Commutative Algebra 30 15 5 Master en Enseignement ...€¦ · algebraic geometry and other fields...

Master en Enseignement Secondaire - Master enEnseignement Secondaire, Filière Mathématiques

Version: 26 December 2020 of 38

Semester 1

Lecture(UE)

Exercise(UE)

ECTS

Module 1.1 : In each of the 4 semesters of the Master in Secondary Education – Mathematics,the students are requested to choose 1 course in Didactics Professional Knowledge: researchfield school, educational system and policy, teaching and learning in the social context. Over the4 semesters, all 3 fields must be covered. The students will be individually advised by the coursedirector, who must validate their choice.

10

Commutative Algebra 30 15 5

Riemannian Geometry 45 5

Partial Differential Equations I 45 5

Probability (Martingale Theory) 45 5

Module 1.2 : In each of the 4 semesters of the Master in Secondary Education – Mathematics,the students are requested to choose 1 course in Didactics Professional Knowledge: researchfield school, educational system and policy, teaching and learning in the social context. Over the4 semesters, all 3 fields must be covered. The students will be individually advised by the coursedirector, who must validate their choice.

10

Student Project (Optional) 60 4

Algorithmic Number Theory (Optional) 30 4

Basics of Discrete Mathematics (Optional) 30 4

Probabilistic Models in Finance (Optional) 30 4

Reading Course "complements to commutative algebra" (Optional) 15 1

Reading Course "complements to Riemannian Geometry" (Optional) 15 1

Reading Course "complements to Partial DifferentialEquations" (Optional)

15 1

Reading Course "complements to Probability (MartingaleTheory)" (Optional)

15 1

Module 1.3 : In each of the 4 semesters of the Master in Secondary Education – Mathematics,the students are requested to choose 1 course in Didactics Professional Knowledge: researchfield school, educational system and policy, teaching and learning in the social context. Over the

5



Lecture(UE)

Exercise(UE)

ECTS

4 semesters, all 3 fields must be covered. The students will be individually advised by the coursedirector, who must validate their choice.

Applied Didactics I 30 2

Applied Didactics II 30 3

Internship in a secondary school I 0

Module 1.4 5

Workshop 1 zum Praktikum 1 SEM (Optional) 10 3

Einführung in die Schulpädagogik GENERAL COMPETENCES I(Optional)

28 3


Semester 2

Lecture(UE)

Exercise(UE)

ECTS

Module 1 10

Probability (stochastic analysis) (Optional) 45 5

Homological Algebra 45 5

Partial Differential Equations II 45 5

Graph theory (Optional) 45 5

Module 2 5

Algebraic Number Theory (Optional) 30 4

Introduction to Continuous Time Models in Mathematical Finance(Optional)

30 4

Numerical Analysis (Optional) 30 4

An introduction to mathematical statistics (Optional) 30 4

Reading course "complements to Homological Algebra" (Optional) 15 1



Lecture(UE)

Exercise(UE)

ECTS

Reading course "complements to Partial Differential EquationII" (Optional)

15 1

Reading course "complements to Graph Theory" (Optional) 15 1

Reading course "complements to Probability (stochastic analysis)(Optional)

15 1

Module 3 5

Hands-on experiences with mathematical didactics I 30 3

Learning and teaching mathematics I 30 2

Module 4 : General Professional Competence II 5

General Professional Competence II-IV 1 5

Mehrsprachigkeit im Sprach- und Fachunterricht 28 3

Workshop zur Nachbereitung des orientierenden Schulpraktikums fürStudierende im 2. Semester

10 2

Workshop zur Nachbereitung des orientierenden Schulpraktikums fürStudierende im 2. Semester (Gruppe 2)

8 2

Semester 3

Lecture(UE)

Exercise(UE)

ECTS

Module 3.1 10

Introduction into Algebraic Geometry (Optional) 30 5

Riemann Surfaces (Optional) 30 5

Lie Algebras and Lie Groups 30 5

Combinatorial Geometry 30 5

Module 3.2 5

Continuous-Time Stochastic Calculus and Interest Rate Models 30 5



Lecture(UE)

Exercise(UE)

ECTS

Numerical solution of partial differential equations and applications(Optional)

30 5

Gaussian processes and applications 30 5

Advanced Mathematical Statistics (Optional) 30 5

Module 3.3 0

Internship in secondary school II 0 0

Applied Didactics IV (Optional) 30 3

Applied Didactics III (Optional) 30 2

Module 3.4 5

Teaching Children with Special Educational Needs (Optional) 28 3

Digitale Schule (Semester 3) (Optional) 28 5

Einführung in die Pädagogische Psychologie (Optional) 28 3



Workshop Professionell Auftreten (Optional) 56 1

Semester 4

Lecture(UE)

Exercise(UE)

ECTS

Module 4.1 5

Hands-on experiences with mathematical didactics II 0 3

Learning and teaching mathematics II 30 2

Module 4.2 20

Master Thesis 1 20



Lecture(UE)

Exercise(UE)

ECTS

Module 4.3 : General Professionnal Competence IV - FHSE 3

PROFESSIONELL Auftreten (Sem4) (Optional) 32 1



Semester 1

Commutative Algebra

Module: Module 1.1 (Semester 1)

ECTS: 5

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

Course learningoutcomes:

"The successful students possesses deepened and extended knowledge of the topics treatedin Commutative Algebra."

Description: In number theory one is naturally led to study more general numbers than just the classicalintegers and, thus, to introduce the concept of integral elements in number fields. The rings ofintegers in number fields have certain very beautiful properties (such as the unique factorisationof ideals) which characterise them as Dedekind rings. Parallely, in geometry one studies affinevarieties through their coordinate rings. It turns out that the coordinate ring of a curve is aDedekind 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 Dedekindrings. Along the way, we shall introduce and demonstrate through examples basic concepts ofalgebraic geometry and algebraic number theory. Moreover, we shall be naturally led to treatmany concepts from commutative algebra.

