• Course title: Commutative Algebra
• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-1
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: Yes

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

The successful students possesses deepened and extended knowledge of the topics treated in Commutative Algebra.

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 non-singular (e.g. has no self-intersection).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

Mode: Continuous or Combined

Exam modality for the first sessionThe assessment method is combined assessment. The final mark consists of two parts:

A mark for continuous assessment, counting 40% of the final mark

A mark for the end-of-course exam, counting 60% of the final mark.

The mark for the continuous assessment is the arithmetic mean of the marks for three supervised exercises. In the case of a justified absence, a supervised exercise is replaced by an oral exam.

Exam modality for the retake examOral exam

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, Addison-Wesley Publishing Company.

• Course title: Probability Theory and Mathematical Statistics I
• Number of ECTS: 5
• Course code: MA_DS-31
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

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.  

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 real-valued random variable? What are the median and quartiles of a random variable?The main inequalities: the Markov inequality, the Bienaymé-Tchebychev 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.

First sessionCombined assessment (end of course assessment + continuous assessment)Retake examOral exam

• Course title: Functional Analysis
• Number of ECTS: 6
• Course code: F1_MA_MAT_MMCS2-2
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

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, Han-Banach Theorem, Uniform Boundedness Principle or Closed Graph Theorem, and with spectral theory of compact (self-adjoint) linear operators. 

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. 

Functional analysis aims to study infinite-dimensional 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 “real-life problems” involve non-linear partial differential equations with infinite-dimensional 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.

Exam modality for the first sessionWritten test on week of October 28, duration 1 hour, 30% of the final grade,Final written exam (scheduled by the administration), 60% of the final grade,Written homework (problems to be determined), 10% of the final grade.Exam modality for the retake examRetake exam (scheduled by the administration) will be in written form.AbsenceA make-up will be possible only for an unavoidable reason supported by a proof.

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

• Course title: Riemannian Geometry
• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-2
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

On successful completion of the course, the student should be able to:compute with tensors, such as metrics and curvature, in coordinates or coordinate-freedemonstrate detailed knowledge of the exponential map, including criteria for completeness, conjugate points, and Jacobi fieldsunderstand 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

This foundational Master-level 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.  

Oral or written exam

LiteraturM. P. do Carmo, Riemannian Geometry, Birkhäuser (1992)B. O’Neill, Semi-Riemannian Geometry with applications to Relativity, Elsevier (1983)John M. Lee, Introduction to Smooth Manifolds (2nd edition), Springer (2012)

• Course title: Partial Differential Equations I
• Number of ECTS: 7
• Course code: F1_MA_MAT_GM-3
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

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.

On successful completion of the course the student should be able to: Apply methods of Fourier Analysis to the discussion of constant coefficient differential equationsWork freely with the classical formulas in dealing with boundary value problems for the Laplace equationProve 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 equationsGive a pedagogic talk for peers on a related topic 

Fourier transform, the classical equations, spectral theory of unbounded operators, distributions, fundamental solutions.

Written exam

LiteraturRudin: Functional analysisJost: Postmodern analysisFolland: Introduction to partial differential equations.Reed-Simon: Methods of mathematical physics I-IV

• Course title: Scientific Python
• Number of ECTS: 1
• Course code: F1_MA_MAT_MMCS2-4
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

IntroductionGetting startedBasics of PythonArray computations with numpyArray computations with numpy (cont.)Plotting with matplotlibTabular data manipulation with pandasTabular data manipulation with pandas (cont.)Writing good quality and robust Python code

First sessionYou should feel comfortable writing basic scientific programs in Python, and be able to participate fully in future courses that require an element of programming.Retake examRetake exam not possible, course must be retaken.

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/scientific-python/ 

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.

• Course title: Introduction to Graph Theory
• Number of ECTS: 3
• Course code: MA_DS-8
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

On successful completion of the course the students should be able tounderstand 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 coursebe autonomous in learning in the field of Graph Theory.

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 self-contained 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

First sessionWritten exam and homework, and possibly algorithm implementation project during the semester.  Retake exam Writen exam 

Note / Literature / BibliographyR. Diestel, Graph Theory, Springer, 2017D. Jungnickel, Graphs, Networks and algorithms, Springer 2017

• Course title: Elementary mathematics from a higher viewpoint I
• Number of ECTS: 4
• Course code: F1_SECEDUC-26
• Module(s): Specialisation – Mathematics 1
• Language: FR, EN
• Mandatory: No

Les étudiant·e·s vont revisiter leurs connaissances des mathématiques de fin de secondaire. En parallèle, ils·elles aborderont ces sujets sous un angle nouveau, grâce à leur maîtrise des mathématiques de niveau supérieur. Les étudiant·e·s du MSE-M Master devront être capable de répondre aux questions possibles de leur futur·e·s élèves.

Le cours explorera les thématiques des mathématiques de fin de secondaire sous un angle plus abstrait. Les concepts universitaires seront utilisés pour revisiter les mathématiques du secondaire et pour approfondir la théorie derrière chaque formule. Des questions de compréhension et des liens entre différents sujets seront également abordés.Attention: les thématiques abordées sont divisées en deux groupes et sont traitées en alternance chaque année.

Exam modalities for the first session

3 midterms (50% of the final mark) and 1 written exam in january (50 % of of the final mark).

Exam modalities for the retake exam

1 written exam (100 % of final mark)

Absence plan

if justified absent during midterms, an oral exam will be organised about the same material, if justified absence during the written exam, an oral exam will take place

• Course title: Student project 1
• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-7
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

On successful completion of the project the student should be able to:analyse complex taskspropose 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.

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.

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 / Literature / BibliographyDepends on the project.

• Course title: Student seminar 1
• Number of ECTS: 1
• Course code: F1_MA_MAT_GM-8
• Module(s): Specialisation – Mathematics 1
• Language: EN
• Mandatory: No

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

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).

The evaluation is based on the quality of the talk delivered by the student

Note / Literature / Bibliography Depends on the subject and is communicated by the chosen supervisor

• Course title: Didactics of Algebra
• Number of ECTS: 2
• Course code: F1_SECEDUC-31
• Module(s): Didactics – Mathematics 1
• Language: FR, EN, DE
• Mandatory: Yes

Transitioning from arithmetic to abstract algebra is a significant mathematical journey for students, and the teacher’s role is to guide them through this complex process. Elementary algebra presents specific didactical challenges that require careful consideration. The primary objective of this course is to analyze these challenges and discuss strategies for addressing them. A secondary objective is to revisit certain algebraic topics from an advanced perspective to enhance mathematical understanding.

By the end of the course, students will be able to critically reflect on the didactical challenges associated with teaching elementary algebra. They will develop the ability to propose solutions to both mathematical and didactical problems related to algebra.

The course will primarily reference “Didaktische Probleme der elementaren Algebra” by Malle, Wittmann, and Bürger. Additionally, a selection of recent papers on the didactics of algebra will be examined. The mathematical content will be drawn from a variety of sources. A complete reference list and detailed subtopic descriptions will be provided to students on Moodle.

