but aims to explore results that are at the limit of current known understanding. In particular, we’ll discuss some open problems and try to illustrate the process of modern research. The subjects are chosen so that they can be treated with a handson approach, and this approach and experience are as important for this course as the actual content. 
Description
The course will cover a selection of themes from combinatorial aspects of geometry.Themes include general theorems about convex sets in n dimensional real space (and Helly type theorems), Minkovski’s first theorem for lattices, and Ramsey theory (graph coloring problems). 
Assessment
Oral exam and classwork 
Note
Literature Varied. (A mix of lecture notes and articles.)

Details
 Number of ECTS: 2
 Course code: MAMATH144
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
On successful completion of the project the student should be able to: cooperate effectively in a team, analyse complex tasks propose solution strategies, break up a longer project into subsequent steps, apply a variety of methods in one project, present a task and its solution in a scientific way.

Description
The student group project consists of project work that is carried out by a group of (usually two or three) students under the supervision of a professor or a postdoc.At the beginning of the project, supervisor and students define tasks to be carried out by the students, corresponding to the volume of 50 working hours (2 ECTS). The student(s) need to notify the Study Director of the project and the tasks at the latest on 15 October.The project outcome is a pdf document written by the student. Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December. 
Assessment
The students are marked for their project work.In the case of a retake exam, a new project has to be done. The retake can be done in other groups and with other supervisors. 
Note
Literature Depends on the project.

Details
 Number of ECTS: 6
 Course code: MAMATH147
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Know examples of and be able to explain the continuity of mathematics from classical problems (as taught in school) to modern research questions. Understand the relevance of knowing modern mathematics for being able to teach an integral picture of mathematics at secondary schools. Be able to work with mathematical notions going beyond secondary school and Bachelor level. Master the basics of local fields and be able to compute with padic numbers, as generalisation of the real numbers. Master the fundamentals of the theory of quadratic forms, be able to explain its origin in the study of conics, know and be able to apply important theorems, be able to handle examples. Master the basics of elliptic curves, know about their classical origin, know about their application in cryptography, understand their relevance for current number theory research, know and be able to apply important theorems, be able to handle examples.

Description
This course leads from classical mathematics (real numbers, conics, “classical” geometry, plane curves) to some topics in modern number theory and geometry and underlines the continuity from classical geometry (as taught in school) and classical number theory to the modern points of view.It covers padic numbers and more generally local fields as analogues of the real numbers, quadratic forms (arising from the study of conic sections) and elliptic curves (arising from the study of certain integrals), as well as some of their relevance for modern mathematics.Having their origin in the study of conics, the theory of quadratic forms is a modern theory situated in both geometry and number theory with plenty of applications. It turns out that for a full classification of quadratic forms, one needs to introduce analogues for the real numbers: the socalled padic numbers, or, more generally, local fields. In the first part of the lecture, the theory of quadratic forms is introduced, number theory applications are treated, padic numbers are dealt with, and the classification theorem is fully proved.The second part of the course is concerned with elliptic curves. These are curves arising from the study of certain integrals. They are relevant in everyday life for their fundamental role in Elliptic Curves Cryptography (e.g. used in ID cards, passports). In the language of modern geometry, they are curves of genus one with a rational point. For number theory, they appear in many of the most important questions of current research, e.g. the BirchandSwinnertonDyer conjecture, which is one of the 7 Millenium Problems. In the course, elliptic curves are introduced in modern geometric language, thus introducing this language, and several important number theoretic and geometric properties are proved, such as the addition law (making them into a group, a complicated generalisation of the integers), and statements on their rational points (number theoretic “Diophantine” question).The course will be a classical lecture, complemented by integrated exercises and contributions by the students via short talks.Students from the Master in Secondary Education will be asked to focus in their contributions on how to link the topics of their lectures with High School mathematics. 
Assessment
Students will be rated for their contributions via short talks, the presentation of homework and the performance in short supervised exercises. The number and the lengths of the short talks will depend on the number of participants and will be fixed in the beginning of the course 
Note
Literature Serre: A course in arithmetic, Springer Silverman: The arithmetic of elliptic curves, Springer Anni, Deo, Wiese: Lecture notes for Topics in Number Theory and Geometry, distributed during the lecture

Details
 Number of ECTS: 3
 Course code: MICS238
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The objective of this lecture first consists in providing a structured approach to students in terms of optimization problem modeling. Next various solving techniques based on exact methods (A*, B&B, LP), approximated ones (heuristics, metaheuristics, problem relaxation) and hybrids are described. Students are also taught how to validate the proposed solution by having a scientific approach

Description
Introduction to optimisation and decision problems Linear programming, graphical interpretation and primal simplex Branch&Bound, A* The scheduling problem List algorithms, greedies, heuristics MetaHeuristics and Evolutionary computation 
Assessment
Final Exam: 100%

