• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-6
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

The purpose of the course is to introduce students to various aspects of algorithmic number theory linked to public-key cryptography.

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

The lecture will introduce some of the most important algorithms used for computing with integers modulo m, including the Chinese Remainder Theorem, will introduce a series of theorems (Fermat’s small theorem, theorem of Lagrange, exponentiation method, etc) that allow to accelerate computations in these rings. Fields, especially finite fields, will be introduced from an algorithmic angle. Usual probable primality tests will be described. They will provide the context for the introduction of Legendre symbols, the quadratic reciprocity law, etc. Usual integer factorization methods will be introduced as well, including the method of Fermat up to the quadratic field sieve. A series of cryptographic primitives will be given both for a finite group in general, and for specific groups in particular, arising from finite fields, or from elliptic curves defined over finite fields. If time allows, some methods for the computation of the number of points of an elliptic curve over a finite field will be given. This series of lectures will be completed by some broader conferences presenting the material in a more general setting.

One single exam (probably in written form, depending on the number of students registered for the exam).

Note / Literature / Bibliography “How big is big? How fast is fast? A hands-on tutorial on mathematics of computation” by Franck Leprévost (available at the documentation center of the university and on Amazon). https://www.amazon.fr/How-Big-Fast-Hands-Mathematics/dp/B087SGXLSH/ref=sr_1_1?__mk_fr_FR=%C3%85M%C3%85%C5%BD%C3%95%C3%91&crid=1ZWKY4WZEN0DO&keywords=Franck+Lepr%C3%A9vost&qid=1649316964&sprefix=franck+lepr%C3%A9vost%2Caps%2C219&sr=8-1  

• Number of ECTS: 2
• Course code: F1_SECEDUC-15
• Module(s): General Mathematics 1
• Language: FR
• Mandatory: No

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

Working through the program from a higher standpointsecondary school analysis and geometry

Le programme cadre pour l’enseignement des mathématiques dans l’enseignement secondaire classique et général :les domaines de l’analyse et de la géométrieles compétences à développerExplicitation des finalités de l’enseignement des mathématiques à l’aide de questions d’examenPréparation au concours de recrutementConcepts-clé pour l’enseignement des mathématiques activités mathématiques de baseapprentissage autonomesituations d’apprentissage ouvertes et complexesrésolution de problèmes

• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-1
• Module(s): General 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 For Master in Mathematics The final mark is the arithmetic mean of a mark from continuous assessment and a mark for the final exam.For Master in Secondary Education – Mathematics The final mark is the arithmetic mean of a mark from continuous assessment and a mark for a didactical paper.

Note / 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.

• Number of ECTS: 6
• Course code: F1_MA_MAT_FM-1
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

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 

Radon-Nikodym Theorem, conditional expectations, martingales, stopping times, optional stopping theorems, Doob’s inequalities, martingale convergence theorems, martingale central limit theorem, Brownian motion.

Written exam

Rick Durrett: Probability Theory and Examples Yuval Peres, Peter Morters: Brownian Motion

• Number of ECTS: 5
• Course code: MA_DS-31
• Module(s): General 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.

• Number of ECTS: 3
• Course code: F1_SECEDUC-16
• Module(s): General Mathematics 1
• Language: FR
• Mandatory: No

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

Être conscient du fait que chaque choix didactique a toujours un impact sur l’enseignement et l’apprentissage des contenus qui en dépendent Être capable de formuler des questions visant différents niveaux taxonomiques Être capable d’anticiper les problèmes de compréhension éventuels des élèves Être capable de déceler les moments clés d’une leçon/d’un apprentissage Être capable de planifier une séquence de leçons comprenant une introduction, des activités, des transitions, des temps-élèves, des exemples et exercices et les questions à poser ainsi que les réponses éventuelles des élèves Être capable de formuler des questions se prêtant pour une évaluation du type formatif/sommatif Être capable de planifier une leçon de façon que l’atteinte des objectifs d’apprentissage puisse se faire de plusieurs manières différentes