Depending on the previous knowledge of the audience, the lecture will cover all or parts of thefollowing topics:

- General concepts in the theory of commutative rings

+ rings, ideals and modules

+ Noetherian rings

+ tensor products

+ localisation

+ completion

+ dimension

- Number rings

+ integral extensions



+ ideals and discriminants

+ Noether's normalisation theorem

+ Dedekind rings

+ unique ideal factorisation

- Plane Curves

+ affine space

+ coordinate rings and Zariski topology

+ Hilbert's Nullstellensatz

+ resultant and intersection of curves

+ morphisms of curves

+ singular points

Teaching modality: Lecture course and problem sessions

Language: Anglais

Mandatory: Oui

Evaluation:Continuous Assessment

Remark: Lecture notes, exercise sheets (available on Moodle)

Literature :

There are many books on commutative algebra, for example:

- E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry.

- Dino Lorenzini. An Invitation to Arithmetic Geometry, Graduate Studies in Mathematics, Volume9, American Mathematical Society.

- M. F. Atiyah, I. G. Macdonald. Introduction to Commutative Algebra, Addison-WesleyPublishing Company.

Professor: WIESE Gabor, SGOBBA Pietro

Riemannian Geometry




ECTS: 5

Objective: 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 puton the Mathematical Method, i.e., the optimal technique to learn and apply mathematics, inparticular in order to solve real life problems using mathematical tools. This method is the mostimportant of the objectives of any study program in mathematics.


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

- Explain the main definitions and results of Differential and Riemannian Geometries

- Comment on new concepts, like the category of smooth manifolds, smooth scalar observables,their derivatives, vector bundles, tensor fields, Lie derivatives, covariant derivatives, Christoffelsymbols, curvature, torsion, Riemannian manifolds, Levi-Civita connection, parallel transport,geodesics, as well as various types of curvature on Riemannian manifolds

- Apply the new techniques and solve related 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

Description: Differential and Riemannian Geometries have applications in numerous fields of science,including Einstein's general theory of relativity, string theory, black holes and galaxy clusters,probability, engineering, economics, modeling and design, wireless communication and imageprocessing, biology, chemistry, geology… The main concept in Differential Geometry aredifferential manifolds - roughly, higher dimensional analogs of curves and surfaces. To be ableto use efficiently these new spaces, a generalization of fundamental mathematical notions,such as derivatives and integrals, is required. As for Riemannian manifolds, they are differentialmanifolds that come equipped with a metric, which then allows for concepts like length, areaand volume on the manifold considered. This metric is also an essential ingredient of thedefinition of the curvature of a Riemannian or pseudo-Riemannian manifold. Curvature is themost important aspect of (pseudo-) Riemannian manifolds: they are curved spaces like, forinstance, the Universe. To understand the fundamental concept of curvature, a prior excursionin the world of connections (a kind of derivatives) on vector bundles (manifolds with additionalstructure) is indispensable.

Teaching modality: Interactive lectures and exercise sessions

Language: Anglais

Mandatory: Oui

Evaluation: Evaluation: Oral or written exam

Remark: BIBLIOGRAPHY

. M. P. do Carmo , Riemannian Geometry, Birkhäuser (1992)

2. W. Klingenberg , Riemannian Geometry, de Gruyter (1995)



3. John M. Lee, Riemannian Manifolds, Springer (1997)

4. P. Petersen, Riemannian Geometry, Springer (2006)

Professor: PONCIN Norbert

Partial Differential Equations I


ECTS: 5

Objective: The goal of the course it to get acquainted with Partial differential equations (PDE) as a powerfultool for modeling problems in science, providing functional analytic techniques in order to dealwith PDE.


On successful completion of the coursethe 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 Laplaceequation

- Prove acquaintance with the basic properties of harmonic functions (maximum principle, meanvalue 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.

Teaching modality: Lecture course

Language: Anglais

Mandatory: Oui

Evaluation: Written exam

Remark: 1. Rudin: Functional analysis

2. Jost: Postmodern analysis

3. Folland: Introduction to partial differential equations.

4. Reed-Simon: Methods of mathematical physics I-IV



Professor: OLBRICH Martin, PALMIROTTA Guendalina

Probability (Martingale Theory)


ECTS: 5

Objective: Introduction to basic concepts of modern probability theory


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: Filtrations, conditional expectations, martingales, stopping times, optional stopping, Doobinequalities, martingale convergence theorems, canonical processes, Markov semigroups andprocesses, Brownian motion


Language: Anglais

Mandatory: Oui


Remark: H. Bauer, Wahrscheinlichkeitstheorie

D. Williams, Probability with Martingales

Professor: CAMPESE Simon

Student Project


ECTS: 4

Language: Anglais

Mandatory: Non

Professor: MERKOULOV (MERKULOV) Serguei, THALMAIER Anton



Algorithmic Number Theory


ECTS: 4


On successful completion of the course, the student should be able to:- Explain the main algorithms for primality testing, factorizing large integers, solving thediscrete logarithm problem, both in the multiplicative group of finite fields, as well as in thecontext of elliptic curves defined over finite fields- Read and understand some scientific articles published in the domain, and ask relevantquestions- Give a talk for peers on related topics- Organize his approach to general problems in an algorithmic way

Language: Anglais

Mandatory: Non

Professor: LEPREVOST Franck

Basics of Discrete Mathematics


ECTS: 4


More important than the actual content of the course (given below), is the development of thestudent's mathematical maturity. Upon completing this course a student should be able to:1. recall basic concepts and tools which should be present in any science curriculum, not onlyin mathematics and computer science, but also in engineering and other applied sciences,2. formulate and solve mathematically several logical and combinatorial problems arising inscience as well as in quotidian life, and3. recognize the genuine pleasure in tackling problems and the blissful joy by attaining theirsolution.