Details
 Number of ECTS: 5
 Course code: MAMATH114
 Module(s): General Mathematics 3
 Language: FR, EN
 Mandatory: No

Lecturer
Coming soon 
Course learning outcomes
On successful completion of the Internship in a Company, the student should be able to: Identify employment opportunities in the local job market and possible career tracks; Apply insights gained during the internship into the functioning and the operational business of companies and institutions outside the academic world; Perform work in an institution outside the academic world at a level adequate to a Master student in mathematics; Apply and adapt the theoretical skills and techniques learned in the Master programme to the practical needs of the chosen company or institution. 
Description
The objective of the Internship in a Company is to allow the student to get acquainted with a professional work situation outside the academic context.The Internship is typically carried out in the summer break between the 1st and the 2nd year and counts for the 3rd semester. A typical duration is 2 months.It is the student’s responsibility to find a company. In order to establish a first contact, the student can (but need not) make use of an existing list of contacts at companies.After the student and a company agreed on the internship, the student contacts the Study Director and the Study Programme Administrator. The latter will proceed with the establishment of an internship convention between the University and the company, which will be signed by the Study Director, a representative of the company and the student. The model contract of the University need be used. It is accompanied by two Annexes: Annex 1 is an evaluation form that the company representative fills in after the internship; Annex 2 is a description of the planned tasks of the intern. Annex 2 has to be approved by the Study Director in order to ensure that the internship benefits the student in her/his studies.An academic supervisor, who is an academic employee at the University, is named. The academic supervisor is the contact point for both the student and the company during the internship and (s)he keeps an eye on the execution of the internship.Remark: By law, internships carried out in Luxembourg of a duration of at least 4 weeks have to be remunerated by the company. Internships can also be carried out in other countries, but then the student must make sure that (s)he has the legal right to work in that country. 
Assessment
The course is either succeeded or failed, but not marked.For succeeding, the student needs to get an overall positive evaluation by the company supervisor (Annexe 1 to the convention), and the student needs to hand in a report after the internship, which must be judged satisfactory by the academic supervisor and the SPD. The report needs to cover at least the following topics: Short description of the company and the work environment; Description of the tasks, objectives and the performed work (possibly with examples); How the student was integrated in the company/institution; An indication if the work performed by the student was successful with respect to the objectives; How the performed work fits in her/his Master studies; How the student liked it, if (s)he would recommend other students to choose this internship, possibly with recommendations. 
Note
Summer semester – AprilJune Student looks for a company; can ask Study Director (SPD) or Study Programme Administrator (SPA) for company contacts Student informs SPD and SPA about choice of company SPD finds an academic supervisor SPA prepares internship contract with company SPD checks Annex 2 Summer Typical time for the internship (usually around 2 months) After internship Company representative fills in Annex 1 and sends it to the University Student writes her/his report and sends it to the SPD SPD gets feedback from academic supervisor SPD informs the SPA about the validation of the internship or its failure

Details
 Number of ECTS: 6
 Course code: MAMATH176
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Gaining familiarity with the language and concepts and of algebraic geometry building on commutative algebra. Applying the general theory to concrete simple examples.

Description
We will introduce basic notions of algebraic geometry starting with Hilbert’s Nullstellensatz, algebraic sets, affine and projective varieties over algebraically closed fields. We will then introduce the modern language of sheaf theory and introduce schemes. We will then discuss divisors, line bundles and vector bundles on schemes. 
Assessment
Written exam 
Note
Literature Robin Hartshorne, Algebraic geometry Ravi Vakil, The rising sea Igor Shafarevitch, Basic algebraic geometry 1

Details
 Number of ECTS: 5
 Course code: MAMATH177
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Basic introduction to path integral techniques and MonteCarlo methods in Analysis and Finance.

Course learning outcomes
Upon successful completion of the course students should be able to evaluate functionals of Brownian motion and relate them to PDE; manipulate FeynmanKac formulas; derive stochastic representations of classical initial value problems; derive stochastic representations of classical boundary value problems; calculate MonteCarlo formulas for Greek parameters in financial models 
Description
Stochastic flows associated to second order differential operators, stochastic differential equations and Ldiffusions, FeynmanKac formulas and Dirichlet problems, boundary value problems (elliptic and parabolic), spectral problems of Schrödinger operators, differentiation of heat semigroups, computation of price sensitivities (Greeks). 
Assessment
Written exam 
Note
Bass, Richard F. Diffusions and elliptic operators. SpringerVerlag, 1998. Bass, Richard F. Probabilistic techniques in analysis. SpringerVerlag, 1995. Durrett, Richard. Brownian motion and martingales in analysis. Wadsworth, 1984. Varadhan, S. R. S. Lectures on diffusion problems and partial differential equations. Tata Institute, 1980. Course Notes provided in class

Details
 Number of ECTS: 5
 Course code: MAMATH93
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The objective is to learn the basic concepts and techniques associated with Gaussian processes, that are omnipresent in modelling random phenomena (finance, physics, machine learning, statistics, etc.).

