Content

- Different types of model families (such as the linear model, generalised linear model, non-linear model).  Estimation and inference inside such models.
- Application of model selection methods (forward, backward selection, Cp, AIC, BIC,  FIC, …)
- Aspects of goodness-of-fit testing (Neyman smooth test, order selection test, …)
- Models with random effects (in linear, non-linear, generalised linear models)
- Smoothing methods (penalized spline estimators in additive models, partially linear models)
- Models and methods for high-dimensional data (lasso, penalisation approaches, shrinkage estimation, …)

Course material

Used course materials:  "Statistical Modelling" (G. Claeskens, Acco course notes).

Possible extra material such as data sets will be made available via Toledo.

Format: more information

The student is expected to actively participate to the course. It is expected that the student learns the methodology and applies it using the statistical software package R to, for example, obtain estimators, conduct inference, construct graphs. Students are expected to solve exercises.

Explanation

FEATURES OF THE EVALUATION

The exam of 'Statistical Modelling' consists of two parts: the first part is a take-home project, which contains practical or theoretical questions and might require data analysis (for which the software package R is used) related to the course topics. The second part is a written exam during the exam period.

The deadline for the take-home project will be determined by the lecturer and communicated via TOLEDO.The set deadline is strict and cannot be changed. For the take-home part of the exam a computer and the course notes may be used. The take-home project cannot be retaken for the third exam period. For the on campus written exam during the exam period a calculator may be used, but no computer and no course notes.

DETERMINATION OF FINAL GRADES

The grades are determined by the lecturer as communicated via TOLEDO and stated in the examination schedule. The result is calculated and communicated as a number on a scale of 20.
The take-home project counts for 5 points (out of 20) of the final grade. The on campus written exam counts for 15 points (out of 20) of the final grade. The take-home project is part of the final exam and its grade will not be communicated separately.

If the student does not participate in the take-home project or when the set deadline was not respected, the final grade of the course will be NA (not taken) for the whole course.

Information about retaking exams

The grade obtained for the take-home project carries over to the third exam period.The take-home project cannot be retaken in the third exam period. The set deadline cannot be changed.

The features of the evaluation and determination of grades for the written exam are similar to those of the first examination opportunity, as described above.

The result is calculated and communicated as a number on a scale of 20.
The take-home project counts for 5 points (out of 20) of the final grade. The on campus written exam counts for 15 points (out of 20) of the final grade.

If the student does not participate in the take-home project or when the set deadline was not respected, the final grade of the course will be NA (not taken) for the whole course in the third examination period.

Content

• storage and processing of large datasets
• big data platforms such as Hadoop and Spark
• streaming and unstructured data techniques
• using predictive models in a big data context
• working with graph data

Course material

Lecture slides, additional background reading

Content

• data science process
• supervised and unsupervised methods
• anomaly detection
• ensemble methods
• data science tools
• deep learning
• text mining
• graph analytics

Course material

Lecture slides, additional background reading

Explanation

FEATURES OF THE EVALUATION

• The evaluation consists of a discussion paper (50% of the marks) and a closed-book written exam with both multiple-choice and open questions (50% of the marks).
• If the student does not participate in one of the partial evaluations, the final grade of the course will be NA (not attended) for the whole course.

DETERMINATION OF FINAL GRADE

• The grades are determined by the lecturer as communicated via Toledo and stated in the examination schedule. The result is calculated and communicated as a whole number on a scale of 20.

SECOND EXAMINATION OPPORTUNITY

• The features of the evaluation and determination of grades are identical to those of the first examination opportunity, as described above.
• At the second examination opportunity, assignments are no longer part of the evaluation.

Information about retaking exams

The features of the evaluation and determination of grades are identical to those of the first examination opportunity, as described in the tab 'Explanation'.

Inhoud

Totale belasting van dit opo bedraagt gemiddeld 75 uur.

Academische component:

Tijdens een introductie wordt een brainstorm gehouden over de relatie tussen het service learning project en een opleiding Wetenschappen. Tijdens het terugkommoment wordt deze relatie duidelijker geëxpliciteerd aan de hand van de uitwisseling van de persoonlijke ervaringen van de studenten. Door deelname aan het service learning project zal de student het belang van bepaalde theoretische aspecten die in de opleiding aan bod komen, bijvoorbeeld rond duurzaamheid, beter begrijpen door de verankering ervan in de dagelijkse praktijk zelf vast te stellen.

• De student krijgt tijdens de introductie inleidend inzicht in theoretische kaders omtrent technologie en maatschappij, duurzaamheid en kwetsbaarheid algemeen (vanuit interdisciplinair perspectief).
• Tijdens de introductie wordt de essentie van ‘reflectie’ onderwezen en geoefend. Een praktijkdagboek wordt opgestart.

De studenten krijgen ter voorbereiding op het terugkommoment een tekst te lezen en integreren deze tijdens de dialoog van het terugkommoment. Deze tekst handelt over bepaalde visies op wetenschap en maatschappij die oriënterend kunnen zijn voor de keuzes die gemaakt worden, zowel op maatschappelijk vlak als op individueel vlak wat de inzet en het engagement van de wetenschapper betreft (honest broker, ethiek, mensbeeld, human scale development).

Praktijkcomponent:

• Kennisname van bestaande organisaties en initiatieven in het veld, gericht op de doelgroepen.
• (Passieve) observatie ter inleving in de situatie.
• Actieve, dienstverlenende participatie in de door de student gekozen organisatie, gericht op de met de organisatie afgesproken doelen.

Reflectiecomponent:

• De student dient vooraleer het ISP kan worden goedgekeurd, een voorstel van project in bij het docententeam waarbij ook de concrete stageplanning is uitgewerkt (periode, organisatie, tentatieve dienstverlenende doelen en gedetailleerde belasting).
• Gedurende de activiteiten houdt de student een dagboek bij in het ePF om concrete ervaringen te noteren.
• Eerste Reflectie in het ePF in samenspraak met de stagebegeleider, via terugkoppeling vanuit observatie naar de leeractiviteiten die zullen nodig zijn om de stage-doelen van de student en de organisatie te realiseren – Deze reflectie krijgt vormende feedback van het docententeam
• Tweede Reflectie in het ePF: Individuele reflectie via terugkoppeling vanuit de actieve stage naar de theoretische kaders - Deze reflectie krijgt vormende feedback van het docententeam.
• Terugkommoment - Afsluitende reflectie (deel van eindevaluatie): 10’ presentatie en verdere dialoog met het docententeam, de lokale begeleider en medestudenten over wat de student op welke vlakken heeft ervaren en geleerd, integratie van de aangeleverde visietekst, explicitering van de link tussen de opleiding en service learning.