• Course title: Préparation de leçons de mathématiques
• Number of ECTS: 3
• Course code: F1_SECEDUC-16
• Module(s): Didactics – Mathematics 1
• Language: FR
• Mandatory: Yes

Maîtriser la conception de situations d’apprentissage.

À l’issue de ce cours, les étudiants devront être capables de concevoir des séquences d’apprentissage structurées et cohérentes en mathématiques, en s’appuyant sur une formulation claire et mesurable des objectifs d’apprentissage. Ils devront maîtriser diverses techniques de questionnement leur permettant de stimuler efficacement la réflexion des élèves et être capables d’intégrer des stratégies de différenciation pédagogique adaptées dans leurs leçons, assurant ainsi un enseignement qui répond aux besoins variés des apprenants.

Ce cours est conçu comme une préparation essentielle au stage pratique pour les futurs enseignants de mathématiques. Il vise à développer les compétences nécessaires à la conception et à la mise en œuvre de leçons efficaces. Les étudiants seront formés aux aspects essentiels de la planification, de la didactique et de l’évaluation en mathématiques. Ils apprendront à structurer des séquences d’apprentissage, à formuler des objectifs clairs, à développer des stratégies de questionnement efficaces, et à intégrer des méthodes d’évaluation formative. 

Première session d’examenL’évaluation du cours combinera contrôle continu (20%) et examen final (80%). Le contrôle continu évaluera la participation en classe et les travaux réguliers, réalisés individuellement ou en groupe. L’examen final comprendra une préparation écrite de leçons (60% de la note finale) et une soutenance orale (20%).Évaluation lors d’une deuxième session (rattrapage)L’évaluation reposera entièrement sur un examen final (100% de la note). Celui-ci consistera en une préparation écrite de leçons (75% de la note) suivie d’une soutenance orale (25%).  

Note / Literature / BibliographyAstolfi, J.-P. (2024). L’erreur, un outil pour enseigner. Paris : ESF Éditeur.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.Jonnaert, P., & Vander Borght, C. (2003). Créer des conditions d’apprentissage. Bruxelles : De Boeck.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.Meyer, H. (1989). Unterrichtsmethoden. I: Theorieband und II: Praxisband. Berlin: Cornelsen Scriptor.Morissette, R. (2002). Accompagner la construction des savoirs. Montréal: Chenelière/McGraw-Hill.Rieunier, A. (2018). Préparer un cours. Tome 1. Applications pratiques. Paris : ESF.Rieunier, A. (2012). Préparer un cours. Tome 2. Les stratégies pédagogiques efficaces. Paris : ESF.Sill, H.-D. (2018). Grundkurs Mathematikdidaktik. UTB.Sturm, R. (2016). Schritt für Schritt zum guten Mathematikunterricht. Klett Kallmeyer.

• Course title: Internship in a secondary school I
• Number of ECTS: 0
• Course code: F1_SECEDUC-17
• Module(s): Didactics – Mathematics 1
• Language: FR
• Mandatory: Yes

Validated in S2

• Course title: Einführung in die Schulpädagogik
• Number of ECTS: 3
• Course code: F3_SECEDUC-197
• Module(s): General Competences I – Forschungsfeld Schule
• Language: DE, EN
• Mandatory: Yes

Die Studierenden haben gelernt:Sinn, Absicht und Realität von Schule unter einem historischen, pädagogischen udn soziologischen Blickwinkel zu analysierenDen Lehrerberuf als Profession zu verstehenDen Zusammenhang von Bildung, Bildungszielen und Bildungsplänen zu erkennenLernen als sozialen Prozess zu beschreibenVerschiedene Ideen davon, was “guten Unterricht” ausmacht zu diskutieren Die Studierenden sind in der Lage:Professionelle Standards von Lehrerhandeln umzusetzenDie Bedingungen des luxemburgischen Schulsystems zu analysieren und verschiedene Theorien zur Unterrichtsqualität und Diagnostik anzuwenden.

Die Vorlesung “Einführung in die Schulpädagogik” analysiert die (luxemburgische) Schule als eine historisch gewachsene Institution, die ganz unterschiedlichen Zwecken dient bzw. dienen soll. Dabei stehen pädagogische, soziologische und historische Erklärungsansätze im Mittelpunkt der Beschreibung schulischer Wirklichkeiten. Zudem werden der Lehrerberuf sowie die Schulentwicklung (Curricula, Bildungsziele, etc.) auf ihre professionellen Begründungen hin vorgestellt und hinterfragt. Die Vorlesung führt ebenfalls in die wichtigsten Ideen zum Thema “Lernen” und zur Unterrichtsqualität ein.

End-of-term Assessment: Klausur.

Bibliographie:Ludwig Haag, Sibylle Rahm, Hans Jürgen Apel, Werner Sacher (Hrsg.): Studienbuch Schulpädagogik. Verlag Julius Klinkhardt 2013. Hanna Kiper, Hilbert Meyer, Wilhelm Topsch (Hrsg.): Einführung in die Schulpädagogik. Cornelsen Verlag 2011. Ilona Esslinger-Hinz, Anne Sliwka (Hrsg.): Schulpädagogik. Beltz Verlag 2011.

• Course title: Informationveranstaltung / Réunion de rentrée – Cours Communs
• Number of ECTS: 0
• Course code: F3_SECEDUC-171
• Module(s): General Competences I – Forschungsfeld Schule
• Language:
• Mandatory: No

• Course title: Workshop zum orientierenden Schulpraktikum (MES MATH)
• Number of ECTS: 0
• Course code: F3_SECEDUC-217
• Module(s): General Competences I – Forschungsfeld Schule
• Language: DE
• Mandatory: No

Die Studierenden …

kennen die wichtigsten diagnostischen Instrumente der Unterrichtsbeobachtung und ihre Funktion.

reflektieren vor dem Hintergrund ihrer eigenen Schulerfahrungen, Aspekte und Gütekriterien von Lehr-Lernsituationen nach Andreas Helmke.

kennen die Merkmale guten Unterrichts nach Hilbert Meyer.

kennen die Grundstruktur eines Unterrichtsentwurfs.

kennen die Grundstruktur eines Stundenverlaufplans sowie die verschiedenen Phasierungsschritte von Unterricht.

kennen die Basisformen des Lehrens und Lernens sowie gängige Unterrichtsmethoden und Methoden des kooperativen Lernens.

reflektieren die wesentlichen Eckpunkte und Rahmenbedingungen, die sie bei der Vorbereitung, Durchführung und Reflexion eigener Unterrichtsversuche beachten müssen.

Das Tutorium fokussiert auf die praxisorientierte Umsetzung diagnostischer Verfahren zur Vorbereitung der Unterrichtshospitation wie auch auf Gestaltungsmöglichkeiten von Unterricht zur Vorbereitung der eigenen Unterrichtspraxis im Rahmen des orientierenden Praktikums.Darüberhinaus werden eigene Wissens- und Erfahrungsstände der Studierenden gemeinsam mit den das Praktikum begleitenden Lehrpersonen reflektiert, um individuelle Entwicklungsziele und Forschungsinteressen für das Masterstudium zu identifizieren.Das Tutorium ist an das Seminar Mehrsprachigkeit im Sprach- und Fachunterricht: L’importance des différents langages en mathématiques (appliquée au système éducatif luxembourgeois) angekoppelt.