Présentation de leçons et de types d’introductions Taxonomie de Bloom / Anderson & KrathwohlDiscussion d’exercices et de feuilles de travailIntroduction aux différentes formes de temps-élèvesPréparation de leçons (avec un accent sur le cycle inférieur et moyen) – Comment faire ? Good practice :Public cibleImportance du sujet pour l’élèveObjectifs à atteindreMatériel disponible/à utiliserMéthodes didactiquesMoyens techniques, Visualisation, Apps, Jeux,RéflexionIntroductions et RappelsEnchaînements et TransitionsQuestions à poser, types de questions et taxonomieMoments clésExercicesÉvaluation – pourquoi, comment, types d’évaluation, fréquenceRessourcesProblèmes et conséquences des choix didactiques opérésPréparation au stage en janvier

Examen final : Remise d’une préparation écrite de leçons (avec présentation) Évaluation continue à l’aide de travaux hebdomadaires (à remettre individuellement, en binôme ou en groupe – suivant la tâche imposée)

  Litérature Barzel, B., Büchter, A., Leuders, T. (2007), Mathematik Methodik, Handbuch für die Sekundarstufe I und II, Cornelsen Hattie, J. (2010), Visible Learning, Routledge Leuders, T. (2020), Mathematikdidaktik (9e édition), Cornelsen Leuders, T., Prediger, S. (2016), Flexibel differenzieren und fokussiert fördern im Mathematikunterricht, Cornelsen Malle, G. (1993), Didaktische Probleme der elementaren Algebra, Vieweg Padberg, F., Wartha, S. (2017), Didaktik der Bruchrechnung (5e édition), Springer Sill, H.-D. (2018), Grundkurs Mathematikdidaktik, utb. Sturm, R. (2016), Schritt für Schritt zum guten Mathematikunterricht, Klett Kallmeyer

• Number of ECTS: 6
• Course code: F1_MA_MAT_MMCS2-2
• Module(s): General 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.

There will be an exam after the first half of the semester, and a final exam at the end of semester. Attendance is mandatory. Homework assignments will be posted as recommended problems, but not graded. Selected problems will be discussed in class. If you miss a lecture, you are responsible for obtaining lecture notes and for determining if any announcements were made. 

Note / 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

• Number of ECTS: 8
• Course code: F1_MA_MAT_GM-2
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: Yes

On successful completion of the course, the student should be able to: compute with tensors, such as metrics and curvature, in coordinates or coordinate-free demonstrate detailed knowledge of the exponential map, including criteria for completeness, conjugate points, and Jacobi fields understand multiple interpretations of curvature, for example, as the obstruction to a local parallel framing, or how it influences the spreading of geodesics via the Jacobi equation

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

Literatur M. 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)

• Number of ECTS: 6
• Course code: F1_MA_MAT_FM-2
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

Introduction to basic concepts of Stochastic Analysis.

Continuous martingales, stochastic integration, quadratic variation, Itô calculus, theorem of Girsanov, stochastic differential equations, Markov property of solutions, connection of stochastic differential equations and partial differential equations, martingale representation theorems, chaotic expansions, Feynman-Kac formula.

Written exam

I. Karatzas, S. Shreve: Brownian motion and stochastic calculus. 2nd edition. Springer, 1991 B. Oksendal:  Stochastic differential equations. Springer, 2003 D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. Springer Grundl., 1999

• Number of ECTS: 7
• Course code: F1_MA_MAT_GM-3
• Module(s): General 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 equations Work freely with the classical formulas in dealing with boundary value problems for the Laplace equation Prove acquaintance with the basic properties of harmonic functions (maximum principle, mean value property) and solutions of the wave equation (Huygens property) Solve Cauchy problems for the heat and the wave equations Give a pedagogic talk for peers on a related topic 

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

Written exam

Literatur Rudin: Functional analysis Jost: Postmodern analysis Folland: Introduction to partial differential equations. Reed-Simon: Methods of mathematical physics I-IV

• Number of ECTS: 6
• Course code: F1_MA_MAT_FM-3
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

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

On successful completion of the course, the student should be able to: Derive and apply formulas for option pricing and hedging strategies Carry out calculations based on arbitrage arguments Calculate the price of European call and put options using the Cox, Ross and Rubinstein model Apply the techniques of Snell envelopes to evaluate American options Derive the classical Black-Scholes formulas as limiting case of a sequence of CRR markets

Discrete financial markets, the notion of arbitrage, discrete martingale theory, martingale transforms, complete markets, the fundamental theorem of asset pricing,  European and American options, hedging strategies, optimal stopping, Snell envelopes, the model of Cox, Ross and Rubinstein.

written exam

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

• Number of ECTS: 1
• Course code: F1_MA_MAT_MMCS2-4
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

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

You should feel comfortable writing basic scientific programs in Python, and be able to participate fully in future courses that require an element of programming.

