• Course title: Commutative Algebra
• 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 CombinedExam modality for the first sessionThe assessment method is combined assessment. The final mark consists of two parts:A mark for continuous assessment, counting 40% of the final markA mark for the end-of-course exam, counting 60% of the final mark.The mark for the continuous assessment is the arithmetic mean of the marks for three supervised exercises. In the case of a justified absence, a supervised exercise is replaced by an oral exam.Exam modality for the retake examOral exam

Note / Literature / BibliographyLecture notes, exercise sheets (available on Moodle)E. Kunz, Introduction to Commutative Algebra and Algebraic GeometryDino Lorenzini. An Invitation to Arithmetic Geometry, Graduate Studies in Mathematics, Volume 9, American Mathematical SocietyM. F. Atiyah, I. G. Macdonald. Introduction to Commutative Algebra, Addison-Wesley Publishing Company.

• Course title: Discrete-time stochastic processes
• 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 evolutionsKnow and apply classical martingale convergence theoremsDescribe 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.

Exam modalities for the first session Combined Assessment (40% Homeworks, 60% Written In-person Final Exam).Exam modalities for the retake examCombined Assessment (40% Homeworks, 60% Written In-person Final Exam).Absence PlanIn the case of justified absence in the final exam a replacement exam during the same semester will be offered.  

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

• Course title: Probability Theory and Mathematical Statistics I
• 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.

First session

Combined assessment (end of course assessment + continuous assessment)Retake examOral exam

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

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

Note / Literature / Bibliography Introductory Functional Analysis with Applications, by Erwin Kreyszig.Functional Analysis, Spectral Theory and Applications, by Manfred Einsiedler and Thomas Ward.An Introduction to Fourier Analysis, by Russell L. Herman

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

Maîtriser la conception de situations d’apprentissage.

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

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

Première session d’examen

L’évaluation du cours combinera contrôle continu (20%) et examen final (80%). Le contrôle continu évaluera la participation en classe et les travaux réguliers, réalisés individuellement ou en groupe. L’examen final comprendra une préparation écrite de leçons (60% de la note finale) et une soutenance orale (20%).

Évaluation lors d’une deuxième session (rattrapage)

L’évaluation reposera entièrement sur un examen final (100% de la note). Celui-ci consistera en une préparation écrite de leçons (75% de la note) suivie d’une soutenance orale (25%).  

Note / Literature / Bibliography

Astolfi, J.-P. (2024). L’erreur, un outil pour enseigner. Paris : ESF Éditeur.

Barzel, B., Büchter, A., & Leuders, T. (2007). Mathematik Methodik, Handbuch für die Sekundarstufe I und II. Cornelsen.

Büchter, A., & Leuders, T. (2005). Mathematikaufgaben selbst entwickeln, Lernen fördern – Leistung überprüfen. Cornelsen.

Hattie, J. (2010). Visible Learning. Routledge.

Jonnaert, P., & Vander Borght, C. (2003). Créer des conditions d’apprentissage. Bruxelles : De Boeck.

Leuders, T. (2020). Mathematikdidaktik (9e édition). Cornelsen.

Leuders, T., & Prediger, S. (2016). Flexibel differenzieren und fokussiert fördern im Mathematikunterricht. Cornelsen.

Malle, G. (1993). Didaktische Probleme der elementaren Algebra. Vieweg.

Meyer, H. (1989). Unterrichtsmethoden. I: Theorieband und II: Praxisband. Berlin: Cornelsen Scriptor.

Morissette, R. (2002). Accompagner la construction des savoirs. Montréal: Chenelière/McGraw-Hill.

Rieunier, A. (2018). Préparer un cours. Tome 1. Applications pratiques. Paris : ESF.

Rieunier, A. (2012). Préparer un cours. Tome 2. Les stratégies pédagogiques efficaces. Paris : ESF.

Sill, H.-D. (2018). Grundkurs Mathematikdidaktik. UTB.

Sturm, R. (2016). Schritt für Schritt zum guten Mathematikunterricht. Klett Kallmeyer.

• Course title: Riemannian Geometry
• 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)

• Course title: Stochastic Analysis
• 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, 1991B. Oksendal:  Stochastic differential equations. Springer, 2003D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. Springer Grundl., 1999

• Course title: Partial Differential Equations I
• 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

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

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

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

This course covers the basics of scientific programming with Python. It is aimed at people who have done some programming before, perhaps on an undergraduate course, but need a refresher before starting their Masters or Doctoral degrees at the University.https://jhale.github.io/scientific-python/ 

A coursework will be distributed at the end of the class. To pass the course and receive the ECTS credits you must complete the coursework.