Course learning outcomes
On successful completion of the course, the student should be able to: Explain the language, basic concepts and techniques associated with Gaussian variables, vectors, and processes Identify, analyse, and prove relevant properties of models based on a Gaussian structure Solve exercises involving a Gaussian structure 
Description
Gaussian random variables (characteristic function, CLT, stability properties, Stein's lemma)Gaussian random vectors (definition, characteristic function, existence, uniqueness in law, multivariate CLT, density, Hermite polynomials)Gaussian random processes (definition, modifications, uniqueness in law, function of positive type, existence, Brownian motion, continuity)Fractional Brownian motion (definition, existence, Hölder regularity). 
Assessment
Written exam 
Note
Literature Will be discussed in class

Details
 Number of ECTS: 1
 Course code: MAMATH179
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Description
The Complements to Student Group Project extends the Student Group Project. It can only be taken in conjunction with the Student Group Project. By taking the Complements, the volume of work on the chosen project in the frame of the Student Group Project is increased by 50%. Consequently, students are expected to invest around 25 hours in addition to their time investment for the regular Student Group Project. Moreover, the outcome is also expected to be more substantial.It is possible that not all participants in a project in the frame of the Student Group Project take the complements course. In that case, the students taking the complements write a separate part of the final project report in addition to the jointly written report.

Details
 Number of ECTS: 5
 Course code: MAMATH99
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The objectives of the course are to introduce some advanced discretization techniques for the numerical solution of partial differential equations arising in engineering and applied sciences. The schemes will be explained in details as well as their mathematical properties (e.g. order of accuracy, stability). In addition, these methods will be implemented by using Matlab and tested on concrete problems.

Course learning outcomes
On successful completion of the course the student should be able to: Explain the mathematical foundation of advanced discretization techniques for PDEs Master their concrete implementation on nontrivial engineering boundaryvalue problems Adapt them according to the problem under consideration 
Description
Complements the Finite Element MethodFinite difference schemes in spaceFinite difference schemes fort he discretization of timedependent PDEsIntroduction to integral equations 
Assessment
The students will have to provide some reports that will be evaluated. In addition, a final written examination will be organized. 
Note
Support / Arbeitsunterlagen / Support: Lecture notes (french), exercise sheets (english) Littérature / Literatur / Literature X. Antoine, Numerical solution of PDEs, lecture notes X. Antoine, Numerical Analysis, course at the University of Luxembourg G. Allaire, Analyse Numérique et Optimisation, Presses de l’Ecole Polytechnique P.A. Raviart et J.M. Thomas, Introduction à l’Analyse Numérique des Equations aux Dérivées Partielles, Dunod

Details
 Number of ECTS: 5
 Course code: MA_DS18
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
Upon completion a successful student should be able to Construct classical estimators of unknown probability distribution Derive asymptotic properties of empirical distribution functions Perform tests for unknown distribution functions Construct nonparametric estimators of the density Estimate components of a nonlinear regression model Understand the concept of minimax estimation

Course learning outcomes
Understanding the concepts of empirical distribution functions, kernel estimation and minimax theory Performing derivation of standard estimation and testing methods in nonparametric statistics 
Description
Estimation of probability measuresWeak limit theorems for empirical measuresKolmogorovSmirnov and Cramervon Mises testsEstimation of the densityNadarajaWatson estimator and general kernel estimatorsNonlinear regression modelsMinimax theory 
Assessment
First session Solving biweekly exercises (40%) and a final exam (60%) will build the overall grade.Retake examWritten exam. 
Note
Literatur A.B. Tsybakov (2009): “Introduction to Nonparametric Estimation”, Springer. A. Van der Vaart and J. Wellner (1996): “Weak Convergence and Empirical Processes”, Springer.

Details
 Number of ECTS: 5
 Course code: MAMATH80
 Module(s): General Mathematics 3
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
The successful candidate will understand the basic theoretical concepts of datacentric aspects and will be able to work on datacentric problems. The aim is to continue the work done in the project within the framework of a Master's thesis.