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.

• Number of ECTS: 6
• Course code: F1_MA_MAT_MMCS2-5
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

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

At the end of the course, a student will be able to understand the main methods, the rigorous mathematical analysis and algorithms developed in numerical analysis to master their concrete implementation on a computer to formulate some basic physics or engineering applications in view of their treatment by a numerical method and its computer solution.

The main results about the theory of numerical methods will be explained. During the course, the student will implement algorithms in Python to propose some concrete, robust and efficient scientific computing solutions to concrete problems. The course content is the following:Notion of error in numerical analysisPolynomial interpolation and approximationNumerical derivation and integrationNumerical solution of Ordinary Differential EquationsDirect and iterative methods for solving linear systemsNonlinear equations in one and several variablesProject presentation + Complementary topics (optimization, eigenvalues problems)

The evaluation will be based on continuous assessment (weekly programming exercises – 50 % of the grade) and a project presentation with a report (50 % of the grade).

X. Antoine, Numerical Analysis, course at the University of Luxembourg Ascher, Uri M and Greif, Chen, A First Course in Numerical Methods, SIAM, 2011. Heath, Michael T, Scientific computing: an introductory survey, revised second edition SIAM, 2018.

• Number of ECTS: 1
• Course code: F1_MA_MAT_GM-5
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

• Number of ECTS: 3
• Course code: MA_DS-8
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

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

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 session Written exam and homework, and possibly algorithm implementation project during the semester.  Retake exam Writen exam 

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

• Number of ECTS: 4
• Course code: F1_MA_MAT_GM-7
• Module(s): General Mathematics 1
• Language: EN
• Mandatory: No

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

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 / Bibliography Depends on the project.

• Number of ECTS: 1
• Course code: F1_MA_MAT_GM-8
• Module(s): General 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

• Number of ECTS: 2
• Course code: F1_SECEDUC-18
• Module(s): General Mathematics 3
• Language: FR
• Mandatory: No

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

Working through the program from a higher standpointsecondary school analysis and geometry

Le programme cadre pour l’enseignement des mathématiques dans l’enseignement secondaire classique et général les domaines de l’algèbre et des probabilitésles compétences à développerExplicitation des finalités de l’enseignement des mathématiques à l’aide de questions d’examenPréparation au concours de recrutementConcepts-clé pour l’enseignement des mathématiques :activités mathématiques de baseapprentissage autonomesituations d’apprentissage ouvertes et complexesrésolution de problèmes

• Number of ECTS: 3
• Course code: F1_SECEDUC-19
• Module(s): General Mathematics 3
• Language: FR
• Mandatory: No

Savoir concevoir des situations d’apprentissage Savoir évaluer les apprentissages

Être capable de planifier une séquence de leçons comprenant une introduction, des activités, des transitions, des temps-élèves, des exemples et exercices et les questions à poser ainsi que les réponses éventuelles des élèves Être capable d’anticiper les problèmes de compréhension éventuels des élèves Être capable de déceler les moments clés d’une leçon/d’un apprentissage Être capable de planifier une leçon permettant d’atteindre les objectifs d’apprentissage de plusieurs manières différentes Être capable de formuler des questions se prêtant pour une évaluation du type formatif/sommatif Être capable de concevoir une épreuve sommative adaptée aux objectifs d’apprentissage visés qui respecte le principe de la structure et de la répartition des points d’un devoir en classe imposé par le programme du niveau d’enseignement en question Connaître les critères de correction des devoirs en classe en mathématiques Être capable d’évaluer les tests et examens de mathématiques Être capable d’utiliser les résultats de l’évaluation de manière diagnostique et remédiative

Préparation détaillée de leçons de mathématiques (avec un accent sur le cycle supérieur et moyen)Planification des unités et séquences de mathématiques avec choix de méthodes et de formes sociales appropriéesIntroduction aux différents types d’évaluationConception d’épreuves et de devoirs en classe en mathématiquesTravail avec des grilles de compétences/indicateursGestion de l’erreur dans le processus d'apprentissage en mathématiquesMesures de différenciationPréparation au stage en janvier