Toelichting

De evaluatie gebeurt door het docententeam op basis van een gesprek (presentatie) en het ePF dat de student samenstelt. Dit ePF brengt volgende elementen naar voor:

-het maatschappelijk engagement van de student (dagboek) en de beoordeling door de stagebegeleider in de partnerorganisatie en de begeleidende docent (procesevaluatie - beoordeling omvat volgende criteria: aanwezigheid, tijdigheid, inzet, respectvolle houding, waardevolle inbreng, heldere communicatie)

-de kwaliteit van reflecties en verslagen (individueel – verslag en self-assessment)

Bepaling van het eindresultaat

Het opleidingsonderdeel wordt beoordeeld door het docententeam met inbreng van de partnerorganisatie, zoals meegedeeld via Toledo.

EEen negatieve beoordeling voor de praktijkcomponent resulteert automatisch in een fail voor het hele opo.

Het resultaat wordt bekendgemaakt als een pass/fail.
 

Toelichting bij herkansen

Het ePF kan herwerkt worden om kwaliteitsvoller de gevraagde elementen te illustreren.  Na een negatief oordeel voor de praktijkcomponent is geen herkansing mogelijk.

Content

In this course an overview of the generalized linear model is presented as the unifying framework for many commonly used statistical models. The emphasis is on the methods in categorical data analysis, model building and interpretation. We start with a recapitulation on previous knowledge on statistical inference and linear regression. The generalized linear model framework is presented.  Examples are the Poisson model for counts and the logitic model for binary outcome variables. Overdispersion is discussed in the Poisson and logistic model. Topics like quasi-likelihood, complete and semi-separation, the information-theoretic approach to model selection are discussed. The methods are illustrated with many practical examples in R.

Is also included in other courses

G0A18B : Generalized Linear Models

Explanation

• The students will have to do a project divided in tutorial groups. Each tutorial group has to write a report with a detailed discussion of the analysis. A report should contain no more than 3 sheets in total, including the title page, i.e., if you print your report it should not exceed 3 sheets two sided. Every page should only have one column of text.  Please use an A4 page format, a times new roman 12 font and a 1.2 spacing between lines. The title page of the report should contain the number of the group and a list with the names and student numbers of all the members of the group. Please note that these reports are part of your evaluation, thus follow these instructions very carefully. The project will account for 5 points and the exam for 15 points.
• Students enrolled in the on-campus program will have a written multiple-choice exam worth 15 out of 20 points. This exam will include a correction for guessing. If students cannot attend the regular exam for a justified reason, they will be given an oral exam at a time specified by the professor.
• The project is a compulsory task and an integral part of the exam. Students who do not complete or contribute to the project with their group will receive an NA for both the regular and second-chance exams. If a group does not complete the project task or fails to submit it on time, all members of the group will receive an NA for both the regular and second-chance exams.

Information about retaking exams

If a student has to do a second chance exam then the grade of the project will still be considered for the final grade in the second chance exam as well.

Content

The course will treat fundamentals, basic properties and use of modern nonparametric techniques:
Kernel estimators, local polynomial estimators, penalized likelihood techniques, spline approximations and spline smoothing, orthogonal series and wavelet techniques, among others.
These so-called smoothing techniques are applied in a variety of application areas in medicine (e.g. in nonparametric estimation of the hazard or survival function), in engineering (kernel estimators, neural networks, classification and pattern recognition, unsupervised learning, image analysis,...), in econometrics and economics (e.g. nonparametric estimation of a trend or volatility, moving averages), in social sciences (e.g. non- and semiparametric models to describe heterogeneity).
 
A table of content for the course can read as follows:
1. overview of nonparametric methods for estimating a density: kernel estimation methods, nearest-neighbour methods, maximum-likelihood-based methods, orthogonal series method, wavelets, ...
2. kernel estimators of densities: basic properties (bias, variance, mean squared error), asymptotic properties, asymptotic normality, rates of convergence (and their meaning/interpretation), selection of smoothing parameters (via cross-validation, plug-in, bootstrap or resampling procedures, ...).
3. nonparametric estimation of a regression function: the  cases of fixed and random design, homoscedasticity and heteroscedasticity, Nadaraya-Watson estimator, Gasser-Müller estimator, weighted least-squares methods, local polynomial fitting, splines, P-splines, wavelets, ... The impact and choices of parameters in each of these techniques will be discussed.
4. nonparametric estimation of hazard functions and applications (e.g. in survival analysis).
5. multivariate regression models: additive modelling and backfitting algorithms, dimension reduction techniques.
6. nonparametric smoothing and deconvolution problems (e.g. measurement errors).
7. nonparametric estimation of boundaries and frontiers with applications in image analysis and econometrics (for example).
8. modelling dependencies and nonparametric techniques, for example, use of nonparametric techniques in time series context.
9. other applications of nonparametric techniques: classification techniques, neural networks, statistical learning and data mining, modelling dependencies, ....
 
Parts 1---3 are basic items and will  be treated each year.  A further selection of minimal 2 items from items 4—9 will be made and this selection can possibly alter from year to year.

Format: more information

The contents of the three basic items will be presented to the students. A selection  from the set Items 4---9 will be covered.
Since this course is preparing the students to a research-oriented direction, it is also required that the students get acquainted to the literature in this domain.  As such each student will be asked to give a presentation (seminar) during the semester. The topic of this presentation must be linked to the use of nonparametric methods (discussed in the course) in an specific problem or area of application. The topic, possibly proposed by the student, has to be discussed and approuved by the instructor. 
Students will also be asked to use available statistical software on modern nonparametric techniques (software available as packages of statistical software, e.g. R) to get acquainted with the discussed methods.  A possibility is to do this as part of the presentation.

Content

- Basic Concepts from algebra and topology: finitely generated algebras, dimension theory of rings and topological spaces, Zariski topology, relation with quotients and localization, algebraic versus topological dimension, basic examples.