Description: The content includes but is not limited to :1. Discrete calculus: sums and recurrences; manipulation of sums; multiple sums; generalmethods; finite calculus; summation by parts.2. Binomial coefficients: basic identities; binomial theorem; multinomial coefficients;Vandermonde's convolution; Newton series.3. Generating functions: basic maneuvers; solving recurrences; exponential generatingfunctions.4. Special numbers: Stirling numbers; Eulerian numbers; Harmonic numbers; Harmonicsummations; Bernoulli numbers; Fibonacci numbers.5. Asymptotics: O Notation, Euler's summation formula.




Course slides

Language: Anglais

Mandatory: Non


Remark: Littérature / Literatur / Literature (recommended but not mandatory) :• R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2ndedition, 2003.• W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications,Academic Press, 2nd edition, 2001.• C. Mariconda and A. Tonolo, Discrete Calculus. Methods for Counting, Springer, 2016.• D. E. Knuth, The Art of Computer Programming, Volume 1: Fundamental Algorithms, 3rdedition, Addison Wesley, 1997.

Professor: MARICHAL Jean-Luc

Probabilistic Models in Finance


ECTS: 4

Objective: Introductory course to basic concepts of Mathematical Finance, also suitable for students whoare not going to choose their specialization in Finance. The goal is to deepen the knowledge ofmodern probability theory by studying applications of general interest in an actual field of appliedmathematics.


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 Rubinsteinmodel- Apply the techniques of Snell envelopes to evaluate American options- Derive the classical Black-Scholes formulas as limiting case of a sequence of CRR markets

Description: Discrete financial markets, the notion of arbitrage, discrete martingale theory, martingaletransforms, complete markets, the fundamental theorem of asset pricing, European andAmerican options, hedging strategies, optimal stopping, Snell envelopes, the model of Cox, Rossand Rubinstein.


Language: Anglais

Mandatory: Non




Remark: Literature :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

Professor: THALMAIER Anton

Reading Course "complements to commutative algebra"


ECTS: 1

Language: Anglais

Mandatory: Non

Professor: WIESE Gabor, SGOBBA Pietro

Reading Course "complements to Riemannian Geometry"


ECTS: 1

Language: Anglais

Mandatory: Non


Reading Course "complements to Partial Differential Equations"


ECTS: 1

Language: Anglais

Mandatory: Non



Professor: OLBRICH Martin

Reading Course "complements to Probability (Martingale Theory)"


ECTS: 1

Language: Anglais

Mandatory: Non

Professor: CAMPESE Simon

Applied Didactics I


ECTS: 2

Mandatory: Oui

Professor: HAUSTGEN Marc Paul, COLLETTE-CLERBAUT Frédérique

Applied Didactics II


ECTS: 3

Mandatory: Oui

Professor: BARTHOLME Carine

Internship in a secondary school I


ECTS: 0

Language: Français

Mandatory: Oui

Evaluation: Validated in S2



Workshop 1 zum Praktikum 1 SEM

Module: Modul: Forschungsfeld Schule (Semester 1)

ECTS: 3


Lernziele: Die Studierenden…… sind vertraut mit den theoretischen Grundlagen allgemeiner Didaktik und können diese inunterrichtspraktische Anwendungsfelder übertragen.… haben ein Basiswissen pädagogischer Diagnostik erworben und können dieseshandlungsleitend in Unterrichtshospitation und Unterrichtsplanung umsetzen.… haben ihren eigenen Wissensstand und ihre Kompetenzen in Theorie und Unterrichtspraxisreflektiert und auf dieser Basis persönliche Entwicklungsziele und Forschungsinteressen fürihr Studium identifiziert.

Description: ACHTUNG: Es werden drei Parallelveranstaltungen (11:30-13:00 / 13:15-14:45 / 15:45.17:15)angeboten – bitte melden Sie sich nur zu einem Workshop an.Kursbeschreibung:Der Workshop dient einerseits der Einführung in die Allgemeine Didaktik und fokussiert sowohlauf theoretische Grundlagen und praxisorientierte Umsetzung diagnostischer Verfahren zuVorbereitung der Unterrichtshospitation wie auch auf Gestaltungsmöglichkeiten von Unterrichtzur Vorbereitung der eigenen Unterrichtspraxis im Rahmen des orientierenden Praktikums.Andererseits werden eigene Wissens- und Erfahrungsstände der Studierenden gemeinsam mitden das Praktikum begleitenden Lehrpersonen reflektiert, um individuelle Entwicklungsziele undForschungsinteressen für das Masterstudium zu identifizieren.Die Workshopinhalte und Lernziele sind ebenso wie die zu Grunde gelegte Literaturmit den Didaktiken der jeweiligen Fächer abgestimmt und bilden den allgemeinenbildungswissenschaftlichen Rahmen derselben.Bibliographie: Ein Reader mit Grundlagentexten wird zu Beginn des Semesters über Moodle zurVerfügung gestellt. Dieser ist vor Beginn der Workshops zu bearbeiten.

Language: Allemand, Français

Mandatory: Non

Professor: HARION Dominic3

Einführung in die Schulpädagogik GENERAL COMPETENCES I


ECTS: 3

Objective: Die Studierenden haben gelernt…• Sinn, Absicht und Realität von Schule unter einem historischen, pädagogischen undsoziologischen Blickwinkel zu analysieren• den Lehrerberuf als Profession zu verstehen• den Zusammenhang von Bildung, Bildungszielen und Bildungsplänen zu erkennen• Lernen als sozialen Prozess zu beschreiben.