Bibliografie:

Ein Reader mit Grundlagentexten wird zu Beginn des Semesters über Moodle zur Verfügung gestellt.

• Course title: Digitale Didaktik (FR)
• Number of ECTS: 4
• Course code: F3_SECEDUC-212
• Module(s): General Competences I – Lehren und Lernen
• Language: FR
• Mandatory: No

À la fin du semestre vous aurez appris:

– à planifier dans un premier temps des séquences de cours dans vos matières respectives en utilisant les manuels scolaires et dans un deuxième temps de prendre en considération, d’évaluer les nouvelles technologies numériques pour la même sequence. – à développer un esprit critique dans l’utilisation des outils numériques en classe et à aider les élèves à utiliser de manière autonome les nouvelles technologies.- à découvrir les nouvelles technologies, à améliorer vos propres compétences numériques pour l’apprentissage scolaire/ à faire des expériences dans l’apprentissage immersif (lunettes 3D) et en évaluer les bénéfices.

Après une approche théorique sur l’évolution sociale dans la transimission de l’apprentissge, nous regarderons en détails la didactique numérique actuelle (principalement le modèle SAMR : intégration efficace des nouvelles technologies), puis nous regarderons de plus près les outils numériques utilisés dans des cours de mathématiques et de français (Padlet/Flipped Classroom/ Chat GPT, Canva, IMovie, VN, Visio,etc.), nous discuterons de façon critique leurs avantages, leurs bénéfices et leurs défauts tout en comparant les cours traditionnels aux cours innovants. Nous lierons bien évidemment la théorie à la pratique en planifiant des séquences pour des cours de Mathématiques et de Français.

Contrôle continu 50%Travail pratique de fin de semestre 50%

Bibliographie:

Lors de la première séance, je mettrai  les manuels scolaire dont nous aurons besoin sur Moodle. Au fil du semestre, vous aurez accès au protocole des cours.

• Course title: Algebraic Topology
• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-12
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

On successful completion of the course, the student should be able to:

Explain the main definitions and results of Algebraic Topology;

Comment on new concepts;

Apply the new techniques and solve problems;

Structure the acquired abilities and summarize essential aspects adopting a higher standpoint;

Give a talk for peers or students on a related topic and write scientific texts or lecture notes, observing modern standards in scientific writing, in Didactics and in Pedagogy;

Provide evidence for the mastery of the Mathematical Method. 

Topology is a field of mathematics that deals with the fundamental properties of spaces. In the topological world, two spaces are considered identical if they can be continuously transformed into one another or more generally, if they are related by what topologists call homotopy equivalence. For example, a sphere and a pyramid are identical, as are a donut and a coffee cup… A main goal is therefore to classify spaces up to homotopy equivalence.Algebraic topology uses algebraic methods to solve this and other topological problems. An important method is the search for algebraic homotopy invariants. In particular, we try to assign an algebraic object such as a group to each topological space and to prove that this group is invariant if we replace the space with a homotopy equivalent space. One of the proofs of the fundamental theorem of algebra uses an algebraic invariant.The main homotopy invariants we discuss are the ‘singular homology functor’ and the ‘homotopy functor’. An amazing result called the Hurewicz isomorphism, shows that the first singular homology group and the (abelianized) first homotopy group of a space are two variants of the same information. Van Kampen’s theorem allows us to practice our understanding of category theory and to compute homotopy groups of larger spaces from those of smaller spaces from which they are constructed. The final chapter will highlight the relationship between the coverings of a space and subgroups of its first homotopy group.The objective is to allow the student to familiarize himself with a very active field of mathematics, with broad applications throughout science. Beyond this goal, special emphasizes will be put on the Mathematical Method, i.e., the optimal technique to learn and apply mathematics. This method is the most important of the objectives of any study program in mathematics.Fundamental concepts of homological algebraChain complexes, chain maps, chain homotopies, homology functor, connecting homomorphism, Künneth isomorphism, simplicial and singular homology, homotopy invariance, examples and applications.Homotopy groupsAlgebra and topology (revision), first homotopy group, Hurewicz isomorphism, homotopy functor, homotopy invariance, typical examples and applications.Van Kampen theoremFree product with amalgamation, Van Kampen-Seifert theorem (basic, general and groupoid version), examples and applications (in particular: first homotopy group of a product and of a coproduct).Covering space theoryCoverings of spaces, lifting property, coverings of groupoids, orbit categories, classifications and constructions of coverings, connections with the preceding chapters, examples and applications. 

Written exam 

Literatur

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002 – Mathematics – 544 pages, ISBN-13: 978-0521795401 

Peter May, A concise course in Algebraic Topology, University of Chicago Press, 1999 – Mathematics – 243 pages, ISBN-13: 978-0226511832

Charles Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics 38, 1997, ISBN: 0-521-55987-1

Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5(2), Springer, 1978, ISBN: 978-0-387-98403-2

• Course title: Mathematical Statistics II
• Number of ECTS: 5
• Course code: MA_DS-35
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

In parametric models, a successful student should be is able to establish the main properties (consistency, asymptotic normality, etc) of the most classical estimators, provide (possibly asymptotic) confidence regions and tests between two hypotheses.  

The main problems of mathematical statistics: estimation, testing, the notion of risk. Construction of confidence intervals and tests. False discovery rate. The empirical measure and its applications: the moment method, estimation by empirical quantiles, the Kolmokorov-Smirnov Test. The maximum likelihood estimator. Exponential families in one dimension. Introduction to the Bayes paradigm and elements of decision theory.

First sessionA partial written exam and a final written exam.Retake examOral exam

LiteraturFor probability theory: Real Analysis and Probability, R.M. DudleyFor statistics: Mathematical Statistics, P. Bickel and K. Doksum

• Course title: Numerical solution of partial differential equations and applications
• Number of ECTS: 6
• Course code: F1_MA_MAT_MMCS2-7
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

This course is an introduction to the numerical solution of partial differential equations (PDEs). It contains a theoretical part setting the mathematical foundations necessary for some important numerical methods used to obtain solutions to some classical PDEs, in particular the finite element method and the finite difference method. The theoretical part of the course is supported by the development of a one-dimensional Galerkin finite element code for the Poisson problem and a one-dimensional finite difference code for a scalar hyperbolic transport problem.We largely follow the reference [Qua09] which is available via the National Library’s Website as well as on some less official websites. In particular, we cover the contents of [Qua09, Chapters 1, 2, 3, 4, and 14].