- Affine varieties: algebraic sets, morphisms of algebraic sets, Hilbert's Nullstellensatz, dimension of an affine variety.

- Quasi-projective varieties: raded rings, projective, quasi-projective and quasi-affine varieties, morphisms and regular functions, rational maps, blowing-ups, the local ring of a variety at a point, singular and regular points.

- Intersection theory in projective spaces: Bézout theorem and generalizations, Bertini theorem.

Course material

Course notes + Toledo

Format: more information

Lectures with assignments during the lecture.

Content

- Basic Concepts from algebra and topology: finitely generated algebras, dimension theory of rings and topological spaces, Zariski topology, relation with quotients and localization, algebraic versus topological dimension, basic examples.

- Affine varieties: algebraic sets, morphisms of algebraic sets, Hilbert's Nullstellensatz, dimension of an affine variety.

- Quasi-projective varieties: graded rings, projective, quasi-projective and quasi-affine varieties, morphisms and regular functions, rational maps, blowing-ups, the local ring of a variety at a point, singular and regular points.

- Intersection theory in projective spaces: Bézout theorem and generalizations, Bertini theorem.

Course material

Course notes + Toledo

Explanation

There will be one take-home exam during the semester.

The final exam is also take-home and consists either of classical exam questions or of submission of a short expository paper on a topic of own choice related to the course and agreed upon by the instructor. This paper has to contain, beside some clear introductory theory, non-trivial explicit examples, agreed upon by the instructor, worked out to illustrate the theory.

In order to pass, the student must obtain at least the score 10/20. The take-home exam during the semester will count 5 points, the final exam will count 15 points. If the student has failed to pass, for the second-chance examination no points will be carried forward from the take-home exam or the final exam. The student will be given the chance to pass the course via, again, a package consisting of a new take-home exam and a new final exam, with the same format and score share.

Content

- Modules over general rings
- Free modules, projective and injective modules, torsion
- Noetherian rings and modules
- Modules over Principal Ideal Domains and applications in advanced linear algebra
- Rings and modules of fractions, localizations
- Tensor product
- Exact sequences
- Introduction to categories and functors

Content

See G0A82a.

Format: more information

Throughout the semester, there will be several take home assignments, for which you will have to hand in an individual report.
Each of these will consist of one or two broad exercises and will be marked.
 

Explanation

Throughout the semester, there will be several take home assignments, for which you will have to hand in an individual report. Each of these will consist of one or two broad exercises and will be marked.  This results in a mark H for the homework assignments out of 20 points.
The actual exam will consist of theory and exercises. You can use all the material from the course and the exercise sessions during the exam. At least one exam question will build on the material which appeared in the take home assignments. This results in a mark E for the exam out of 20 points.
Your final score will be max{E, (3E + H)/4}.

Content

In this course, we treat two basic algebraic invariants of a topological space and show some applications.

The fundamental group of a topological space.
• Homotopy and the definition of the fundamental group
• Retractions and deformation retracts
• Applications to fixed points (Brouwer fixed point theorem)
• The Seifert – van Kampen Theorem
• Borsuk – Ulam Theorem
• Covering spaces and the connection with the fundamental group:
  - lifting of paths and maps
  - equivalence of covering spaces
  - covering transformations and group actions

The singular homology groups of a topological space
• definition of the singular homology groups Hn(X) of a space
• meaning of H0(X) and H1(X)
• induced morphisms
• The Mayer – Vietoris exact sequence in homology
• Applications towards spheres (degree of a map, vector fields on spheres)

Course material

Study cost: 1-10 euros (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)

For part I on the fundamental group, the book Topology of James R. Munkres (Pearson Education International) is used. Copies of this book are available through the students association

Part II on singular homology a text is made available via Toledo and copies can aslo be obtained through the students association

Explanation

The exam is an open book exam.

Content

Introduction to representation theory of finite groups:

- Definition and examples
- Irreducible representations and complete reducibility
- Lemma of Schur
- Characters: definition and properties

Nilpotent, Solvable and Polycyclic Groups

• Nilpotent groups:
- Upper and lower central sequence
- Definition of nilpotent group and examples
- Finite nilpotent groups
- Finitely generated (torsion free) nilpotent groupsSolvable groups

• Polycyclic and polycyclic-by-finite groups:
- Definition of poly-P and P-by-Q groups
- Nilpotent groups are polycyclic
- Hirsch length
- The max-condition
- Fitting subgroup

An introduction to homological algebra

• Homological algebra
- Exact sequences
- (Co)chain complexes and their (co)homology
- Ext and Tor

• Application: an introduction to the cohomology of groups
- Extn and the definition of the cohomology of groups.
- Fixed points and the zeroth cohomology group
- Semi-direct product and the first cohomology group
Example: the group of isometries of an Euclidian space
- Group extensions and the second cohomology group,
with an application to crystallographic groups.

Course material

Study cost: 1-10 euros (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)

Course notes are available via the students association 

Content

See G0A85a.

Course material

Study cost: Not applicable (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)

Course notes + Toledo

Explanation

The exam is an open book exam.

Content

- Cultural background: history of Fermat's Last Theorem, Fermat's proof of the case n = 4, Lamé's erroneous proof

- Update on commutative algebra: norms, traces, discriminants, Dedekind domains, unique ideal factorization, class groups and class numbers

- Number fields and rings of integers: quadratic numbers fields, integral bases, ramification indices and degrees, norms of ideals

- Geometric representation: Minkowski's lemma + applications, geometric representations, logarithmic representations 

- Finiteness theorems: finiteness of the class number, Dirichlet's unit theorem, Dedekind's theorem on ramification, Hermite's theorem 

- Connections with Galois theory: Frobenius elements, decomposition and intertia groups, Chebotarev's density theorem (without proof) + applications

- Cyclotomic fields: cyclotomic polynomials, primes in arithmetic progressions, Fermat's Last Theorem for regular exponents 

Course material

Course notes + Toledo.

Content

See G0A99a.

Course material

Exercise sets + Toledo.