• Verschiedene Ideen davon, was „guten Unterricht" ausmacht zu diskutierenDie Studierenden sind in der Lage …• professionelle Standards von Lehrerhandeln umzusetzen• die Bedingungen des luxemburgischen Schulsystems zu analysieren• verschiedene Theorien zur Unterrichtsqualität und Diagnostik anzuwenden


Die Studierenden haben gelernt…• Sinn, Absicht und Realität von Schule unter einem historischen, pädagogischen undsoziologischen Blickwinkel zu analysieren• den Lehrerberuf als Profession zu verstehen• den Zusammenhang von Bildung, Bildungszielen und Bildungsplänen zu erkennen• Lernen als sozialen Prozess zu beschreiben.• Verschiedene Ideen davon, was „guten Unterricht" ausmacht zu diskutierenDie Studierenden sind in der Lage …• professionelle Standards von Lehrerhandeln umzusetzen• die Bedingungen des luxemburgischen Schulsystems zu analysieren und verschiedeneTheorien zur Unterrichtsqualität und Diagnostik anzuwenden

Description: Die Vorlesung „Einführung in die Schulpädagogik" analysiert die (luxemburgische) Schuleals eine historisch gewachsene Institution, die ganz unterschiedlichen Zwecken dient bzw.dienen soll. Dabei stehen pädagogische, soziologische und historische Erklärungsansätze imMittelpunkt der Beschreibung schulischer Wirklichkeiten. Zudem werden der Lehrerberuf sowiedie Schulentwicklung (Curricula, Bildungsziele etc.) auf ihre professionellen Begründungen hinvorgestellt und hinterfragt. Die Vorlesung führt ebenfalls in die wichtigsten Ideen zum Thema„Lernen" und zur Unterrichtsqualität ein.Bibliographie:Ludwig Haag, Sibylle Rahm, Hans Jürgen Apel, Werner Sacher (Hrsg.): StudienbuchSchulpädagogik. Verlag Julius Klinkhardt 2013Hanna Kiper, Hilbert Meyer, Wilhelm Topsch (Hrsg.): Einführung in die Schulpädagogik.Cornelsen Verlag 2011Ilona Esslinger-Hinz, Anne Sliwka (Hrsg.): Schulpädagogik. Beltz Verlag 2011

Language: Allemand

Mandatory: Oui

Evaluation: Abschlussklausur

Professor: LENZ Thomas



ECTS: 3


Lernziele: Die Studierenden…… sind vertraut mit den theoretischen Grundlagen allgemeiner Didaktik und können diese inunterrichtspraktische Anwendungsfelder übertragen.… haben ein Basiswissen pädagogischer Diagnostik erworben und können dieseshandlungsleitend in Unterrichtshospitation und Unterrichtsplanung umsetzen.



… haben ihren eigenen Wissensstand und ihre Kompetenzen in Theorie und Unterrichtspraxisreflektiert und auf dieser Basis persönliche Entwicklungsziele und Forschungsinteressen fürihr Studium identifiziert.

Description: ACHTUNG: Es werden drei Parallelveranstaltungen (11:30-13:00 / 13:15-14:45 / 15:45.17:15)angeboten – bitte melden Sie sich nur zu einem Workshop an.Kursbeschreibung:Der Workshop dient einerseits der Einführung in die Allgemeine Didaktik und fokussiert sowohlauf theoretische Grundlagen und praxisorientierte Umsetzung diagnostischer Verfahren zuVorbereitung der Unterrichtshospitation wie auch auf Gestaltungsmöglichkeiten von Unterrichtzur Vorbereitung der eigenen Unterrichtspraxis im Rahmen des orientierenden Praktikums.Andererseits werden eigene Wissens- und Erfahrungsstände der Studierenden gemeinsam mitden das Praktikum begleitenden Lehrpersonen reflektiert, um individuelle Entwicklungsziele undForschungsinteressen für das Masterstudium zu identifizieren.Die Workshopinhalte und Lernziele sind ebenso wie die zu Grunde gelegte Literaturmit den Didaktiken der jeweiligen Fächer abgestimmt und bilden den allgemeinenbildungswissenschaftlichen Rahmen derselben.Bibliographie: Ein Reader mit Grundlagentexten wird zu Beginn des Semesters über Moodle zurVerfügung gestellt. Dieser ist vor Beginn der Workshops zu bearbeiten.

Language: Français, Allemand

Mandatory: Non




Semester 2

Probability (stochastic analysis)

Module: Module 1 (Semester 2)

ECTS: 5

Objective: Introduction to basic concepts of Stochastic Analysis

Description: Continuous martingales, stochastic integration, quadratic variation, Itô calculus, theoremof Girsanov, stochastic differential equations, Markov property of solutions, connectionof stochastic differential equations and partial differential equations, martingalerepresentation theorems, chaotic expansions, Feynman-Kac formulas


Language: Anglais

Mandatory: Non


Remark: I. Karatzas, S. Shreve: Brownian motion and stochastic calculus. 2 ndedition. Springer,1991

B. Oksendal: Stochastic differential equations. Springer, 2003

D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. Springer Grundl., 1999

Professor: PECCATI Giovanni

Homological Algebra


ECTS: 5

Objective: 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 puton the Mathematical Method, i.e., the optimal technique to learn and apply mathematics, inparticular in order to solve real life problems using mathematical tools. This method is the mostimportant of the objectives of any study program in mathematics.