Évaluation continue à l’aide de travaux hebdomadaires (à remettre individuellement, en binôme ou en groupe – suivant la tâche imposée) Examen final : Remise d’une préparation écrite de leçons suivie d’un examen oral

Litérature Antibi, A. (2003), La constante macabre, Math’Adore Antibi, A. (2021), La folie de l’évaluation, Math’Adore Astolfi, J.-P. (2015), L’erreur, un outil pour enseigner, ESF Editeur Barzel, B., Büchter, A., Leuders, T. (2007), Mathematik Methodik, Handbuch für die Sekundarstufe I und II, Cornelsen Büchter, A., Leuders, T. (2005), Mathematikaufgaben selbst entwickeln, Lernen fördern – Leistung überprüfen, Cornelsen Hattie, J. (2010), Visible Learning, Routledge Leuders, T. (2020), Mathematikdidaktik (9. Auflage), Cornelsen Leuders, T., Prediger, S. (2016), Flexibel differenzieren und fokussiert fördern im Mathematikunterricht, Cornelsen Sill, H.-D. (2018), Grundkurs Mathematikdidaktik, utb. Sturm, R. (2016), Schritt für Schritt zum guten Mathematikunterricht, Klett Kallmeyer

• Number of ECTS: 6
• Course code: MAMATH-173
• Module(s): General 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.

Written examination

Literature "Lie groups and Lie algebras" by Eckhard Meinrenken, 83 pages (free to download) “Prerequisites from Differential Geometry” by Sergei Merkulov (free to download)

• Number of ECTS: 6
• Course code: MAMATH-174
• Module(s): General 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 cover a selection of themes from combinatorial aspects of geometry.Themes include general theorems about convex sets in n dimensional real space (and Helly type theorems), Minkovski’s first theorem for lattices, and Ramsey theory (graph coloring problems). 

Oral exam and classwork

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

• Number of ECTS: 2
• Course code: MAMATH-144
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

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

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

The students are marked for their project work.In the case of a retake exam, a new project has to be done. The retake can be done in other groups and with other supervisors.

Literature Depends on the project.

• Number of ECTS: 6
• Course code: MAMATH-147
• Module(s): General 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.

Students will be rated for their contributions via short talks, the presentation of homework and the performance in short supervised exercises. The number and the lengths of the short talks will depend on the number of participants and will be fixed in the beginning of the course

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

• Number of ECTS: 3
• Course code: MICS2-38
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

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

Introduction to optimisation and decision problems Linear programming, graphical interpretation and primal simplex Branch&Bound, A* The scheduling problem List algorithms, greedies, heuristics Meta-Heuristics and Evolutionary computation

Final Exam: 100%

• Number of ECTS: 5
• Course code: MAMATH-114
• Module(s): General Mathematics 3
• Language: FR, EN
• Mandatory: No

On successful completion of the Internship in a Company, the student should be able to: Identify employment opportunities in the local job market and possible career tracks; Apply insights gained during the internship into the functioning and the operational business of companies and institutions outside the academic world; Perform work in an institution outside the academic world at a level adequate to a Master student in mathematics; Apply and adapt the theoretical skills and techniques learned in the Master programme to the practical needs of the chosen company or institution.

The objective of the Internship in a Company is to allow the student to get acquainted with a professional work situation outside the academic context.The Internship is typically carried out in the summer break between the 1st and the 2nd year and counts for the 3rd semester. A typical duration is 2 months.It is the student’s responsibility to find a company. In order to establish a first contact, the student can (but need not) make use of an existing list of contacts at companies.After the student and a company agreed on the internship, the student contacts the Study Director and the Study Programme Administrator. The latter will proceed with the establishment of an internship convention between the University and the company, which will be signed by the Study Director, a representative of the company and the student. The model contract of the University need be used. It is accompanied by two Annexes: Annex 1 is an evaluation form that the company representative fills in after the internship; Annex 2 is a description of the planned tasks of the intern. Annex 2 has to be approved by the Study Director in order to ensure that the internship benefits the student in her/his studies.An academic supervisor, who is an academic employee at the University, is named. The academic supervisor is the contact point for both the student and the company during the internship and (s)he keeps an eye on the execution of the internship.Remark: By law, internships carried out in Luxembourg of a duration of at least 4 weeks have to be remunerated by the company. Internships can also be carried out in other countries, but then the student must make sure that (s)he has the legal right to work in that country.