Content

Hilbert spaces and Banach spaces

• Reminders on Hilbert spaces, orthogonal projections and orthonormal bases
• Definitions, examples and basic properties of Banach spaces
   

Baire category and its consequences

• Baire category theorem
• Boundedness and continuity of linear maps
• Open mapping theorem
• Closed graph theorem
• Principle of uniform boundedness

Bounded operators on a Hilbert space

• Hermitian adjoint
• Compact operators
• Invertible operators, spectrum of an operator
• Spectral theory of compact selfadjoint operators
• Spectral theory of arbitrary selfadjoint operators

Weak topologies and locally convex vector spaces

• Dual Banach space
• Hahn-Banach extension theorem
• Topological vector spaces, seminormed spaces
• Weak topologies
• Hahn-Banach separation theorem
• Banach-Alaoglu theorem
• Krein-Milman theorem
• Markov-Kakutani fixed point theorem

Amenability of groups

• Invariant means on groups
• Examples and counterexamples to amenability
• Various characterizations of amenability
• Abelian groups are amenable

Course material

Lecture notes.
Book: G.K. Pedersen, Analysis Now. Graduate Texts in Mathematics, Volume 118. Corrected second printing. Springer-Verlag, New York, 1995.

Format: more information

There will be a two-hour lecture each week, in which new concepts will be introduced and several results will be proved. Additionally, there will be an exercise session each week, in which the students further develop the topics of the course and apply the material in different situations.

Content

Exercises and problem assignments related to the different topics of the course.

Course material

Lecture notes.
Book: G.K. Pedersen, Analysis Now. Graduate Texts in Mathematics, Volume 118. Corrected second printing. Springer-Verlag, New York, 1995.

Explanation

Detailed information will be provided via Toledo.

Information about retaking exams

For the second exam opportunity the grades on the take-home assignments are transferred. There is no possibility to have a new take-home assignment.

Content

Basic theory of Riemann surfaces.
(1) Definitions and examples of Riemann surfaces.
(2) Functions on a Riemann surface. Local and global properties.
(3) Integration on a Riemann surface. Holomorphic and meromorphic 1-forms. Residue theorem. Surface integrals and Stokes theorem.
(4) Divisors and meromorphic functions.
(5) Jacobi variety and Abel's theorem.
(6) Riemann-Roch theorem.

Introduction to one advanced topic. Possible topics are:
(7) Jacobi inversion theorem.
(8) Theta functions.
(9) Sheaves and Cech cohomology.

Course material

- Recommended literature: Wilhelm Schlag, A Course in Complex Analysis and Riemann Surfaces, American Mathematical Society, 2014
- Course notes
- Toledo

Content

Basic theory of Riemann surfaces.
(1) Definitions and examples of Riemann surfaces.
(2) Functions on a Riemann surface. Local and global properties.
(3) Integration on a Riemann surface. Holomorphic and meromorphic 1-forms. Residue theorem. Surface integrals and Stokes theorem.
(4) Divisors and meromorphic functions.
(5) Jacobi variety and Abel's theorem.
(6) Riemann-Roch theorem.

Introduction to one advanced topic. Possible topics are:
(7) Jacobi inversion theorem.
(8) Theta functions.
(9) Sheaves and Cech cohomology.

Course material

- Recommended literature: Rick Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, Vol 5., American Mathematical Society, Providence, RI, 1995.
- Course notes
- Toledo

Content

see G0B07a

Format: more information

Discussion

Weekly exercise sessions integrated with the lectures, in which the students further develop the topics of the course and apply the material in different situations.

Content

Below a general overview of possible themes and subjects is described. According to the specific background and interests of the students, emphasis can be modulated and possibly extra topics or applications might be covered.

Spectral theory in Banach algebras

• Banach algebras: definition, examples, basic properties
• The spectrum of an element in a unital Banach algebra: definition, examples, general properties of the spectrum, spectral radius

Gelfand's theory of commutative Banach algebras and C*-algebras

• The Gelfand transform for commutative Banach algebras
• C*-algebras: definition, examples, special elements (unitary, self-adjoint, normal) and their spectrum
• The continuous functional calculus for normal elements in a C*-algebra
• Gelfand-Naimark theorem

C*-algebras

• Positivity for elements and functionals
• Non-unital C*-algebras; approximate units
• Universal C*-algebras from generators and relations
• States and representations; GNS construction
• Pure states and irreducible representations
• Construction and study of special C*-algebras (e.g. group C*-algebra, irrational rotation algebra)
• Inductive limits

von Neumann algebras

• the weak, s−weak, strong and s−strong topologies on the bounded operators on a Hilbert space
• Defintion of von Neumann algebras, elementary examples
• Bicommutant theorem
• Kaplansky density theorem
• enveloping von Neumann algebras
• Borel functional calculus
• Construction and study of special examples (e.g. group von Neumann algebra)

Course material

Concise lecture notes are provided by the lecturer. Those notes have to be elaborated by the student using the literature.

Content

• Differentiable manifolds
• Tangent vectors and vector fields
• Bundles
• Differential forms and integration
• The exterior derivative and Stokes' theorem
• de Rham cohomology
• Lie groups
• Foliations

Course material

1. Tu, Loring. An introduction to  Manifolds. Universitext, 2010 (Second edition).
2. Lee, John. Introduction to Smooth Manifolds, Springer Graduate Texts in Mathematics 218 (Second edition).

Language of instruction: more information

The course will be taught in English

Content

The exercise sessions will be devoted to solving and discussing problems proposed by the instructor. The students will be asked to work on certain problems before the exercise session.

See G0B08a for more details.

Course material

See G0B08a

Explanation

There will be Take Home Tasks along the course, each consisting of exercises/problems related to certain parts of the course. The student will write solutions (in English) of the Take Home Tasks, and the solutions will be graded. For the solutions it is allowed to use notes of the lecture and of the exercise sessions, and the portion of the recommended literature corresponding to the material treated in class. Students are allowed to discuss the problems with each other, but are supposed to write up their solutions individually. Solutions that are literally copied from peers will not be tolerated and every reference that is used should be quoted.

30% of the final grade will be based on the take home tasks, and 70% on the final exam. In particular, a student who does not take the final exam can obtain at most 6/20 as grade for the course.

The final grade is meant to reflect  to what extent the student assimilated the basic notions of differential geometry, and is  able to work with them and apply them in concrete situations.