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

- Explain the main definitions and results of Classical and Modern Homological Algebra



- Comment on new concepts, like the 2-category of cochain complexes in modules, theconnecting homomorphism, Künneth theorem, the Mayer-Vietoris exact sequence, homotopyequivalences and deformation retracts, the Koszul, Cech, Hochschild, Chevalley-Eilenberg, andde Rham cohomologies…

- Expound homological algebra in abelian categories

- Apply the new techniques and solve problems, e.g., compute the de Rham cohomology ofspheres, prove the isomorphism between the de Rham and Cech cohomologies…

- 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

Description: Homological algebra is roughly speaking a collection of techniques that allow scientists to extractinformation encoded in a widespread type of sequences of objects and arrows. It originatesfrom topology, became an independent field of Mathematics in the mid-1940s, and developedin close connection with modern category theory. Nowadays, homological algebra plays animportant role in numerous areas of Mathematics, e.g., in algebraic number theory, algebraic anddifferential geometry, higher algebra, graph complexes and deformation theory, operad theory,mathematical and theoretical physics… The first part of the course covers the basic conceptsand the important tools. The second part is devoted to a consolidation of the knowledge acquiredvia exercises and applications, whereas the third part builds on the previous skills and providesinsight into the modern approach to Homological Algebra in the framework of abelian categorytheory.

Teaching modality: Interactive lectures and exercise sessions

Language: Anglais

Mandatory: Oui

Evaluation: Evaluation: Oral or written exam

Remark: Course and lecture notes under construction

C. A. Weibel, An introduction to homological algebra, Cambridge studies in advancedmathematids 38, 1997, ISBN 0-521-55987-1

P.J. Hilton, U. Stammbach, A course in homological algebra, Springer, 1997,0-387-94823-6

S. Mac Lane , Homology, Springer, 1995, 3-540-58662-8




Partial Differential Equations II


ECTS: 5

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

Description: Distributions as generalized functions continued, Sobolev spaces, elliptic regularity,

elliptic operators on compact manifolds, some non-linear equations.


Language: Anglais

Mandatory: Oui


Remark: 1. Jost: Postmodern analysis

2. Folland: Introduction to partial differential equations

3. Reed-Simon: Methods of mathematical physics I-IV

4. Aubin: Nonlinear analysis on manifolds

Professor: OLBRICH Martin, PALMIROTTA Guendalina

Graph theory


ECTS: 5


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

- Illustrate the main results and concepts with well-chosen examples

- Master 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



Description: Contenu / Inhalte / Content :Graphs are structures that are ubiquitous in Mathematics and its applications. Through apresentation of selected topics, the course aims to be an introduction to certain modern aspectsof graph theory. It is designed as a self-contained course, but some familiarity with graphtheoretic notions is of course an asset.The exercises sessions are organized to reinforce the understanding of the topics covered inthe course, and to get acquainted with the proof techniques and heuristics that are useful inGraph Theory.According to time and taste, topics covered will be chosen (and are not limited to) among thefollowing ones:- Matching, covering and packing problems- Colorings- Edge and vertex connectivity- Flows and Networks- Ramsey theory for graphs- Expander graphs and applications- Graph minors and reconstruction problems- Infinite Graph Theory

Teaching modality: Lecture course + exercises sessions

Language: Anglais

Mandatory: Non

Evaluation: oral exam

Remark: Littérature / Literatur / Literature :Reinhard Diestel. Graph Theory, 5th edition. Graduate texts in mathematics 173, Springer, 2017

Professor: TEHEUX Bruno

Algebraic Number Theory


ECTS: 4

Objective: Introduce the students to Algebraic Number Theory.


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

- appreciate the role played by abstract algebraic number theory for the solution ofconcrete Diophantine equations,

- define number rings and enumerate their fundamental properties,

- solve certain Diophantine equations,

- outline the proofs of the fundamental results of the lecture,



- apply the quadratic reciprocity law,

- calculate class numbers and rings of integers in simple cases.

Description: Explicit Diophantine equations

Number rings

Cyclotomic fields

Quadratic reciprocity

Geometry of numbers

Teaching modality: Lecture with integrated exercises

Language: Anglais

Mandatory: Non

Evaluation: Oral exam

Remark: Support and Literature:

to be announced in the lecture

Professor: PERUCCA Antonella

Introduction to Continuous Time Models in Mathematical Finance


ECTS: 4

Objective: Introduction to continuous time models in mathematical finance (Black-Scholes model)

Description: Arbitrage, risk-neutral measures, option pricing, hedging, Black-Scholes-Mertonequation, call-put parity, connections with partial differential equations, forwards andfutures, american options, exotic options , change of numéraire, Garman-Kohlhagenformula, term-structure models, Vasicek model, Heath-Jarrow-Morton model, forwardLIBOR model


Language: Anglais

Mandatory: Non




Remark: M. Baxter, A. Rennie: Financial calculus. An introduction to derivative pricing. CUP, 1996

F. E. Benth: Option Theory with Stochastic Analysis. Springer, 2004

R. J. Elliot, P. E. Kopp: Mathematics of Financial Markets. 2 nded., Springer Finance, 2004

J. C. Hull: Options, futures, and other derivatives. 6th ed., Prentice-Hall, 2005

S. E. Shreve: Stochastic calculus for finance. II: Continuous-time models. Springer, 2004

J. Michael Steele: Stochastic Calculus and Financial Mathematics. Springer, 2001


Numerical Analysis


ECTS: 4

Language: Français

Mandatory: Non

Professor: ANTOINE Xavier

An introduction to mathematical statistics


ECTS: 4

Objective: The course provides an introduction to Mathematical Statistics and addresses more specificallythe problems of estimation and testing in parametric models.


A student should be able to

ntest the goodness-of-fit of the distribution of the data to some parametric models,

estimate and test a hypothesis in parametric models. Build a confidence region for theparameter

Description: 1. Introduction the statistical paradigm and statistical models.2. The main statistical issues I. Estimation, risk, asymptotic normality, confidence intervals. The

main probabilistic tools: Markov's inequality, Hoeffding's inequality, Berry-Esseen's theorem,the central limit theorem, Slutsky's lemma, the delta method.