A brief survey of partial differential equationsThis chapter briefly introduces the notion of linear PDEs and their classification into elliptic, parabolic and hyperbolic equations. We mention some classical examples, mainly issued from physics and engineering, such as the transport equation, the Laplace equation, the heat equation and the wave equation.Elements of functional analysisWe introduce the main notions and theoretical results of functional analysis that are extensively used in the numerical analysis of partial differential equations. We consider the Riesz representation theorem regarding representation of continuous linear forms on a Hilbert space, and we survey the notions of bilinear form, of continuous injection of a Hilbert space into another, the notion of derivative in the sense of Fréchet, some elements of the theory of distributions, the basic properties of the Lebesgue and Sobolev spaces, and the notion of adjoint operator.Elliptic equationsWe illustrate boundary-value problems for elliptic equations (in one and several dimensions), present their variational reformulations, treat the boundary conditions and analyze their well-posedness. Several examples of physical interest are introduced, in particular the Poisson equation, starting with the one-dimensional case, for various boundary conditions. We consider some variational formulations of these problems, and then turn to the boundary-value problems associated to the Poisson equation in the two-dimensional case. We establish that under some regularity condition the weak formulation is equivalent to the strong one. For general elliptic problems, the Lax-Milgram theorem ensures that the weak formulation is well-posed.The Galerkin finite element method for elliptic problemsWe formulate Galerkin’s method for the numerical discretization of elliptic boundaryvalue problems and analyze its existence, uniqueness, stability and convergence features in an abstract functional setting. We then introduce the Galerkin finite elements method, first in one dimension, and then in several dimensions.The Galerkin finite element method – a numerical exampleThis section introduces students to the development of a one-dimensional Galerkin finite element solver for the Poisson problem using Python. We focus on building the solver within a provided Jupyter notebook, providing hands-on experience with the computational and algorithmic aspects of the finite element method.Finite differences for hyperbolic equationsThe aim of this final chapter is to study classical finite differences methods for approximating solutions of first-order hyperbolic equations. We start with exposing some hyperbolic equations starting with the scalar transport problem in one dimension which we analyze by the method of characteristics. We also establish an a priori estimate by the energy method. We then turn to systems of linear hyperbolic equations in one dimension and give the example of the wave equation. Then we introduce the finite difference method, together with its variants: the forward/centered Euler scheme, the Lax-Friedrichs scheme, the Lax-Wendroff scheme, in the case of the scalar transport problem, before to move to more general cases, for which we analyze the consistency, stability, convergence, dissipation and dispersion properties of the finite difference methods.Finite differences – a numerical exampleThis section introduces students to the development of a one-dimensional finite difference solver for a hyperbolic transport problem using Python. We focus on building the solver within a provided Jupyter notebook, providing hands-on experience with the computational and algorithmic aspects of the finite difference method.

First sessionAssessment is via coursework (30%) and final examination (70%). The due date for the coursework will be set at a later time.Retake policyFailure on the coursework can be compensated through the examination, and vice versa, as long as the final mark for the course is greater than or equal to 10.In the event that the final mark is less than 10:The student may request a retake the examination once.The student may not resubmit the coursework.If the final mark remains less than 10 the student has failed, and can take the entire course again in the next semester that it is offered.

Reference[Qua09] Alfio Quarteroni. Numerical models for differential problems. Vol. 2. Springer, 2009.

• Course title: Algebraic Number Theory
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-14
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

Introduce the students to Algebraic Number Theory. Reinforce students’s understanding of commutative algebra. Upon successful completion of the course, the students will understand the basic notions of algebraic number theory and be able to solve simple problems in that area. They will have an awareness that many questions in algebraic number theory require to consider local and global information and their interplay to solve them. Moreover, they will be able to prove elementary results in analytic number theory.

 

Review of integrality and Dedekind domains, including the factorisation of prime ideals. Proof that the ring of integers of a number field is a Dedekind domain. Study of some properties of the ring F_q[t],Valuations, places, rational function fields.Zeta functions of F_q[t], Q and F_q(t) and some arithmetic consequences of their properties,Gauss reciprocity over number fields and global function fields and related subjects,Extensions of Dedekind domains and their fraction fields,Decomposition groups, inertia groups and basic properties,Cyclic extensions over number fields and global function fields,Dirichlet L-series and primes in arithmetic progressions,Study of rings of integers I: Minkowski Theory and Class Number,Study of rings of integers II: Dirichlet’s Unit Theorem,Time permitting: An introduction to Drinfeld modules

Quizzes (20%)Each theme will be followed by an in-class quiz of up to 10 minutes, testing basic definitions and simple questions about the theory covered in class.Exercise Sheets (35%)Each week, students will work on an exercise sheet that tests their ability to reflect on the class material and to apply it to solve problems in Algebraic Number Theory.Final Exam (45%)A final written exam will assess their understanding of the course material covered in the semester.

LiteratureClass notes will be provided. However, students may benefit from consulting various sources on Algebraic Number Theory. References with relevant material for this course include:Jürgen Neukirch, Algebraic Number Theory, Springer Berlin, Heidelberg, 1999Serge Lang, Algebraic Number Theory, Springer New York, NY, 2013Pierre Samuel, Algebraic Theory of Numbers, Dover, 2008,Michael Rosen, Number Theory in Function Fields, Springer New York, NY, 2013Henning Stichtenoth, Algebraic Function Fields and Codes, Springer Berlin, Heidelberg, 2009Sara Arias-de-Reyna, Gabor Wiese, Algebraic Number Theory, Online notes

• Course title: Partial Differential Equations II
• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-15
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

Learning tools in order to deal with PDE, understanding the interplay between local and global problems and techniques.

Distributions as generalized functions continued, Sobolev spaces, elliptic regularity, elliptic operators on compact manifolds, some non-linear equations. 

Written exam

LiteraturJost: Postmodern analysisFolland: Introduction to partial differential equations Reed-Simon: Methods of mathematical physics I-IV Aubin: Nonlinear analysis on manifolds

• Course title: Advanced Graph Theory
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-16
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

 

On successful completion of the course, the student should be able to Illustrate the main results and concepts with well-chosen examplesMaster the proofs and techniques of the theory Solve exercises related to the topics covered in class Give an overview of the course content, focusing on his/her own taste and favorite topics Communicate his/her own pleasure in solving mathematical problem 

Graphs are structures that are ubiquitous in Mathematics and its applications. Through a presentation of selected topics, the course aims to be an introduction to certain modern aspects of graph theory. Some basic knowledge of the concept of graphs is a prerequisite for this course.According to time and taste, topics covered will be chosen among (and are not limited to) the following ones:Matching, covering and packing problemsEdge and vertex coloringsGraph polynomialsRamsey theory for graphsInfinite Graph Theory 

Oral exam or written exam according to number of participants

LiteraturReinhard Diestel. Graph Theory, 5th edition. Graduate texts in mathematics 173, Springer, 2017

• Course title: Riemann Surfaces
• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-13
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

At the end of the course the students should have a basic understanding of what Riemann surfaces are and how they can be studied from different points of view, especially the complex and topological one. They should be able to understand and solve simple problems involving the material from the course.