Content

 
Riemannian and pseudo-Riemannian geometry
- metrics,
- connection theory (Levi-Cevita),
- geodesics and complete spaces
- curvature theory (Riemann-Christoffel tensor, sectional curvature, Ricci-curvature, scalar curvature),
- tensors
- Jacobi vector fields.
 
Global and local isometries
- space forms,
- symmetric spaces.

Immersions (and introduction to submanifold theory)

Submersions (and the Riemannian structure of the complex projective space)
 
Selected topics of differential geometry
classification of real space forms, Hadamard's theorem and variational calculus on Riemannian manifolds
 
 

Course material

• Wolfgang Kühnel : Differential Geometry : Curves - Surfaces - Manifolds, Student Mathematical Library, volume 16. American Mathematical Society, 2002
• M.P. do Carmo, Riemannian Geometry, Birkhäuser, 1992
• Barrett O'Neill, Semi-Riemannian geometry. With applications to relativity, Academic press (1983)

Explanation

The exam is written and consists of several exercises.

Content

PART 1:

• Symplectic linear algebra.
• Symplectic manifolds. The physical motivation of symplectic geometry: classical mechanics.
• Lagrangian submanifolds, coisotropic submanifolds. Normal form theorems: Darboux's, Weinstein's and Gotay's theorems.

PART 2:

• Lie algebra cohomology and representations.
• Hamiltonian actions and moment maps. Existence and uniqueness theorems.
• The Marsden-Weinstein symplectic reduction theorem. The convexity theorem of Atiyah and Guillemin-Sternberg.

Course material

•  Ana Cannas da Silva, "Lectures on symplectic geometry", Springer Verlag. Available at http://www.math.ethz.ch/~acannas/Papers/lsg.pdf
•  Eckhart Meinrenken, "Symplectic geometry", lecture notes available from http://www.math.toronto.edu/mein/teaching/lectures.html

Language of instruction: more information

The course will be held in English

Explanation

There will be Take Home Tasks during the course, each consisting of exercises/problems related to certain parts of the course. The student will write solutions (in English) of the Take Home Tasks, and the solutions will be graded. For the solutions it is allowed to use notes of the lecture and of the exercise sessions, and the portion of the recommended literature corresponding to the material treated in class. Students are allowed to discuss the problems with each other, but are supposed to write their solutions individually. Solutions that are literally copied from peers will not be tolerated and every reference that is used should be quoted.

30% of the final grade will be based on the take home tasks, and 70% on the final exam (January exam or September exam). 
In particular, a student who does not take the final exam can obtain at most 6/20 as grade for the course. 

The final grade is meant to reflect  to what extent the student assimilated the basic notions of symplectic geometry, and is  able to work with them and apply them in concrete situations.

Content

The goal of this course is to introduce concepts and statistical procedures for advanced contemporary data-analysis. Classical statistical techniques such as maximum likelihood and least squares estimation make strong assumptions that need to be satisfied by the data. However, in practical applications these assumptions are often violated. Modern statistical procedures aim to relax these stringent assumptions to obtain more reliable statistical inference. Moreover, standard statistical models are unrealistic or too restrictive for many of the complex types of data encountered in practice, such that more advanced models are needed to fit these data. :  In this course modern statistical methods and procedures are introduced, such as advanced resampling techniques (based on bootstrap or subsampling), robust statistical inference, methods for high-dimensional  data (screening, sparsity) and functional data, for instance. The practical use of these methods will be discussed as well.

Course material

Course notes

Content

During the exercise and computer sessions the material exposed during the lectures will be further illustrated and used in various contexts, and the application of the methods to real data will be discussed.

Course material

Course notes and datasets

Content

A project will be made in small groups (usually 2 or 3 students). In the project the students explain and illustrate a statistical method or procedure based on a scientific article or book chapter.

Course material

Course notes, scientific article or book chapter.

Explanation

A project will be made in small groups (usually 2 or 3 students). The evalation of the project takes place in the last week(s) of the semester and involves an oral presentation if permitted by the circumstances.

The written exam consists of open questions and is closed book. 

Project part and exam each count for 50% of the total course mark.

Information about retaking exams

For the second chance exam, the total score for the course mark consists of 50% project work, and 50 % open questions during the exam. This modality is the same for first and second exam chances.

Students that passed  the project work at the first exam chance can keep their score on this part for the second chance evaluation. Students that failed  the project work at the first exam chance will get a new project assignment for the second exam chance.

Content

The goal of robust statistics is to develop and study techniques for data analysis that are resistant to outlying observations, and are also able to detect these outliers.

In this course we introduce notions of robustness such as the breakdown value and the influence function. We study several robust estimators of univariate location and scale, multivariate location and covariance, linear regression, and principal component analysis.

Course material

Course text and slides.

Format: more information

The course consists of lectures and some computer sessions in which the methods will be applied to real data.

Content

In computer sessions, robust methods will be applied to real data sets and the results will be interpreted. Some properties of the estimators will be verified empirically, for instance by Monte Carlo simulation.

Format: more information

The exercises will take place in computer classes.

Content

The students make a project in which they study the behavior and/or performance of certain robust methods on real and simulated data sets.

Explanation

The evaluation consists of a project and an examination. The specific form of the exam and the grading will be explained in class and on Toledo.

Information about retaking exams

If you received a passing mark for the project, this score can be retained.

Content

1. Linear waves and instabilities in fluids
-      Recapitulation: surface and internal gravity waves, Rayleigh-Taylor instability, classic Kelvin-Helmholtz instability, acoustic waves
-      Hyperbolic waves, dispersive and anisotropic waves
-      Linear surface water waves generated by a moving source
-      Linear shallow water theory: reflection, amplification, refraction
-      Thermal instability: the Bénard problem
-      Waves and instability of continuously stratified parallel flows: Rayleigh’s equation, Taylor-Goldstein equation, Orr-Sommerfeld equation.
-      Critical layer behaviour
-      Transient growth due to non-normality
 
2. Nonlinear waves in fluids
-   Traffic waves: advection equation, kinematic waves, advectiondiffusion equation, Burgers equation, Cole-Hopf transformation
-   One-dimensional gas dynamics (characteristics and Riemann invariants)
-   Shallow water theory (characteristics and Riemann invariants)
-   Multi-valued solutions
-   Shock waves in 1-D gas dynamics (Rankine-Hugoniot conditions)
-   Shocks in shallow water (Rankine-Hugoniot conditions, hydraulic jumps and bores)
-   2-D steady shocks (flow past a wedge)
-   Nonlinearity versus dispersion: KdV equation
 