3. The main statistical issues II. Hypothesis testing. First and second kind errors. Constructionof tests from confidence regions. The pivot method.

4. The empirical measure and its applications. The empirical moments, the empiricaldistribution function, the empirical variance. The moment method, Kolmokorov-Smirnov'stests, estimation based on the empirical quantiles. Robustness.

5. The maximum likelihood estimator (MLE). Example and counter-examples. Consistency ofthe MLE.

6. One dimensional exponential families. Properties of the MLE.7. Tests in parametric models. The Neyman-Pearson's test, the likelihood ratio test. The

monotone likelihood ratio property.

Mandatory: Non


Remark: Literature : Mathematical Statistics (vol 1), Bickel, P. and Doksum, K.

Professor: BARAUD Yannick

Reading course "complements to Homological Algebra"


ECTS: 1

Language: Anglais

Mandatory: Non


Reading course "complements to Partial Differential Equation II"


ECTS: 1

Language: Anglais

Mandatory: Non

Professor: OLBRICH Martin

Reading course "complements to Graph Theory"


ECTS: 1

Language: Anglais



Mandatory: Non

Professor: TEHEUX Bruno

Reading course "complements to Probability (stochastic analysis)


ECTS: 1

Language: Anglais

Mandatory: Non


Hands-on experiences with mathematical didactics I


ECTS: 3

Language: Anglais

Mandatory: Oui


Learning and teaching mathematics I


ECTS: 2

Language: Anglais

Mandatory: Oui

Professor: COLLETTE-CLERBAUT Frédérique

General Professional Competence II-IV

Module: Module 4 : General Professional Competence II (Semester 2)

ECTS: 5

Description: - Research fields "school"



- Multilingualism and heterogeneity

- Social Context of teaching and learning

Language: Français

Mandatory: Oui

Mehrsprachigkeit im Sprach- und Fachunterricht

Module: Modul: Mehrsprachigkeit und Heterogenität (Semester 2)

ECTS: 3

Objective: Die Studierenden haben gelernt…• ... dass eine Sprache aus mehreren Varietäten bestehen und dass Äußerungen aus

mehreren Sprachen bestehen können.• … dass in der Schule eine bestimmte Sprachvarietät, die „Standardsprache" vorausgesetzt

wird, diese aber nicht bei allen Schüler/innen vorausgesetzt werden kann• ... dass Sprachen aus bestimmten formalen Einheiten bestehen (Phoneme, Morpheme,

Wörter, Sätze, …), die jeweils bestimmte Funktionen in der sprachlichen Äußerungübernehmen.

• … dass es verschiedene Möglichkeiten gibt, Mehrsprachigkeit in den Unterricht zu integrieren(Scaffolding, Interlanguaging, CLIL, Sprachkontraste, Übersetzungen, …)

Die Studierenden sind in der Lage …• … die stillschweigende Voraussetzung der „Standardsprache" zu erkennen und

angemessene, „sprachsensible" Formen des Umgangs im Unterricht zu finden.• … verschiedene Aspekte sprachsensiblen Unterrichts in Ihren Unterricht zu integrieren.• … Lehrwerke und Unterrichtsmaterialien nach sprachsensiblen Kriterien zu analysieren und

im Unterricht einzusetzen.

Description: Schule wird bis heute als eine Institution mit einem „monolingualen Habitus" gesehen. DieVorlesung hinterfragt diese institutionelle Einsprachigkeit und führt in verschiedene Aspektevon Mehrsprachigkeit ein: Was unterscheidet eine einsprachige von einer mehrsprachigenÄußerung? Wie sind mehrsprachige Praktiken strukturiert?Aus didaktischer Perspektive wird gefragt, welche Faktoren den Spracherwerbs- bzw.Sprachlernprozess von Schüler/innen in schulischen und außerschulischen Kontextenbeeinflussen und wie diese Lernprozesse im Sprachenunterricht unterstützt werden können.Hierzu werden verschiedene pädagogische Haltungen und didaktische Mittel präsentiert.Ein wesentlicher Teil der Auseinandersetzung sprachlicher und didaktischer Mittel wird anTextaufgaben aus dem Bereich Mathematik erarbeitet, sowie an Aufgabenstellungen in denSprachenfächern.• Schleppegrell, M. (2004). The Language of Schooling: A Functional Linguistics Perspective.

Mahwah, New Jersey: Lawrence Erlbaum Associates Publishers. (= d er zentraleEinstiegstext)

• García, O., & Wei, L. (2014). Translanguaging: Language, Bilingualism and Education. NewYork: Palgrave Macmillan.

• Kniffka, G., & Siebert-Ott, G. (200). Deutsch als Zweitsprache: Lehren und Lernen.Paderborn: Schöningh.



• Wessel, L. (2015). Fach- und sprachintegrierte Förderung durch Darstellungsvernetzung undScaffolding. Ein Entwicklungsforschungsprojekt zum Anteilbegriff. Heidelberg u.a.: SpringerSpektrum.