The course is either succeeded or failed, but not marked.For succeeding, the student needs to get an overall positive evaluation by the company supervisor (Annexe 1 to the convention), and the student needs to hand in a report after the internship, which must be judged satisfactory by the academic supervisor and the SPD. The report needs to cover at least the following topics: Short description of the company and the work environment; Description of the tasks, objectives and the performed work (possibly with examples); How the student was integrated in the company/institution; An indication if the work performed by the student was successful with respect to the objectives; How the performed work fits in her/his Master studies; How the student liked it, if (s)he would recommend other students to choose this internship, possibly with recommendations.

Summer semester – April-June Student looks for a company; can ask Study Director (SPD) or Study Programme Administrator (SPA) for company contacts Student informs SPD and SPA about choice of company SPD finds an academic supervisor SPA prepares internship contract with company SPD checks Annex 2 Summer  Typical time for the internship (usually around 2 months) After internship Company representative fills in Annex 1 and sends it to the University Student writes her/his report and sends it to the SPD SPD gets feedback from academic supervisor SPD informs the SPA about the validation of the internship or its failure      

• Number of ECTS: 6
• Course code: MAMATH-176
• Module(s): General 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, algebraic sets, affine and projective varieties over algebraically closed fields. We will then introduce the modern language of sheaf theory and introduce schemes. We will then discuss divisors, line bundles and vector bundles on schemes.

Written exam

Literature Robin Hartshorne, Algebraic geometry Ravi Vakil, The rising sea Igor Shafarevitch, Basic algebraic geometry 1

• Number of ECTS: 5
• Course code: MAMATH-177
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

Basic introduction to path integral techniques and Monte-Carlo methods in Analysis and Finance.

Upon successful completion of the course students should be able to evaluate functionals of Brownian motion and relate them to PDE; manipulate Feynman-Kac formulas; derive stochastic representations of classical initial value problems; derive stochastic representations of classical boundary value problems; calculate Monte-Carlo formulas for Greek parameters in financial models

Stochastic flows associated to second order differential operators, stochastic differential equations and L-diffusions, Feynman-Kac formulas and Dirichlet problems, boundary value problems (elliptic and parabolic), spectral problems of Schrödinger operators, differentiation of heat semigroups, computation of price sensitivities (Greeks).

Written exam

Bass, Richard F. Diffusions and elliptic operators. Springer-Verlag,  1998. Bass, Richard F. Probabilistic techniques in analysis. Springer-Verlag,  1995. Durrett, Richard. Brownian motion and martingales in analysis. Wadsworth, 1984. Varadhan, S. R. S. Lectures on diffusion problems and partial differential equations. Tata Institute, 1980. Course Notes provided in class

• Number of ECTS: 5
• Course code: MAMATH-93
• Module(s): General 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 processes Identify, analyse, and prove relevant properties of models based on a Gaussian structure Solve 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).

Written exam

Literature Will be discussed in class

• Number of ECTS: 1
• Course code: MAMATH-179
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

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

• Number of ECTS: 5
• Course code: MAMATH-99
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

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

On successful completion of the course the student should be able to: Explain the mathematical foundation of advanced discretization techniques for PDEs Master their concrete implementation on nontrivial engineering boundary-value problems Adapt them according to the problem under consideration

Complements the Finite Element MethodFinite difference schemes in spaceFinite difference schemes fort he discretization of time-dependent PDEsIntroduction to integral equations

The students will have to provide some reports that will be evaluated. In addition, a final written examination will be organized.