3. Linear MHD waves in plasmas
-   Recapitulation (Alfvén waves and slow and fast magnetosonic waves in uniform plasmas of infinite extent in ideal MHD)
-   Damping of Alfvén waves in resistive uniform plasmas
-   MHD waves of uniform cylindrical plasmas
-   Nonuniformity and resonant waves
-   Equilibrium flows and resonant overstabilities

Content

1. Linear waves and instabilities in fluids
-      Recapitulation: surface and internal gravity waves, Rayleigh-Taylor instability, classic Kelvin-Helmholtz instability, acoustic waves
-      Hyperbolic waves, dispersive and anisotropic waves
-      Linear surface water waves generated by a moving source
-      Linear shallow water theory: reflection, amplification, refraction
-      Thermal instability: the Bénard problem
-      Waves and instability of continuously stratified parallel flows: Rayleigh’s equation, Taylor-Goldstein equation, Orr-Sommerfeld equation.
-      Critical layer behaviour
-      Transient growth due to non-normality
 
2. Nonlinear waves in fluids
-   Traffic waves: advection equation, kinematic waves, advectiondiffusion equation, Burgers equation, Cole-Hopf transformation
-   One-dimensional gas dynamics (characteristics and Riemann invariants)
-   Shallow water theory (characteristics and Riemann invariants)
-   Multi-valued solutions
-   Shock waves in 1-D gas dynamics (Rankine-Hugoniot conditions)
-   Shocks in shallow water (Rankine-Hugoniot conditions, hydraulic jumps and bores)
-   2-D steady shocks (flow past a wedge)
-   Nonlinearity versus dispersion: KdV equation
 
3. Linear MHD waves in plasmas
-   Recapitulation (Alfvén waves and slow and fast magnetosonic waves in uniform plasmas of infinite extent in ideal MHD)
-   Damping of Alfvén waves in resistive uniform plasmas
-   MHD waves of uniform cylindrical plasmas
-   Nonuniformity and resonant waves
-   Equilibrium flows and resonant overstabilities

Explanation

An oral exam is organised where open questions are discussed. Part of the score is also earned during the year in a permanent evaluation system. 

Content

1. General description: the Sun, observations in different wavelengths, Sunspots, the solar cycle, the solar magnetic field, the coronal

heating problem, actives regions, solar flares, coronal loops, the solar wind, coronal mass ejections, space weather.

2. Elements of plasma physics: motion of charged particles, gyration, the E×B drift, the ∇B drift, gravitational drift, magnetic mirrors.

3. Magnetohydrodynamics (MHD): one-fluid and two-fluid MHD, Hall MHD, the plasma β, the Alfvén Mach number, magnetic flux tubes,

conservation of magnetic flux, the frozen-in theorem, quasi neutrality in plasmas, magnetic pressure and tension, conductivity in a plasma,

the displacement current, field aligned currents, MHD waves, shocks and discontinuities, Alfvén and fast waves, the Rankine-Hugoniot

relations.

4. Coronal and solar wind plasma: macroscopic or fluid models (the Parker model, the Weber-Davis model, force free magnetic field models,

magnetic field reconstruction techniques), microscopic or kinetic models (collisional, collisionless, homogeneous or inhomogeneous).

5. Kinetic modeling: particle velocity distributions, observations, Vlasov-Boltzman formalism of plasma waves, wave-particle interaction, anisotropic

Distributions (temperature anisotropy in the solar wind, beams in the fast wind, counterstreams in coronal mass ejections and shocks).

6. Spectral theory: motivation for collisionless and collision-poor plasma models, plasma waves and characteristics (collisionless dissipation, Landau, cyclotron,

high and low-frequency waves, MHD waves), instabilities and enhanced fluctuations in plasmas with free energy: kinetic anisotropy,  inhomogeneities, etc.

applications in the corona, solar wind and planetary magnetospheres.

Content

Two assignments are given during the semester. The subjects of these assignments depends and can vary form coronale heating, over sunspots to coronal seismology.

Course material

Recent papers on solar physics are provided, depending on the assignment.

Format: more information

Two assignments are given during the semester. A report has to be handed in for both of these. Each report is marked on 4 points, i.e. 20% of the total end score.

Explanation

Closed book exam with about 5-open questions about the treated material.

The reports of the two tasks that are given during the semester are marked on 4 points each. The weigth of the exam thus amounts to 12 points.

In case of a re-sitis the points on the report are transferred. So it is not possible to make new tasks in that case.

Content

The course is organized in modules. The basic modules consist of lectures combined with (home assignment and group)  worksessions, and will cover:
 
1. Introduction
a. Developing numerical codes
   – Computer code development, programming techniques, code maintenance, optimization
   – Concepts of verification, validation, sensitivity analysis, error and uncertainty quantification
 b. Spatial and temporal discretization techniques.
   – Spatial discretizations: Basic concepts for discrete representations. Finite difference, finite element, and spectral methods. An application: solving a Sturm-Liouville model problem and handling boundary conditions (eigenoscillations of a planar stellar atmosphere).
   – Temporal discretizations: Explicit versus implicit time integration strategies. Semi-discretization, predictor-corrector and Runge-Kutta schemes.
 
2. Towards computational gas dynamics.
• The advection equation and handling discontinuous solutions numerically. Stability, diffusion, dispersion, and order of accuracy, demonstrated with linear advection problems. Extension to linear hyperbolic systems and solution of the Riemann problem. Nonlinear scalar equations and shocks: solving Burgers equation. Non-conservative versus conservative schemes.
• Isothermal hydrodynamics and basic stellar wind models. Governing equations, Rankine-Hugoniot conditions, Prandtl-Meyer shock relation. Rarefactions, integral curves and Riemann invariants. Application to transonic stellar winds: Parker solar wind solution, isothermal rotating transonic winds, shocked accretion flows.
 