A primary goal of various kinds of geometric theories is to describe shapes of arbitrary dimensions from a mathematical point of view. One way to do so is to build such shapes out of pieces of a euclidean space R^n, obtaining smooth manifolds. In some cases, smooth manifolds can be covered instead with pieces of a complex affine space C^n: when the complex dimension is 1, such a manifold is a Riemann surface (its real dimension will be 2). From a topological point of view, we will be looking simply at a sphere, a doughnut, or a surface with more “holes”; however, the mere existence of a complex structure provides the foundations for an extremely rich theory. Riemann surfaces lie at the crossroads of many fields of research of modern mathematics: topology, complex and algebraic geometry, number theory and harmonic analysis, among others. The course will cover the basic notions of the theory, starting from the definition of a complex manifold, and move on to the classification of Riemann surfaces via the study of their divisors. This will allow the students to acquire familiarity with some ubiquitous tools from complex geometry, such as the Riemann-Roch theorem. We will put some focus on examples along the way. If time allows, we will see how some constructions from the complex analytic theory can be reinterpreted in the language of modern algebraic geometry, bridging the gap between the two worlds, and we will touch some more advanced topics that could inspire the students’ further work.

Weekly exercise will be provided and the students will be asked to present some of them orally. This will provide for continuous evaluation that will make up for 1/3 of the grade. The remaining 2/3 will be given by the result of a final oral exam.

LiteraturRick Miranda, Algebraic Curves and Riemann Surfaces.Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces.Oleksandr Iena, Riemann surfaces. Lecture Notes. Winter Semester 2015/2016.The above texts are available on a-z.lu. More references will be provided during the course.

• Course title: Student Project 2
• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-18
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

On successful completion of the Student Project 2, 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.

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 March. 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 May. The project file will undergo an automatic plagiarism check.

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.

• Course title: Student seminar 2
• Number of ECTS: 2
• Course code: F1_MA_MAT_GM-19
• Module(s): Specialisation – Mathematics 2
• Language: EN
• Mandatory: No

On successful completion of the Student Seminar 2, the students should be able to:Fully benefit from seminar talksAcquire good insight into a field by means of individual workGive themselves lectures on specific topicsShare their knowledge with others

Every participant chooses a supervisor among the academic staff of the Department of Mathematics and, jointly with the supervisor, a topic for a talk.The audience of the talk consists at least of the participants of the seminar, 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 75 minutes (time for questions not included).A typewritten version of this (these) lecture(s) is requested.The students’ seminar talks will take place during April and May. The exact timetable depends on the number of registered students and is made during the semester.The supervisor together with the course responsible marks the students’ performance and the written text. 

Continuous evaluation. The mark is based on the talk and the written text delivered by the student.If a student fails, a new talk and a new text can be delivered in the following semester. A supervisor change is allowed.

• Course title: Didactical Seminar I
• Number of ECTS: 2
• Course code: F1_SECEDUC-29
• Module(s): Didactics – Mathematics 2
• Language: FR, EN, DE
• Mandatory: Yes

The students train their mathematical-didactical capacities and their collaborative attitude. They benefit from the experience and insights of the main lecturer and the guest lecturers (for example, MSE-M Alumni that provide their recommendations on how to navigate the first years within the Luxembourgish school system).

The aim of reflective teaching is continuously questioning the teacher’s professional practice with the cycle of “planning, acting/observing and reflecting/adapting”. In this workshop, we reflect on various aspects of the teacher’s role. We address both general didactical principles and specificities of the Luxembourgish school system.

The students are requested to give a presentation on a topic of school mathematics (the grade depending on their mathematical-didactical performance). Moreover, it is expected that the students take part in the training offered by Thierry Meyrath (one time participation to a Scienteens Lab workshop).

Each year we focus on a different theme, and we have different guest lecturers. The literature is announced in class and is available on Moodle.

• Course title: Learning and Teaching Mathematics I
• Number of ECTS: 3
• Course code: F1_SECEDUC-30
• Module(s): Didactics – Mathematics 2
• Language: FR, EN, DE
• Mandatory: Yes

Maîtriser la conception d’un enseignement qui se caractérise par l’activation cognitive et le soutien constructif. 

À l’issue de ce cours, les étudiants devront être capables de :concevoir des situations d’apprentissage en tenant compte des connaissances préalables, des conditions d’apprentissage et, si possible, des centres d’intérêt et motivations des élèves afin de les amener à évoluer vers des objectifs d’apprentissage définis ;concevoir des exercices favorisant l’activation cognitive et les adapter au niveau d’apprentissage des élèves tout en initiant les élèves à différentes stratégies d’apprentissage ;concevoir des situations d’apprentissage qui conduisent à une réflexion approfondie sur les contenus et poser des questions qui permettent de pousser les élèves à une réflexion approfondie sur les contenus. 

Ce cours vise à développer les compétences nécessaires à la conception et à la mise en œuvre de leçons de mathématiques. Les étudiants apprendront les principes d’un enseignement de qualité issus de la recherche actuelle en didactique des mathématiques. Dans le cadre de la préparation écrite d’une leçon, l’accent de ce cours sera mis sur le choix des exercices favorisant l’activation cognitive et la différenciation. Ainsi, les étudiants apprendront à documenter continuellement les conclusions d’une réflexion de qualité sous forme d’un portfolio. 

L’évaluation du cours combinera contrôle continu (travaux réguliers) et examen final (production écrite finale et une soutenance orale). Les détails (et les dates limites pour la remise des travaux) seront annoncés sur Moodle au début du semestre.

Note / LiteraturLa liste des références bibliographiques sera publiée sur la page Moodle de ce cours. 

• Course title: Mehrsprachigkeit im Sprach- und Fachunterricht (MATH): L’importance des différents langages en mathématiques (appliquée au système éducatif luxembourgeois)
• Number of ECTS: 3
• Course code: F3_SECEDUC-182
• Module(s): General Professional Competence II – Multilingualism and Heterogeneity
• Language: FR
• Mandatory: No

Les étudiants seront amenés à :Faire des réflexions sur l’utilisation d’une langue et les définitions précises des mots dans le cadre d’un cours de mathématiques.Etudier les différences entre certains mots quand ils sont utilisés en langue courante ou en contexte mathématique.L’impact de la langue d’instruction sur un cours de mathématiques.Etablir un dictionnaire mathématique trilingue (FR-ALL-ENG) adapté à l’enseignement secondaire luxembourgeois.

Quand on pense aux mathématiques, on pense souvent à un tableau noir rempli de symboles obscurs et incompréhensible aux laïcs. Et pourtant, le rôle des langues n’est pas sans importance en mathématiques : nous communiquons les mathématiques avec des mots du langage courant et même si les preuves mathématiques contiennent des signes propres aux mathématiques, autour de ces signes et symboles se trouve du texte écrit en une certaine langue. Les littéraires peuvent s’en passer des mathématiques, mais les mathématiciens doivent utiliser les langues, et ceci correctement. D’autant plus, que beaucoup de mots de la langue courante prennent un sens très précis et strict en mathématiques. L’histoire se complique quand la langue d’instruction du cours de mathématiques n’est pas la langue maternelle du mathématicien et/ou de l’élève.Ces idées constituent le point de départ de ce cours. Nous allons nous pencher sur trois thèmes principaux :L’importance des mots en mathématiques et leur définition précise dans leur contexte mathématique.Les difficultés et avantages des classes de mathématiques multilingues.Un dictionnaire mathématique trilingue (ou plus) adapté au contexte luxembourgeois.