Course learning outcomes
On successful completion of the course the student should be able to: Explain and apply basic theoretical concepts on selected aspects of data processing. Develop appropriate solutions for datacentered problems. Consolidation of the acquired competences in the subject area through a Master's thesis. 
Description
In this course, the term ‘data’ is seen centric and we will look at data from different perspectives. We will discuss selected aspects of Data Preparation and Preprocessing, Data Statistics, Data Security, Data Privacy, Data Management, Big and Small Data, Data Retrieval, Data Visualization, and Data Analytics. 
Assessment
50% of the final mark is done via a workshop (6 December), 50% from a final examThose who do not pass the course have the opportunity to retake the exam in the summer term. 
Note
Literature Elmasri, Navathe: Fundamentals of Database Systems. Pearson Addison Wesley. 2006. Han, Kamber: Data Mining – Concepts and Techniques. Morgan Kaufmann. 2011. Manning, Raghavan, Schütze: Introduction to Information Retrieval. Cambridge University Press.Ware: Information Visualization. Morgan Kaufmann. 2012. Witten, Kamber: Data Mining: Practical Machine Learning Tools and Techniques. Morgan Kaufmann. Aggarwal, Yu: PrivacyPreserving Data Mining – Models and Algorithms. Springer. 2008. Marz: Big Data: Principles and best practices of scalable realtime data systems. Manning. 2015. as well as different articles, reports, and journals contributions.
Course offer for General Mathematics (Académique), Semestre 4

Details
 Number of ECTS: 5
 Course code: MAMATH181
 Module(s): General Mathematics 4
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Description
In this course, the students will learn and present various subjects of higher mathematics. The students can be supervised by different staff members of the DMATH. 
Assessment
Continuous evaluation The students will be evaluated on their presentations.

Details
 Number of ECTS: 5
 Course code: MAMATH107
 Module(s): General Mathematics 4
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Objectives
General acquaintance with the theory of Siegel modular forms.

Description
Siegel modular forms are a natural generalization of elliptic modular forms. These modular forms play an increasingly important role in number theory, algebraic geometry and mathematical physics. The course gives an introduction to and overview of Siegel modular forms. Besides basic topics like the Satake compactification, Hecke operators, and modular forms associated to lattices, we also treat the construction of Siegel modular forms of degree 2 and 3 via invariant theory, a cohomological approach using counting curves over finite fields, and congruences. 
Assessment
Oral exam

Details
 Number of ECTS: 5
 Course code: MAMATH149
 Module(s): General Mathematics 4
 Language: EN
 Mandatory: No

Lecturer
Coming soon

Details
 Number of ECTS: 5
 Course code: MAMATH138
 Module(s): General Mathematics 4
 Language: EN
 Mandatory: No

Lecturer
Coming soon 
Course learning outcomes
On successful completion of the course the student should be able to: Demonstrate proper understanding of the notion of complex manifold Identify the most important examples Master the main techniques of the theory Prove good knowledge of the key theorems Apply the new techniques to geometric problems Give a pedagogic talk for peers on a related topic Write clear and concise lecture notes, including appropriate exercises and applications 
Description
The aim of the lecture course is to give an introduction to complex geometry. In particular the course will treat many examples. After the course the students should understand Hermitian vector bundles and the basics of Kahler manifolds and the Hodge decomposition. The students should be acquainted with the basic theorems and examples. 
Assessment
Oral examination (a presentation) and 12 homework sets 
Note
Literature R.O. Wells: Diﬀeren9al Analysis on Complex Manifolds (main text) S.S. Chern: Complex Manifolds without Potential Theory Griﬃths & Harris: Principles of Algebraic Geometry F. Warner: Foundations of Diﬀerentiable Manifolds D. Huybrechts: Complex Geometry Lecture notes Martin SCHLICHENMAIER, available on the Moodle for download.

Details
 Number of ECTS: 5
 Course code: MAMATH114
 Module(s): General Mathematics 4
 Language: FR, EN
 Mandatory: No

Lecturer
Coming soon

Details
 Number of ECTS: 5
 Course code: MICS217
 Module(s): General Mathematics 4
 Language:
 Mandatory: No

Lecturer
Coming soon 
Objectives
Different problems have different nature. In terms of complexity some problems are called intractable and can not be solved by classical computers. But there are also many other aspects of the nature of optimisation problems such as linearity, convexity, continuity, dynamicity, randomness that may lead the choice of different optimisation techniques

Course learning outcomes
* Characterize problems* Identify the key concepts related to optimisation techniques* Use optimization frameworks* Implement optimization algorithms* Validate optimization algorithms and results* Validate approaches for solving optimization problems 
Description
This lecture confront the students to real instances of such problems. They are first asked to model the problem and next proposed solutions include exact methods, relaxations, approximations, heuristics and metaheuristics. And these practical study cases are supported by the theoretical lectures on Problem Solving (1st semester) 
Assessment
Project: 100%

Details
 Number of ECTS: 20
 Course code: MAMATH30
 Module(s): Master Thesis
 Language: EN
 Mandatory: No

Lecturer
Coming soon