Language: Allemand

Mandatory: Oui

Evaluation: Mitarbeit während des Seminars, semesterbegleitende Zwischenevaluation, abschließendeGruppenarbeit

Professor: WETH Constanze

Workshop zur Nachbereitung des orientierenden Schulpraktikums für Studierendeim 2. Semester


ECTS: 2

Language: Français

Mandatory: Oui


Workshop zur Nachbereitung des orientierenden Schulpraktikums für Studierendeim 2. Semester (Gruppe 2)


ECTS: 2

Language: Allemand

Mandatory: Oui




Semester 3

Introduction into Algebraic Geometry


ECTS: 5

Language: Français

Mandatory: Non

Professor: KALUGIN Alexey

Riemann Surfaces


ECTS: 5

Language: Français

Mandatory: Non

Professor: SCHLENKER Jean-Marc

Lie Algebras and Lie Groups


ECTS: 5

Objective: The purpose of this course is to give an introduction into the theory of finite dimensional Liegroups 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 ofmechanical systems, and geometric structures in nature- Explain the deep interrelations between Lie groups and Lie algebras, as well as the technicaltools behinds these interrelations- Simplify mathematical problems admitting symmetry Lie groups actions to problemsadmitting symmetry actions of their Lie algebras- Master applications to the theory of manifolds and representation theory, which in turn haveapplications in physics, engineering and mechanics



Description: The Lie algebra of a Lie group, the exponential map, , the adjoint representation, actions ofLie groups and Lie algebras on manifolds, the un iversal enveloping algebra, basics of therepresentation theory.


Language: Anglais

Mandatory: Oui

Evaluation: Written examination

Remark: Littérature / Literatur / Literature:

1) "Lie groups and Lie algebras" by Eckhard Meinrenken, 83 pages (free to download)

2) "Prerequisites from Differential Geometry" by Sergei Merkulov (free to download)

Professor: MERKOULOV (MERKULOV) Serguei

Combinatorial Geometry


ECTS: 5


The course requires minimal prerequisites (some linear algebra, Euclidean geometry and basictopology) but aims to explore results that are at the limit of current known understanding. Inparticular, 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 approachand 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 typetheorems), Minkovski's first theorem for lattices, and Ramsey theory (graph coloring problems).


Language: Anglais

Mandatory: Oui

Evaluation: Oral exam and classwork

Professor: PARLIER Hugo



Continuous-Time Stochastic Calculus and Interest Rate Models


ECTS: 5


On successful completion of the course, the student should be able to:- Calculate probabilities and expectations related to the semi-martingale models presented inthe lectures- Carry out calculations based on change of numéraire and no-arbitrage pricing- Compute the prices of interest rate derivatives- Apply stochastic volatility models to deal with implied volatility surfaces

Description: Basic Notions of Fixed Income Markets; Semimartingale Modeling; Stochastic DifferentialEquations; No-Arbitrage Pricing; Change of Numéraire; Short Rate Models; Heath-Jarrow-Morton Framework; Market Models; Stochastic Volatility


Language: Anglais

Mandatory: Oui


Remark: D. Brigo and F. Mercurio (2006) Interest Rate Models : Theory and Practice. Springer VerlagM. Musiela and M. Rutko wski (1997) Martingale Methods in Financial Modeling. Springer Verlag


Numerical solution of partial differential equations and applications


ECTS: 5

Objective: The objectives of the course are to provide to students a global overview of the finite elementmethod. Not only the mathematical foundations will be developed but also the concreteimplementation of finite element approximation techniques in view of their application toengineering problems. Some problems will be fully solved by using some computer programswritten by the students (by using the PDE toolbox of Matlab or some bricks written by students).


On successful completion of the course, the student should be able to:- Explain the mathematical foundations of the finite element method- Master its concrete implementation for nontrivial engineering boundary-value problems- Adapt, if necessary, the finite element method to the problem under consideration

Description: 1. Complements on Sobolev spaces; trace theorems; Green's formulae2. The Lax-Milgram theory ; variational formulations3. The Finite Element Method for stationary elliptic Partial Differential Equations (PDEs):

theoretical aspects



4. The Finite Element Method for stationary elliptic Partial Differential Equations (PDEs):implementation and computational aspects

Teaching modality: Lecture course, exercises of applications

Language: Anglais

Mandatory: Non

Evaluation: The students will have to provide some reports that will be evaluated. In addition, a final writtenexamination will be organized.

Remark: Support / Arbeitsunterlagen / Support:

Lecture notes (french), exercise sheets (english)

Littérature / Literatur / Literature:

1) X. Antoine, Numerical solution of PDEs, lecture notes

2) X. Antoine, Numerical Analysis, course at the University of Luxembourg

3) G. Allaire, Analyse Numérique et Optimisation, Presses de l'Ecole Polytechnique

4) P.A. Raviart et J.M. Thomas, Introduction à l'Analyse Numérique des Equations aux DérivéesPartielles, Dunod.

Professor: ANTOINE Xavier

Gaussian processes and applications


ECTS: 5


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

Language: Anglais

Mandatory: Oui

Professor: NOURDIN Ivan

Advanced Mathematical Statistics


ECTS: 5



Language: Français

Mandatory: Non

Professor: BARAUD Yannick

Internship in secondary school II


ECTS: 0

Language: Français

Mandatory: Oui

Evaluation: validated in S4

Applied Didactics IV


ECTS: 3

Language: Français

Mandatory: Non

Professor: BARTHOLME Carine

Applied Didactics III


ECTS: 2

Language: Français

Mandatory: Non

Professor: HAUSTGEN Marc Paul

Teaching Children with Special Educational Needs

Module: Modul: Lehren und Lernen im sozialen Kontext (Semester 3)

ECTS: 3



Objective: This course (lecture) is aimed to deepen students' knowledge on the latest research aboutthe leaning processes of children with special educational needs (SEN) like dyslexia, specificlanguage impairment, and dyscalculia. This course will also try to equip these future teacherswith strategies aimed to more efficiently teach children with SEN.

Description: Over the year's research has informed the scientific and the educational community about howchildren's learning processes can be affected by neurologically-based difficulties. These specificlearning difficulties can interfere with the learning of basic skills like reading, writing, maths orlanguage learning. They can also interfere with higher level skills such as organization, timeplanning, abstract reasoning, long or short term memory and attention. As teachers, it is essentialto understand the impact specific learning difficulties have on children's learning trajectories andhow we can best ameliorate these difficulties.