25% projet, 75% examen oral

• Course title: Lie Algebras and Lie Groups
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-20
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

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.

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.

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.

First session

End of course assessment

Note / Literature / Bibliography 

“Lie groups and Lie algebras” by Eckhard Meinrenken, 83 pages (free to download)

“Prerequisites from Differential Geometry” by Sergei Merkulov (free to download)

• Course title: Combinatorial Geometry
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-21
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

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

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 hands-on approach, and this approach and experience are as important for this course as the actual content. 

The course will take the form of a topics course, presenting a selection of themes from combinatorial aspects of geometry. Topics may include cube complexes, convex sets in R^n, combinatorial ways of exploring surfaces, topics in graph theory…

First session

3 short in-class tests, worth together a total of 20%; final written exam worth 80%.

Retake exam100% final written exam, no kept grade.

Absence plan: the final grade will be worked out as follows:

Case 1: Student attends all three short tests. The best two out of the three scores will count for 10% each, and the final exam will count for 80%.

Case 2: Student attends two short tests. The two scores will count for 10% each and the final exam will count for 80%.

Case 3: Student attends one short test. The test score will count for 10% and the final exam will count for 90%.

Case 4: Student attends no short test. The final exam will count for 100%.

 

Note / Literature / Bibliography 

Varied. (A mix of lecture notes and articles.)

• Course title: Arithmetic Geometry
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-22
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

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 p-adic 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.

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 p-adic 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 so-called p-adic 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, p-adic 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 Birch-and-Swinnerton-Dyer 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.

Exam modality for the first sessionThe assessment method is continuous assessment. The continuous assessment consists of two parts, each contributing 50% to the final mark:

Talks by participants

Presentation of exercises by participants

Exam modality for the retake examOral exam

 

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

• Course title: Algebraic Geometry
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-23
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

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

We will introduce basic notions of algebraic geometry starting with Hilbert’s Nullstellensatz, affine and projective varieties. We can think of an affine variety as the common zeros of a set of given polynomials in multiple variables. We will further introduce rational maps between varieties leading to the important topic of birational geometry. Finally, we will learn what a singular point of an affine variety is. Along the way, we will look at many simple and concrete examples. 

First sessionWritten exam and supervised homeworkRetake exam 

Note / Literature / Bibliography Robin Hartshorne, Algebraic geometryRavi Vakil, The rising seaIgor Shafarevitch, Basic algebraic geometry 1

• Course title: Gaussian processes and applications
• Number of ECTS: 5
• Course code: F1_MA_MAT_FM-9
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

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.)

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 processesIdentify, analyse, and prove relevant properties of models based on a Gaussian structureSolve exercises involving a Gaussian structure

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)

Exam modalities for the first sessionWritten exam during the exam period (January/February).Exam modalities for the retake examWritten exam during the exam period July).

Note / Literature / BibliographyWill be discussed in class

• Course title: Machine Learning – Reading course
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-12
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

• Course title: Data Science
• Number of ECTS: 5
• Course code: F1_MA_MAT_GM-24
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

The successful candidate will understand the basic theoretical concepts of data-centric aspects and will be able to work on data-centric problems. The aim is to continue the work done in the project within the framework of a Master’s thesis.

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 data-centered problems.

Consolidation of the acquired competences in the subject area through a Master’s thesis.

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.  

First session

50% written assignment or oral interview + 50% practical. 

Retake exam

Those who do not pass the course have the opportunity to retake the exam in the summer term.

Note / Literature / Bibliography 

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: Privacy-Preserving 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 title: Elementary mathematics from a higher viewpoint I
• Number of ECTS: 4
• Course code: F1_SECEDUC-26
• Module(s): Specialisation – Mathematics 3
• Language: FR, EN
• Mandatory: No

Les étudiant·e·s vont revisiter leurs connaissances des mathématiques de fin de secondaire. En parallèle, ils·elles aborderont ces sujets sous un angle nouveau, grâce à leur maîtrise des mathématiques de niveau supérieur. Les étudiant·e·s du MSE-M Master devront être capable de répondre aux questions possibles de leur futur·e·s élèves.

Le cours explorera les thématiques des mathématiques de fin de secondaire sous un angle plus abstrait. Les concepts universitaires seront utilisés pour revisiter les mathématiques du secondaire et pour approfondir la théorie derrière chaque formule. Des questions de compréhension et des liens entre différents sujets seront également abordés.Attention: les thématiques abordées sont divisées en deux groupes et sont traitées en alternance chaque année.

Exam modalities for the first session3 midterms (50% of the final mark) and 1 written exam in january (50 % of of the final mark).Exam modalities for the retake exam1 written exam (100 % of final mark)Absence planif justified absent during midterms, an oral exam will be organised about the same material, if justified absence during the written exam, an oral exam will take place

• Course title: Student Project 3
• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-25
• Module(s): Specialisation – Mathematics 3
• Language: EN
• Mandatory: No

On successful completion of the Student Project 1 (or 3) 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.

The ability to work on a project, either individually or as part of a team, is one of the key skills expected of a mathematician, in and outside academia. The Student Project 1 offers the student the opportunity to improve this skill by allowing them to work on a well-defined and well chosen project, either individually or as a team, under the guidance of a researcher.The Student Project 3 consists of project work that is carried out under the supervision of a researcher.The work is either individual or group work. 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) for each student. The student(s) need to submit the planned topic to the course responsible for approval at the latest on 15 October. The project outcome is a pdf document written by the student(s).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. The project file will undergo an automatic plagiarism and AI check.Timeline for the studentBeginning of semesterConnect to the Moodle classroomFind a supervisor and a topicOn or before 15 October, get approval from the course responsible of the project participants, the supervisor and the topicDuring the semesterWork on the projectSeek feedback from your supervisorEnd of the semesterHand in your project on or before 31 December on Moodle (for plagiarism and AI check)  

First sessionThe students are marked for their project work. In the case of a retake exam, a new project on the same or a different topic has to be done. Retake examThe retake can be done with a different supervisor.

• Course title: Didactics of Algebra
• Number of ECTS: 2
• Course code: F1_SECEDUC-31
• Module(s): Didactics – Mathematics 3
• Language: FR, EN, DE
• Mandatory: Yes

Transitioning from arithmetic to abstract algebra is a significant mathematical journey for students, and the teacher’s role is to guide them through this complex process. Elementary algebra presents specific didactical challenges that require careful consideration. The primary objective of this course is to analyze these challenges and discuss strategies for addressing them. A secondary objective is to revisit certain algebraic topics from an advanced perspective to enhance mathematical understanding.

By the end of the course, students will be able to critically reflect on the didactical challenges associated with teaching elementary algebra. They will develop the ability to propose solutions to both mathematical and didactical problems related to algebra.

The course will primarily reference “Didaktische Probleme der elementaren Algebra” by Malle, Wittmann, and Bürger. Additionally, a selection of recent papers on the didactics of algebra will be examined. The mathematical content will be drawn from a variety of sources. A complete reference list and detailed subtopic descriptions will be provided to students on Moodle.