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

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

Through a presentation of selected topics, the course aims to be an introduction to graph theory, its applications and its algorithmic aspects. It is designed as a self-contained course and focused on problems pertaining to Data Science. Possible topics for the course include, but are not limited toGraphs and digraphs, degree and the degree sequence algorithmConnectedness, distance, shortest paths and connected components algorithmsGraph matching problems and algorithmsElements of algebraic graph theory and PageRank algorithmGraph traversal algorithmsTrees and applicationsMinimum spanning tree algorithmsNetwork flow, min cut – max flow theorem and Ford–Fulkerson algorithmCentrality and betweness measuresCluster analysisRandom Graphs

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

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

• Course title: Algorithmic Number Theory
• 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.

Exam modalities for the first session

Written exam during the exam period (January/February)

Exam modalities for the retake exam

Written exam during the exam period (July)

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

 

• Course title: Numerical Analysis
• 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 ableto understand the main methods, the rigorous mathematical analysis and algorithms developed in numerical analysisto master their concrete implementation on a computerto 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 LuxembourgAscher, 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.

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

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

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

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

• Course title: Student project 1
• 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 taskspropose solution strategies,break up a longer project into subsequent steps,apply a variety of methods in one project,present a task and its solution in a scientific way.

The student project consists of project work that is carried out under the supervision of a professor or a postdoc. The work is either individual or group work. Group work needs the explicit approval of the Study Director.At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS). The student(s) need to notify the Study Director of the project and the tasks at the latest on 15 October.The project outcome is a pdf document written by the student. Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December.

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

Note / Literature / BibliographyDepends on the project.

• Course title: Student seminar 1
• Number of ECTS: 1
• Course code: F1_MA_MAT_GM-8
• Module(s): 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 workDeliver a mathematical lecture on a topic of their choiceShare their knowledge with others

Every participant chooses a supervisor among the academic staff of the Department of Mathematics or among the instructors of the Master in Mathematics and, jointly with the supervisor, a topic for a talk.The audience of the talk consists at least of the participants of the Student Seminar 1, the supervisor and one other academic staff member of the Department of Mathematics (e.g. the supervisor of another talk). The duration of the talk is 40 minutes (time for questions included).

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

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

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

On successful completion of the course, the student should be able to:Explain the main definitions and results of Algebraic Topology;Comment on new concepts;Apply the new techniques and solve problems;Structure the acquired abilities and summarize essential aspects adopting a higher standpoint;Give a talk for peers or students on a related topic and write scientific texts or lecture notes, observing modern standards in scientific writing, in Didactics and in Pedagogy;Provide evidence for the mastery of the Mathematical Method. 

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

Written exam 

LiteraturAllen Hatcher, Algebraic Topology, Cambridge University Press, 2002 – Mathematics – 544 pages, ISBN-13: 978-0521795401 Peter May, A concise course in Algebraic Topology, University of Chicago Press, 1999 – Mathematics – 243 pages, ISBN-13: 978-0226511832Charles Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics 38, 1997, ISBN: 0-521-55987-1Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5(2), Springer, 1978, ISBN: 978-0-387-98403-2

• Course title: Continuous Time Models in Mathematical Finance
• Number of ECTS: 8
• Course code: F1_MA_MAT_FM-5
• Module(s): General Mathematics 2
• Language: EN
• Mandatory: No

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

Arbitrage, risk-neutral measures, option pricing, hedging, Black-Scholes-Merton equation, call-put parity, connections with partial differential equations, forwards and futures, american options, exotic options, change of numéraire, Garman-Kohlhagen formula, term-structure models, Vasicek model, Heath-Jarrow-Morton model, forward LIBOR model.

Written exam

Literatur

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. 2nd ed., 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 

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

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

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

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

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

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

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

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

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

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

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

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

 

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

Quizzes (20%)

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

Literature

Class notes will be provided. However, students may benefit from consulting various sources on Algebraic Number Theory. References with relevant material for this course include:

Jürgen Neukirch, Algebraic Number Theory, Springer Berlin, Heidelberg, 1999

Serge Lang, Algebraic Number Theory, Springer New York, NY, 2013

Pierre Samuel, Algebraic Theory of Numbers, Dover, 2008,

Michael Rosen, Number Theory in Function Fields, Springer New York, NY, 2013

Henning Stichtenoth, Algebraic Function Fields and Codes, Springer Berlin, Heidelberg, 2009

Sara Arias-de-Reyna, Gabor Wiese, Algebraic Number Theory, Online notes

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

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

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

Written exam

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

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

 

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

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

Oral exam or written exam according to number of participants

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

• Course title: Didactical Seminar I
• Number of ECTS: 2
• Course code: F1_SECEDUC-29
• Module(s): General Mathematics 2
• Language: FR, EN, DE
• Mandatory: No

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

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

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

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

• Course title: Learning and Teaching Mathematics I
• Number of ECTS: 3
• Course code: F1_SECEDUC-30
• Module(s): General Mathematics 2
• Language: FR, EN, DE
• Mandatory: No

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

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

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

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

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

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

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

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

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

Literatur

Rick Miranda, Algebraic Curves and Riemann Surfaces.

Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces.

Oleksandr Iena, Riemann surfaces. Lecture Notes. Winter Semester 2015/2016.

The above texts are available on a-z.lu. More references will be provided during the course.

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

On successful completion of the Student Project 2, the student should be able to:

analyse complex tasks,

propose solution strategies,

break up a longer project into subsequent steps,

apply a variety of methods in one project,

present a task and its solution in a scientific way.

The student project consists of project work that is carried out under the supervision of a professor or a postdoc. The work is either individual or group work. Group work needs the explicit approval of the Study Director. At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS). The student(s) need to notify the Study Director of the project and the tasks at the latest on 15 March. The project outcome is a pdf document written by the student. Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 May. The project file will undergo an automatic plagiarism check.

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

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

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

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

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

• Course title: Lie Algebras and Lie Groups
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-20
• 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.

First sessionEnd of course assessment

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

• Course title: Applied Didactics IV – Evaluation methods and supporting mathematics learning
• Number of ECTS: 3
• Course code: F1_SECEDUC-19
• Module(s): General Mathematics 3
• Language: FR
• Mandatory: No

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

Avoir conscience que chaque choix didactique influence toujours l’enseignement et l’apprentissage des contenus qui en découlent. Être en mesure de formuler des questions ciblant différents niveaux taxonomiques. Anticiper les éventuels problèmes de compréhension des élèves. Identifier les moments clés d’une leçon ou d’un apprentissage. Planifier une séquence de leçons comprenant une introduction, des activités, des transitions, des moments consacrés aux élèves, des exemples, des exercices, ainsi que les questions à poser et les réponses attendues des élèves. Formuler des questions adaptées à une évaluation de type formatif ou sommatif. Planifier une leçon de manière à ce que les objectifs d’apprentissage puissent être atteints de plusieurs manières différentes.

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

First sessionExam modalities for the first session: Combined assessment (contrôle continue et examen final sous forme de remise d’un dossier avec présentation et défense orale du dossier)Session de rattrapageExam modalities for the retake exam: remise d’un dossierAbsence plan pour les contrôles continus/ absences lors d’un cours: me contacter au plus tard avant le début du cours pour la remise du dossier: une prolongation peut être obtenue sous condition de pouvoir fournir un certificat médical pour la défense orale: une date alternative peut être trouvée sous condition de pouvoir fournir un certificat médical.

NoteBarzel, B., Büchter, A., Leuders, T. (2007), Mathematik Methodik, Handbuch für die Sekundarstufe I und II, CornelsenHattie, J. (2010), Visible Learning, RoutledgeLeuders, T. (2020), Mathematikdidaktik (9e édition), CornelsenLeuders, T., Prediger, S. (2016), Flexibel differenzieren und fokussiert fördern im Mathematikunterricht, CornelsenMalle, G. (1993), Didaktische Probleme der elementaren Algebra, ViewegPadberg, F., Wartha, S. (2017), Didaktik der Bruchrechnung (5e édition), SpringerSill, H.-D. (2018), Grundkurs Mathematikdidaktik, utb.Sturm, R. (2016), Schritt für Schritt zum guten Mathematikunterricht, Klett Kallmeyer

• Course title: Combinatorial Geometry
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-21
• 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 take the form of a topics course, presenting a selection of themes from combinatorial aspects of geometry. Topics may include cube complexes, convex sets in R^n, combinatorial ways of exploring surfaces, topics in graph theory…

First session3 short in-class tests, worth together a total of 20%; final written exam worth 80%.Retake exam100% final written exam, no kept grade.Absence plan: the final grade will be worked out as follows:Case 1: Student attends all three short tests. The best two out of the three scores will count for 10% each, and the final exam will count for 80%.Case 2: Student attends two short tests. The two scores will count for 10% each and the final exam will count for 80%.Case 3: Student attends one short test. The test score will count for 10% and the final exam will count for 90%.Case 4: Student attends no short test. The final exam will count for 100%. 

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

• Course title: Arithmetic Geometry
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-22
• Module(s): 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.