Reading:

Workshop reading

• Ashcraft, M. H., & Krause, J. A. (2007). Working memory, math performance and mathanxiety. Psychonomic Bulletin & Review. 14: 243-248.

• Jeffes, B. (2016). Raising the reading skills of secondary-age students with severe andpersistent reading difficulties: evaluation of the efficacy and implementation of a phonics-based intervention programme. Educational Psychology in Practice. 32: 73-84.

• Pijl, S. J. Frostad, P., & Mjaavatn, P. E. (2014). Students with special educational needs insecondary education: are they intending to learn or to leave? European Journal of SpecialNeeds Education. 29: 16-28.

• Purpura, D. J., Napoli, A. R., Wehrspann, E. A., & Gold, Z. S. (2017). Causalconnections between mathematical language and mathematical knowledge: A dialogicreading intervention. Journal of Research on Educational Effectiveness. 10: 116-137.

Core reading

• Beck, I., MacKeown, M., & Kucan L. (2013). Bringing Words to Life: Robust VocabularyInstruction. New York: Guilford. Chapter 1.

• Conti-Ramsden, G., Bishop, D., Clark, B., Norbury, C., & Snowling, M. J. (2014). SpecificLanguage Impairment (SLI). The internet RALLI campaign to raise awareness of SLI.Psychology of Language and Communication, 18: 143-148

• Denez, S., & Goswami, U. (2013). Developmental dyscalculia: Fresh perspectives. Trendsin Neuroscience and Education. 2: 33-37.Gathercole, S. E. (2008). Working memory in the classroom. The Psychologist. 21: 382-385

• Planinc, T. R., Kolnik, K. (2016). Working with students with special educational need: viewsand experiences of geography teachers. Dela, 46: 105-122.

• Snowling, M. J., (2014). Dyslexia: A language learning impairment. Journal of the BritishAcademy, 2: 43-58.

• Snowling, M. J., Cain, K., Nation, K., & Oakhill, J. (2009). Reading comprehension: Nature,assessment and teaching. Output of ESRC seminar series.

Further reading

• Bull, E., & Lee, K. (2014). Executive functioning and mathematics achievement. ChildDevelopment Perspectives, 8: 36-41.

• Duff, F. J., Clarke, P. J. (2011). Practitioner Review: Reading disorders: what are theeffective interventions and how should they be implemented and evaluated? Journal of ChildPsychology and Psychiatry, 52:3–12.



• Hehir, T. et al. (2016). A summary of the evidence on inclusive education: Alana• Purpura, D. J., & Reid, E. E. (2016). Mathematics and language: Individual and group

differences in mathematical language skills in young children. Early Childhood ResearchQuarterly, 26: 259-268.

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

• Snowling, M. J. (2013). Early identification and interventions for dyslexia: a contemporaryview. Journal for Research on Special Educational Needs, 13: 7-14.

• United Nations (2016). General comment No. 4. Convention on the Rights of Persons withDisabilities: United Nations.

Language: Anglais

Mandatory: Non

Evaluation: Exam

Professor: ENGEL DE ABREU Pascale

Digitale Schule (Semester 3)


ECTS: 5

Language: Allemand

Mandatory: Non

Professor: BAUMANN Isabell Eva

Einführung in die Pädagogische Psychologie


ECTS: 3

Language: Allemand

Mandatory: Non

Professor: GRUND Axel



ECTS: 3

Language: Allemand



Mandatory: Non




ECTS: 3

Language: Allemand

Mandatory: Non


Workshop Professionell Auftreten


ECTS: 1

Language: Allemand

Mandatory: Non

Professor: ULLMANN Barbara



Semester 4

Hands-on experiences with mathematical didactics II


ECTS: 3

Language: Anglais

Mandatory: Oui


Learning and teaching mathematics II


ECTS: 2


L'étudiant sera capable de

- décrire les aspects principaux liés à la différenciation pédagogique à savoir les contenus, lesstructures, les processus et les productions attendus des élèves

- mettre en œuvre différentes formes de différenciation par flexibilité, adaptation et modification.

- faire le lien entre différenciation pédagogique et évaluation

- élaborer des documents pédagogiques permettant une différenciation en cours demathématiques

Description: · Différenciation pédagogique

o Aspects théoriques

o Elaboration de situations d'apprentissage en vue d'une différenciation interne

o Mise en œuvre pratique dans une classe d'un lycée

· Elaboration de tâches mathématiques différenciantes

Language: Français

Mandatory: Oui

Evaluation: Portfolio de documentation des activités élaborées et mises en œuvre suivis d'un entretien



Remark: Cours distribués lors de chaque séance

Professor: COLLETTE-CLERBAUT Frédérique

Master Thesis


ECTS: 20


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 orconceptual techniques

- Discover original mathematics

- Verify results and apply them

- Write mathematical texts that are consistent with the tradition

Description: The master thesis in mathematics consists of the definition of a research project, thedetailed explanation of research articles and/or monographs aimed at a mathematicsaudience, as well as of potential further developments of these. The project, which shouldcontain parts of original mathematics, will be designed to suit the individual objectives ofthe students, to deepen their competence in a selected field of mathematics, and to opena door towards mathematical research.

Language: Anglais, Français, Allemand

Mandatory: Oui

Evaluation: Supervisor, director of studies

Remark: Admission to the Master Thesis will be granted only to students who acquired at least75 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).

Professor: THALMAIER Anton, MERKOULOV (MERKULOV) Serguei

PROFESSIONELL Auftreten (Sem4)


ECTS: 5



Language: Allemand

Mandatory: Non

Professor: ULLMANN Barbara

Commutative Algebra 30 15 5 Master en Enseignement ...€¦ · algebraic geometry and other fields...

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