• Course title: Applied Didactics IV – Evaluation methods and supporting mathematics learning
• Number of ECTS: 3
• Course code: F1_SECEDUC-19
• Module(s): Didactics – Mathematics 3
• Language: FR
• Mandatory: Yes

Les questions et la taxonomie de Bloom/Anderson & Krathwohl, les problèmes de compréhension, ainsi que les moments clés d’une leçon ou d’un apprentissage sont abordés. La planification d’une leçon ou d’une séquence de leçons, ainsi que l’évaluation formative et sommative sont également au coeur du cours.

Avoir conscience que chaque choix didactique influence toujours l’enseignement et l’apprentissage des contenus qui en découlent. 

Être en mesure de formuler des questions ciblant différents niveaux taxonomiques. 

Anticiper les éventuels problèmes de compréhension des élèves. 

Identifier les moments clés d’une leçon ou d’un apprentissage. 

Planifier une séquence de leçons comprenant une introduction, des activités, des transitions, des moments consacrés aux élèves, des exemples, des exercices, ainsi que les questions à poser et les réponses attendues des élèves. 

Formuler des questions adaptées à une évaluation de type formatif ou sommatif. 

Planifier une leçon de manière à ce que les objectifs d’apprentissage puissent être atteints de plusieurs manières différentes.

Présentation des leçons et diverses méthodes d’introduction ; Utilisation de la taxonomie de Bloom/Anderson & Krathwohl ; Analyse des exercices et des feuilles de travail ; Introduction aux différents types de temps consacrés aux élèves ; Préparation des leçons, mettant l’accent sur le cycle inférieur et moyen : Comment procéder ? Bonnes pratiques : Public cible ; Importance du sujet pour les élèves ; Objectifs à atteindre ; Matériel disponible/à utiliser ; Méthodes pédagogiques ; Outils techniques, visualisation, applications, jeux ; Réflexion ; Introductions et rappels ; Enchaînements et transitions ; Types de questions à poser et utilisation de la taxonomie ; Moments clés ; Exercices ;Évaluation : objectifs, méthodes, types et fréquence ; Ressources ; Problèmes et conséquences des choix didactiques ; Préparation au stage de janvier.

First session

Exam modalities for the first session: Combined assessment (contrôle continue et examen final sous forme de remise d’un dossier avec présentation et défense orale du dossier)

Session de rattrapage

Exam modalities for the retake exam: remise d’un dossier

Absence plan pour les contrôles continus/ absences lors d’un cours: me contacter au plus tard avant le début du cours pour la remise du dossier: une prolongation peut être obtenue sous condition de pouvoir fournir un certificat médical pour la défense orale: une date alternative peut être trouvée sous condition de pouvoir fournir un certificat médical.

Note

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

• Course title: Allgemeine Didaktik II
• Number of ECTS: 6
• Course code: F3_SECEDUC-188
• Module(s): General Competence III – Forschungsfeld Schule
• Language: DE
• Mandatory: No

Die Studierenden …… haben Grundlagen zu Tiefenstrukturen des Unterrichtens, insbesondere der kognitiven Aktivierung und der kognitiven Strukturierung, erarbeitet…. verfügen über ein anwendungsbezogenes Konzeptrepertoire aus den Bereichen Pädagogische Diagnostik und Evaluation…. haben Transfermöglichkeiten vor dem Hintergrund der je eigenen Praktikumserfahrungen reflektiert und auf Basis ihrer individuellen Entwicklungsziele und Forschungsinteressen weiterentwickelt. 

Der Workshop dient der Vertiefung pädagogischer Grundlagen und didaktischer Konzepte im Enseignement Secondaire und damit der Erweiterung und Adaption von Gestaltungsmöglichkeiten der Unterrichtspraxis im Rahmen des vertiefenden Praktikums. Ein besonderer Schwerpunkt liegt dabei auf den individuellen Entwicklungszielen und Forschungsinteressen der Studierenden.

20% Aktive Teilnahme an Seminar und Schulpraktikum40% 2 Zwischenprüfungen in Form einer Klausur und/oder Präsentation40% Praktikumsportfolio

• Course title: Einführung in die Pädagogische Psychologie
• Number of ECTS: 3
• Course code: F3_SECEDUC-129
• Module(s): General Competences III – Mehrsprachigkeit und Heterogenität
• Language: DE, EN
• Mandatory: No

Ein (Pädagogisch-) Psychologisches Fundament für angehende Lehrerinnen und Lehrer zur Verfügung stellenUnterricht aus seiner psychologischen Perspektive analysierenProzesse wissenschaftlichen Erkenntnisgewinns und seiner Bedeutung für die Berufspraxis nachvollziehbar machenZum eigenständigen Denken und professionellem Handeln anregenTypische Mythen und Fehlkonzepte in Bezug auf Lernen und Unterricht aus psychologischer Sicht hinterfragen und ggf. korrigieren

Die Vorlesung führt ein in Gegenstand und Methode der (Pädagogischen) Psychologie als empirische Wissenschaft und ihrer Bedeutung für angehende LehrerInnen.  Insbesondere die Pädagogische Psychologie, aber auch die Allgemeine, Differentielle, Entwicklungs- und Sozialpsychologie tragen bei zum Verständnis menschlichem Erleben und Verhalten in Kontexten der Erziehung, der Bildung und des (formalen) Lernens (bzw. Unterrichtens). Vorgestellt und diskutiert werden zunächst allgemeine Inhalte und Methoden der Psychologie als empirischer Wissenschaft, allgemeine Lernvoraussetzungen (z.B. Informations-verarbeitung, Lernmotivation), Aspekte der Unterrichtsgestaltung (z.B. Klassenführung), Aspekte der sozialen Interaktion (z.B. Freundschaft, Bullying), psychologische Besonderheiten des Lehrerberufs (z.B. Lehrerpersönlichkeit, Burnout) sowie verschiedene Aspekte pädagogisch-psychologischer Diagnostik (z.B. ausgewählte Lernstörungen wie ADHS). Dabei werden sowohl empirische Originalstudien als auch empirisch fundierte Interventionsansätze zur Illustration vertiefend besprochen. Zudem werden Fallbeispiele und Übungen besprochen und durchgeführt.

Schriftliche Klausur

BibliografieGrundlagenliteratur (d.h. online verfügbar und prüfungsrelevant, spez. Kap.)Fritz, A., Hussy, W. & Tobinski, D. (2018). Pädagogische Psychologie. München: Reinhardt/UTB.Fromm, M. (2017). Lernen und Lehren: Psychologische Grundlagen für Lehramtsstudierende. Münster: Waxmann.Götz, T. (2017, Hrsg.). Emotion, Motivation und selbstreguliertes Lernen. Paderborn: Schöningh/UTB.Imhoff, M. (2013). Psychologie für Lehramtsstudierende. Berlin: Springer.Kunter, M. & Trautwein, U. (2013). Psychologie des Unterrichts. Paderborn: Schöningh/UTB.Wild, E. & Möller, J. (2015, Hrsg.). Pädagogische Psychologie. Berlin: Springer.Wisniewski, B. (2019). Psychologie für die Lehrerbildung. Bad Heilbrunn: Klinkhardt/UTB.