Exam modality for the first sessionThe assessment method is continuous assessment. The continuous assessment consists of two parts, each contributing 50% to the final mark:Talks by participantsPresentation of exercises by participantsExam modality for the retake examOral exam 

Note / LiteratureSerre: A course in arithmetic, Springer Silverman: The arithmetic of elliptic curves, SpringerAnni, Deo, Wiese: Lecture notes for Topics in Number Theory and Geometry, distributed during the lecture

• Course title: Algebraic Geometry
• Number of ECTS: 6
• Course code: F1_MA_MAT_GM-23
• Module(s): 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, affine and projective varieties. We can think of an affine variety as the common zeros of a set of given polynomials in multiple variables. We will further introduce rational maps between varieties leading to the important topic of birational geometry. Finally, we will learn what a singular point of an affine variety is. Along the way, we will look at many simple and concrete examples. 

First session

Written exam and supervised homework

Retake exam

 

Note / Literature / Bibliography 

Robin Hartshorne, Algebraic geometry

Ravi Vakil, The rising sea

Igor Shafarevitch, Basic algebraic geometry 1

• Course title: Gaussian processes and applications
• Number of ECTS: 5
• Course code: F1_MA_MAT_FM-9
• 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 processesIdentify, analyse, and prove relevant properties of models based on a Gaussian structureSolve exercises involving a Gaussian structure

Gaussian random variables (characteristic function, CLT, stability properties, Stein’s lemma)Gaussian random vectors (definition, characteristic function, existence, uniqueness in law, multivariate CLT, density, Hermite polynomials)Gaussian random processes (definition, modifications, uniqueness in law, function of positive type, existence, Brownian motion, continuity)Fractional Brownian motion (definition, existence, Hölder regularity)

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

Note / Literature / BibliographyWill be discussed in class

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

• Course title: Data Science
• Number of ECTS: 5
• Course code: F1_MA_MAT_GM-24
• 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.  

First session50% written assignment or oral interview + 50% practical. Retake examThose who do not pass the course have the opportunity to retake the exam in the summer term.

Note / Literature / Bibliography Elmasri, Navathe: Fundamentals of Database Systems. Pearson Addison Wesley. 2006.Han, Kamber: Data Mining – Concepts and Techniques. Morgan Kaufmann. 2011.Manning, Raghavan, Schütze: Introduction to Information Retrieval. Cambridge University Press.Ware: Information Visualization. Morgan Kaufmann. 2012.Witten, Kamber: Data Mining: Practical Machine Learning Tools and Techniques. Morgan Kaufmann.Aggarwal, Yu: Privacy-Preserving Data Mining – Models and Algorithms. Springer. 2008.Marz: Big Data: Principles and best practices of scalable realtime data systems. Manning. 2015.as well as different articles, reports, and journals contributions.

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

On successful completion of the Student Project 1 (or 3) the student should be able to:

analyse complex tasks,

propose solution strategies,

break up a longer project into subsequent steps,

apply a variety of methods in one project,

present a task and its solution in a scientific way.

The ability to work on a project, either individually or as part of a team, is one of the key skills expected of a mathematician, in and outside academia. The Student Project 1 offers the student the opportunity to improve this skill by allowing them to work on a well-defined and well chosen project, either individually or as a team, under the guidance of a researcher.The Student Project 3 consists of project work that is carried out under the supervision of a researcher.The work is either individual or group work. At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS) for each student. The student(s) need to submit the planned topic to the course responsible for approval at the latest on 15 October. The project outcome is a pdf document written by the student(s).Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December. The project file will undergo an automatic plagiarism and AI check.Timeline for the studentBeginning of semesterConnect to the Moodle classroomFind a supervisor and a topicOn or before 15 October, get approval from the course responsible of the project participants, the supervisor and the topicDuring the semesterWork on the projectSeek feedback from your supervisorEnd of the semesterHand in your project on or before 31 December on Moodle (for plagiarism and AI check)  

First session

The students are marked for their project work. In the case of a retake exam, a new project on the same or a different topic has to be done. 

Retake exam

The retake can be done with a different supervisor.

• Course title: Internship (in a company)
• Number of ECTS: 5
• Course code: F1_MA_MAT_MMCS2-13
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

• Course title: Nonparametric Statistics
• 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 toConstruct classical estimators of unknown probability distributionDerive asymptotic properties of empirical distribution functionsPerform tests for unknown distribution functionsConstruct non-parametric estimators of the densityEstimate components of a non-linear regression modelUnderstand the concept of minimax estimation

Understanding the concepts of empirical distribution functions, kernel estimation and minimax theoryPerforming 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 sessionSolving bi-weekly exercises (40%) and a final exam (60%) will build the overall grade.Retake examWritten exam.