Support / Arbeitsunterlagen / Support: Lecture notes (french), exercise sheets (english) Littérature / Literatur / Literature X. Antoine, Numerical solution of PDEs, lecture notes X. Antoine, Numerical Analysis, course at the University of Luxembourg G. Allaire, Analyse Numérique et Optimisation, Presses de l’Ecole Polytechnique P.A. Raviart et J.M. Thomas, Introduction à l’Analyse Numérique des Equations aux Dérivées Partielles, Dunod

• Number of ECTS: 5
• Course code: MA_DS-18
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

Upon completion a successful student should be able to Construct classical estimators of unknown probability distribution Derive asymptotic properties of empirical distribution functions Perform tests for unknown distribution functions Construct non-parametric estimators of the density Estimate components of a non-linear regression model Understand the concept of minimax estimation

Understanding the concepts of empirical distribution functions, kernel estimation and minimax theory Performing derivation of standard estimation and testing methods in nonparametric statistics

Estimation of probability measuresWeak limit theorems for empirical measuresKolmogorov-Smirnov and Cramer-von Mises testsEstimation of the densityNadaraja-Watson estimator and general kernel estimatorsNon-linear regression modelsMinimax theory

First session Solving bi-weekly exercises (40%) and a final exam (60%) will build the overall grade.Retake examWritten exam.

Literatur A.B. Tsybakov (2009): “Introduction to Nonparametric Estimation”, Springer. A. Van der Vaart and J. Wellner  (1996): “Weak Convergence and Empirical Processes”, Springer.

• Number of ECTS: 5
• Course code: MAMATH-80
• Module(s): General 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. 

50% of the final mark is done via a workshop (6 December), 50% from a final examThose who do not pass the course have the opportunity to retake the exam in the summer term.

Literature Elmasri, Navathe: Fundamentals of Database Systems. Pearson Addison Wesley. 2006. Han, Kamber: Data Mining – Concepts and Techniques. Morgan Kaufmann. 2011. Manning, Raghavan, Schütze: Introduction to Information Retrieval. Cambridge University Press.Ware: Information Visualization. Morgan Kaufmann. 2012. Witten, Kamber: Data Mining: Practical Machine Learning Tools and Techniques. Morgan Kaufmann. Aggarwal, Yu: 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.

• Number of ECTS: 5
• Course code: MAMATH-181
• Module(s): General Mathematics 4
• Language: EN
• Mandatory: No

In this course, the students will learn and present various subjects of higher mathematics. The students can be supervised by different staff members of the DMATH.

Continuous evaluation The students will be evaluated on their presentations.

• Number of ECTS: 5
• Course code: MAMATH-107
• Module(s): General Mathematics 4
• Language: EN
• Mandatory: No

General acquaintance with the theory of Siegel modular forms.

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

Oral exam

• Number of ECTS: 5
• Course code: MAMATH-149
• Module(s): General Mathematics 4
• Language: EN
• Mandatory: No

• Number of ECTS: 5
• Course code: MAMATH-138
• Module(s): General Mathematics 4
• Language: EN
• Mandatory: No

On successful completion of the course the student should be able to: Demonstrate proper understanding of the notion of complex manifold Identify the most important examples Master the main techniques of the theory Prove good knowledge of the key theorems Apply the new techniques to geometric problems Give a pedagogic talk for peers on a related topic Write clear and concise lecture notes, including appropriate exercises and applications  

The aim of the lecture course is to give an introduction to complex geometry. In particular the course will treat many examples. After the course the students should understand Hermitian vector bundles and the basics of Kahler manifolds and the Hodge decomposition. The students should be acquainted with the basic theorems and examples.

Oral examination (a presentation) and 1-2 homework sets

Literature R.O. Wells: Differen9al Analysis on Complex Manifolds (main text) S.S. Chern: Complex Manifolds without Potential Theory Griffiths & Harris: Principles of Algebraic Geometry F. Warner: Foundations of Differentiable Manifolds D. Huybrechts: Complex Geometry Lecture notes Martin SCHLICHENMAIER, available on the Moodle for download.

• Number of ECTS: 5
• Course code: MAMATH-114
• Module(s): General Mathematics 4
• Language: FR, EN
• Mandatory: No

• Number of ECTS: 5
• Course code: MICS2-17
• Module(s): General Mathematics 4
• Language:
• Mandatory: No

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

* Characterize problems* Identify the key concepts related to optimisation techniques* Use optimization frameworks* Implement optimization algorithms* Validate optimization algorithms and results* Validate approaches for solving optimization problems

This lecture confront the students to real instances of such problems. They are first asked to model the problem and next proposed solutions include exact methods, relaxations, approximations, heuristics and meta-heuristics. And these practical study cases are supported by the theoretical lectures on Problem Solving (1st semester)

Project: 100%

• Number of ECTS: 20
• Course code: MAMATH-30
• Module(s): Master Thesis
• Language: EN
• Mandatory: No