• Course title: Teaching Children with Special Educational Needs
• Number of ECTS: 3
• Course code: F3_SECEDUC-51
• Module(s): General Competences III – Mehrsprachigkeit und Heterogenität
• Language: EN
• Mandatory: No

This course aims to deepen student’s knowledge and understanding of special educational needs. It provides students with insights into the latest research about the learning processes of children with specific learning difficulties like dyslexia, developmental language disorder or autism spectrum disorder. It will also equip future teachers with strategies to teach children with special educational needs or other learning challenges more effectively.

Over the years, research has informed the scientific and the educational community about how children’s learning processes can be affected by neurologically-based difficulties. These specific learning difficulties can interfere with the learning of basic skills such as reading, writing, maths or language learning. They can also affect higher level skills such as organisation, time planning, abstract reasoning, long- or short-term memory, and attention. It is important for teachers to understand the impact that specific learning difficulties have on children’s learning. To this end, the course will combine relevant theory with best practice. 

Exam/Presentation

Bibliography

 

Beck I., Mackeown M. & Kucan L. (2013). Bringing Words to Life: Robust Vocabulary Instruction. New York: Guilford.

Carroll, J.M., Bowyer-Crane, C., Duff, F., Hulme, C., & Snowling, MJ.  (2011). Effective intervention for language and literacy in the early years.  Oxford: Wiley-Blackwell.

Hulme C. & Snowling M. (2009). Developmental Disorders of Language Learning and Cognition. Oxford UK: Wiley Blackwell. Chapter 2 & 3

Purpura, D. J., & Ganley, C. (2014). Working memory and language: Skill-specific or domain-general relations to mathematics? Journal of Experimental Child Psychology, 122, 104-121.

Purpura, D. J., Napoli, A. R., Wehrspann, E. A., & Gold, Z. S. (in press). Causal connections between mathematical language and mathematical knowledge: A dialogic reading intervention. Journal of Research on Educational Effectiveness.

Purpura, D. J., & Reid, E. E. (2016). Mathematics and language: Individual and group differences in mathematical language skills in young children. Early Childhood Research Quarterly, 26, 259-268.

Purpura, D. J., Schmitt, S. A., & Ganley, C. M. (2017). Foundations of mathematics and literacy: The role of executive functioning components. Journal of Experimental Child Psychology, 153, 15-34.

Sims, D. M., Purpura, D. J., & Lonigan, C. J. (2012). The relation between inattentive and hyperactive/impulsive behaviors and early mathematics skills. Journal of Attention Disorders. doi:10.1177/1087054712464390.

Snowling, M. J., & Hulme, C. (2011). Evidence‐based interventions for reading and language difficulties: Creating a virtuous circle. British Journal of Educational Psychology, 81(1), 1-23.

• Course title: Internship in secondary school II
• Number of ECTS: 0
• Course code: F1_SECEDUC-20
• Module(s): Didactics – Mathematics 4
• Language: FR
• Mandatory: Yes

• Course title: Didactical Seminar I
• Number of ECTS: 2
• Course code: F1_SECEDUC-29
• Module(s): Didactics – Mathematics 4
• Language: FR, EN, DE
• Mandatory: Yes

The students train their mathematical-didactical capacities and their collaborative attitude. They benefit from the experience and insights of the main lecturer and the guest lecturers (for example, MSE-M Alumni that provide their recommendations on how to navigate the first years within the Luxembourgish school system).

The aim of reflective teaching is continuously questioning the teacher’s professional practice with the cycle of “planning, acting/observing and reflecting/adapting”. In this workshop, we reflect on various aspects of the teacher’s role. We address both general didactical principles and specificities of the Luxembourgish school system.

The students are requested to give a presentation on a topic of school mathematics (the grade depending on their mathematical-didactical performance). Moreover, it is expected that the students take part in the training offered by Thierry Meyrath (one time participation to a Scienteens Lab workshop).

Each year we focus on a different theme, and we have different guest lecturers. The literature is announced in class and is available on Moodle.

• Course title: Learning and Teaching Mathematics I
• Number of ECTS: 3
• Course code: F1_SECEDUC-30
• Module(s): Didactics – Mathematics 4
• Language: FR, EN, DE
• Mandatory: Yes

Maîtriser la conception d’un enseignement qui se caractérise par l’activation cognitive et le soutien constructif. 

À l’issue de ce cours, les étudiants devront être capables de :concevoir des situations d’apprentissage en tenant compte des connaissances préalables, des conditions d’apprentissage et, si possible, des centres d’intérêt et motivations des élèves afin de les amener à évoluer vers des objectifs d’apprentissage définis ;concevoir des exercices favorisant l’activation cognitive et les adapter au niveau d’apprentissage des élèves tout en initiant les élèves à différentes stratégies d’apprentissage ;concevoir des situations d’apprentissage qui conduisent à une réflexion approfondie sur les contenus et poser des questions qui permettent de pousser les élèves à une réflexion approfondie sur les contenus. 

Ce cours vise à développer les compétences nécessaires à la conception et à la mise en œuvre de leçons de mathématiques. Les étudiants apprendront les principes d’un enseignement de qualité issus de la recherche actuelle en didactique des mathématiques. Dans le cadre de la préparation écrite d’une leçon, l’accent de ce cours sera mis sur le choix des exercices favorisant l’activation cognitive et la différenciation. Ainsi, les étudiants apprendront à documenter continuellement les conclusions d’une réflexion de qualité sous forme d’un portfolio. 

L’évaluation du cours combinera contrôle continu (travaux réguliers) et examen final (production écrite finale et une soutenance orale). Les détails (et les dates limites pour la remise des travaux) seront annoncés sur Moodle au début du semestre.

Note / LiteraturLa liste des références bibliographiques sera publiée sur la page Moodle de ce cours. 

• Course title: Master Thesis
• Number of ECTS: 20
• Course code: F1_SECEDUC-28
• Module(s): Master Thesis – Mathematics
• Language: FR, EN, DE
• Mandatory: Yes

On successful completion of the Master Thesis, the students should be able to:

Organize a comprehensive literature review

Discuss and communicate scientific ideas

Approach mathematical problems efficiently and identify appropriate theories or conceptual techniques

Discover original mathematics

Verify results and apply them

Write mathematical texts that are consistent with the tradition

The Master Thesis in mathematics consists of the definition of a research project, the detailed explanation of research articles and/or monographs aimed at a mathematics audience, as well as of potential further developments of these. The project, which should contain parts of original mathematics, will be designed to suit the individual objectives of the students, to deepen their competence in a selected field of mathematics, and to open a door towards mathematical research.

Supervisor, director of studies

Admission to the Master Thesis will be granted only to students who acquired at least 75 ECTS credit points during the first three semesters of the Master’s program (in a well-founded case, an exception to this rule might be decided by the study director).

• Course title: Professionell Auftreten: Communication rhétorique pour enseignants
• Number of ECTS: 1
• Course code: F3_SECEDUC-231
• Module(s): Module 4.3 : General Professionnal Competence IV – FHSE
• Language: FR
• Mandatory: No