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

• Course title: Bayesian Statistics
• Number of ECTS: 5
• Course code: MA_DS-22
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

Provide students with the necessary knowledge of Bayesian statistics, both from a theoretical as well as practical viewpoint. 

Handle of the concepts from Bayesian statistics, Bayesian inference, Bayesian modelling, practical handle of Bayesian techniques on Python.

This course will provide a thorough introduction and overview on the most important concepts from Bayesian inference, starting with the Bayesian philosophy in contrast to frequentist statistics. We will discuss various choices of prior distributions, Bayesian inference, Bayesian modelling, model checking and comparison (in particular introducing the concept of Bayes factor), and advanced computation. The latter aspect will especially be dealt with in the practicals, where Markov Chain Monte Carlo methods shall be treated.

First sessionWritten exam with open questions, both theory and exercises, in particular there will be computational exercisesRetake examWritten exam with open questions, both theory and exercises, in particular there will be computational exercises

Note / Literature / Bibliography Box, GEP and Tiao, GC (1992) Bayesian Inference in Statistical Analysis. John Wiley and SonsGelman, A, Carlin, JB, Stern, HS, Dunson, DB, Vehtari, A and Rubin, DB (2013) Bayesian Data Analysis 3rd edition. Chapman & Hall/CRC.Brooks, S, Gelman, A, Jones, GL, Meng, X-L (2011) Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC

• Course title: Continuous-Time Stochastic Calculus and Interest Rate Models
• Number of ECTS: 5
• Course code: F1_MA_MAT_FM-6
• Module(s): General Mathematics 3
• Language: EN
• Mandatory: No

On successful completion of the course, the student should be able to:Calculate probabilities and expectations related to the semi-martingale models presented in the lecturesCarry out calculations based on change of numéraire and no-arbitrage pricingCompute the prices of interest rate derivativesApply stochastic volatility models to deal with implied volatility surfaces

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

First sessionWritten examRetake examWritten exam

Note / Literature / Bibliography D. Brigo and F. Mercurio (2006) Interest Rate Models : Theory and Practice. Springer Verlag Damir Filipovic (2009). Term-structure models. A graduate course. Springer Verlag.

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

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

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

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

• Course title: Master Thesis
• Number of ECTS: 20
• Course code: F1_MA_MAT_GM-29
• Module(s): Master Thesis
• Language: FR
• Mandatory: Yes

• Course title: Reading Course
• Number of ECTS: 5
• Course code: F1_MA_MAT_GM-30
• Module(s): General Mathematics 4
• Language: EN
• Mandatory: No

This is an individual reading course about the background and complementary knowledge for a Master thesis.

The assessment is based on the performance in the meetings with the instructor(s).

• Course title: Extended Reading Course
• Number of ECTS: 10
• Course code: F1_MA_MAT_GM-31
• Module(s): General Mathematics 4
• Language: EN
• Mandatory: No

This is an individual extended reading course about the background and complementary knowledge for a Master thesis.

The assessment is based on the performance in the meetings with the instructor(s).

• Course title: Complex Geometry
• Number of ECTS: 5
• Course code: F1_MA_MAT_GM-28
• 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.

First session

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.

• Course title: Optimisation for Computer Science
• 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%

• Course title: Didactical Seminar I
• Number of ECTS: 2
• Course code: F1_SECEDUC-29
• Module(s): General Mathematics 4
• Language: FR, EN, DE
• Mandatory: No

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

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

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

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

• Course title: Learning and Teaching Mathematics I
• Number of ECTS: 3
• Course code: F1_SECEDUC-30
• Module(s): General Mathematics 4
• Language: FR, EN, DE
• Mandatory: No

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

À l’issue de ce cours, les étudiants devront être capables de :

concevoir des situations d’apprentissage en tenant compte des connaissances préalables, des conditions d’apprentissage et, si possible, des centres d’intérêt et motivations des élèves afin de les amener à évoluer vers des objectifs d’apprentissage définis ;

concevoir des exercices favorisant l’activation cognitive et les adapter au niveau d’apprentissage des élèves tout en initiant les élèves à différentes stratégies d’apprentissage ;

concevoir des situations d’apprentissage qui conduisent à une réflexion approfondie sur les contenus et poser des questions qui permettent de pousser les élèves à une réflexion approfondie sur les contenus. 

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

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

Note / Literatur

La liste des références bibliographiques sera publiée sur la page Moodle de ce cours.