3. Compressible gas dynamics and multi-dimensional applications.
• The Euler equations and finite volume methods. Conservative form, Rankine-Hugoniot shock relations, exact solution of the Riemann problem, Riemann invariants. Basic shock-capturing discretization methods: finite volume methods and the TVDLF algorithm.
Possible advanced topic: Roe solver. Godunov scheme for Euler equations, Approximate Riemann solver, Roe scheme, numerical tests.
• Extensions to multi-dimensional algorithms and example multi-dimensional stellar wind models. Example 2D Euler simulations, emphasizing stellar wind models for various evolutionary phases, for cool to massive stars. Extension to interacting wind models using optically thin radiative losses. Attention to failures of modern schemes that still plague 1D and multi-D Euler simulations.
 
4. Numerical radiative transfer.
• Basic radiative transfer. The governing equations of radiative transfer and the rate equations. Discretization, treatment of angle dependence (with angle quadrature), handling of frequencies and optical depths.
• Specific numerical treatments. Feautrier method, Lambda iteration, Multi-level iteration. Application to stellar winds which are dust or radiative driven.
 
5. Intro to Computational Magneto-Hydro-Dynamics.
• Introduction: the MHD model. Applicability, use in astrophysical context.
• Transmagnetosonic stellar winds and 1D MHD simulations. Weber-Davis MHD wind model, numerical simulations for solar and stellar rotating, magnetized winds, consequences for stellar rotational evolution. MHD shocks, Riemann problem tests.
 
A final module can be chosen depending on the interest of the students, linking to current research trends.

Course material

The lecture sheets are made available through Toledo. Additional course notes are provided online as well. Reference books are (students will not be required to purchase these, no book covers all topics):

• Numerical Methods in astrophysics, Taylor & Francis 2007, Bodenheimer et al.
• Advanced Magnetohydrodynamics, Cambridge University Press 2010, Goedbloed, Keppens, Poedts

Format: more information

Next to the lectures, students will either individually or in pair work out computerassignments, directly related to the topics covered. This will encompass both self-coding for a relevant toy problem and using advanced state-of-the-art software in a modern application. Part of these will be organized in joint computerclass sessions.

Content

Using a combination of self-written and available software to solve selected astrophysical toy problems numerically. The idea is to gain insight in method limitations, as well as get acquainted with its inherent possibilities.

In a first part, the students will be asked to program their own solver.

In a second part, the students perform selected hydrodynamic simulations, and learn how to interpret and visualize their computational data.

Course material

During the second assigment, we make use of opensource modern computational codes, specifically the MPI-AMRVAC code, widely used in astrophysical applications.

Format: more information

Assignments will be formulated and presented in teams, and we foresee access to supercomputer platforms.

Explanation

Permanent assessment, working out project assignments. At least one project will be handed in as a written report, along with the self-written computercode. The team assignment lets the students perform modern computational research, to be reported in a team presentation.

Information about retaking exams

The second exam will be formulated as an extensive take-home computerassignment, where the student ultimately reports on the numerical strategy, (astro)physical application and makes contact with relevant modern literature.

Content

Introduction and motivation
 
    * Definition of space weather
    * Space weather effects
    * Space weather components
    * Predictions and forecasts
 
A tour of the Solar System
 
    * Sun
    * Solar corona
    * Interplanetary space
    * Planetary magnetosphere
 
The Earth Environment
 
    * Magnetosphere
    * Magnetosphere-ionosphere coupling
    * Magnetosphere-thermosphere coupling
 
Solar energetic particles
 
    * Generation of high-energy particles in space weather events
    * Transport of high-energy particles in the solar system
    * Radiation belts
 
Models of space weather
 
    * fluid modeling
    * kinetic effects
 
Following a typical space weather storm
 
    * Coronal Mass Ejections (CME): initiation
    * CME: Inter−planetary evolution
    * Impact on the Earth environement
    * Geo−effectivity of magnetic storms
    * Ground and space based solar observations
    * Radio observations
    * In situ measurements (e.g. ACE, CLUSTER)
    * Unsolved problems
 
Resources and Forecast
 
    * Web-based services from NOAA and ESA
    * Simulation: NASA's community coordinated modeling center (CCMC)
    * Soteria and the SSA.
 
 

Course material

G. Lapenta, Lecture notes.

A. Hanslmeier, The Sun and Space Weather (Springer, 2008)
M. Kallenrode, Space Physics (Springer, 2004)

Format: more information

Lessons from the teaching team, including distinguished experts from space agencies and industry.

Is also included in other courses

G0B32B : Space Weather

Content

Introduction and motivation
 
The students use online web site and computer codes to build experience on space weather. For example:
 
* Use of the CCMC web site to simulate space weather
 
* Study of the astrophysics of the Sun and of the Solar System
 
* Computer simulation of spacecrafts immersed in the environment near the Earth
 
*  Space weather between the Earth and the Moon
 
* A trip to Mars: consequences of radiation and particles

Course material

A. Hanslmeier, The Sun and Space Weather (Springer, 2008)
M. Kallenrode, Space Physics (Springer, 2004)

Format: more information

Student projects guided by experts in the field.

Explanation

The exam is composed of differnt parts:

oral presentation on the project: 30%
report on the project: 40%
practical work done at  home and due before the exam: 30%

Content

For the content in the current academic year, see https://wis.kuleuven.be/english/education/ma-math/SelectedTopics/Seltop

Course material

See Toledo

Content

- Schemes: spectrum of a ring, sheaves, schemes and their relation with classical varieties, local and global properties of schemes.

- Coherent sheaves: locally free sheaves, vector bundles, divisors,  projective morphisms, differentials.

- Cohomology: derived functors, Cech cohomology, Serre Duality, Grothendieck-Riemann-Roch theorem, Semicontinuity theorem.

- Curves and Surfaces: basic classification results in complex and arithmetic algebraic geometry.

- The Weil Conjectures: zta functions of varieties, Deligne’s theorem.

Course material

Course notes + Toledo

Content

- Schemes: spectrum of a ring, sheaves, schemes and their relation with classical varieties, local and global properties of schemes.

- Coherent sheaves: locally free sheaves, vector bundles, divisors,  projective morphisms, differentials.

- Cohomology: derived functors, Cech cohomology, Serre Duality, Grothendieck-Riemann-Roch theorem, Semicontinuity theorem.

- Curves and Surfaces: basic classification results in complex and arithmetic algebraic geometry.

- The Weil Conjectures: zeta functions of varieties, Deligne’s theorem.

Course material

Course notes + Toledo

Explanation

There will be two take-home exams during the semester.

The final exam is also take-home and consists either of classical exam questions or of submission of a short expository paper on a topic of own choice related to the course and agreed upon by the instructor. This paper has to contain, beside some clear introductory theory, non-trivial explicit examples, agreed upon by the instructor, worked out to illustrate the theory.

In order to pass, the student must obtain at least the score 10/20. The take-home exams during the semester will count 5 points each, the final exam will count 10 points. If the student has failed to pass, for the second-chance examination no points will be carried forward from any of the take-home exams or the final exam. The student will be given the chance to pass the course via, again, a package consisting of two new take-home exams and a new final exam, with the same format and score share.

Content

The master thesis comprises the research work including the written report, the thesis, under the guidance of a supervisor from the Department of Mathematics and of his/her research group. In principle, theses are made in one of the domains belonging to the research expertise of the Department, but domains which are closely related to mathematics, are possible as well (e.g. theoretical physics, computer science).
Students are expected to become integrated during some months in the research group. They have to get involved in the current research by following seminars, participating in discussions, studying and, last but not least, by carrying out the specific activities (computations, simulations, exploration and/or construction of proofs, etc.) leading to the solution of the posed research problem. The results have to be reported in a scientific text and have to be presented and defended in front of the fellow students and the staff of the Department.

Master's thesis topic: validity period

If the supervisor, at the end of the 3rd examination period of the second stage, finds that insufficient progress is made, this will be discussed with the student. The chairman of the programme committee will be informed. It may be possible that in that case the choice of the topic lapses and that a new topic must be chosen. Reasons for the cancellation of the topic may be because:

• during the academic year in which the master's thesis is included in the ISP the student has worked, without legitimate reasons, too limited on the master's thesis research, or practical arrangements or agreements have not been fulfilled, so that the master's thesis could not be completed
• the supervisor can not offer the topic in a next academic year (eg the research topic is finished/stopped, guidance will no longer be possible in the research team when needed)

Course material

Depends on the research subject.

Explanation

The evaluation consists of the assessment of both process and product (form and content; manuscript and defense). Four quotes are given: one by the promotor, one of each of the two readers and one for the defense. The relative weight of these four quotations is 10:3:3:4. Each quotation is determined by means of the facultary assessment roster and appreciation scale. Additional informtion on the evaluation of the master's thesis is to be found on the faculty website. Complementary regulations for a master's thesis in Mathematics are to be found on the webpages of the Department of Mathematics.

In order to succeed the master’s thesis, the student must obtain a credit for the supervisor apart, the average of the results of the readers taken together (taking into account the rounding rules) and the defence apart. If for one or more of these components this is not the case, the maximum score will be 9/20.

There is no resit for the intermediate presentation (in January) and the seminar for the research group (in April/May), as outlined in the regulations for a master's thesis in Mathematics.

For passing this course students have to upload an information skills certificate in Toledo. This certificate can be obtained in the Toledo community “Scientific integrity at the Faculty of Science”. Obtaining and submitting the information skills certificate is evaluated by ‘pass/fail’. A student with a ‘fail’ for the certificate, obtains a ‘fail’ for the course, that is converted to a non-tolerable fail. This means that students cannot pass the course and cannot use tolerance credits, if they have not obtained and submitted the certificate.

This course can not be tolerated.

Information about retaking exams

see explanation

Content

For the contents in the current academic year, see https://wis.kuleuven.be/english/education/ma-math/SelectedTopics/Seltop

Course material

See Toledo

Content

• Multivariate data, covariance, checking normality assumption;
• Transformation to normality by the Box-Cox transform;
• Dimension reduction methods;
• Cluster analysis: hierarchical and partitioning, graphical displays;
• Topics in regression analysis: interactions, categorical predictors, heteroskedasticity, variable selection criteria, multicollinearity, ridge regression, outliers and leverage points, prediction models
• Classification techniques: evaluation measures, misclassification rate, k-nearest neighbor classification, logistic regression, modern classification techniques

Course material

Course notes

Content

Weekly organised exercise sessions in the PC lab where the new methods (see OLA G0O00a) are illustrated and practised by means of the statistical software R. Some homework assignments need to be made as well.

Course material

Course notes and datasets

Content

The projects consist of a thorough statistical analysis of real data. The results need to be presented in a written report.

Course material

Course notes and excercise material

Explanation

The evaluation consists of two projects and an examination.  The projects involve data analysis tasks. For each project an individual written report is handed in with the analysis results presented in a scientific manner and an appendix describing the complete workflow. The written examination consists of a closed book part with open questions and an open book part on the computer which involves the analysis of a dataset.

The project part and exam part each count for 50% of the total course mark. Students should pass both parts to get a pass mark for the course.

Information about retaking exams

For the second chance exam, the project part and exam part again each count for 50% of the total course mark. This modality is the same for first and second exam chances.

Students that passed  the project work at the first exam chance can keep their score on this part for the second chance evaluation. Students that failed  the project work at the first exam chance will get a new project assignment for the second exam chance.

Inhoud

Galoistheorie
- Herhaling over velduitbreidingen
- Ontbindingsvelden en primitieve elementen
- Galoisuitbreidingen
- De Galoisgroep en de Galoiscorrespondentie
- Galoisgroep van een vergelijking met graad n en verband met oplosbaarheid door middel van radicalen (stelling van Galois)
- Onoplosbaarheid van de algemene vergelijking van de 5-de graad


Groepentheorie
- Stellingen van Sylow
- Toepassing: de hoofdstelling van de algebra
- Presentaties van groepen; direct en vrij product

Studiemateriaal

Cursustekst + Toledo.

Toelichting werkvorm

Hoorcollege met opdrachten tijdens de les

Inhoud

Zie G0P53a 

Studiemateriaal

Idem Hoorcollege + Toledo.

Toelichting werkvorm

Oefeningen.

Inhoud

De studenten maken in de loop van het semester een schriftelijk werkstuk in groepen van twee of drie personen. Het kan gaan om het oplossen van een reeks oefeningen, het uitwerken van een bijkomend stuk theorie of een soortgelijke opdracht. De studenten zullen tijdens de hoorcolleges feedback ontvangen op het ingediende werkstuk.

Toelichting

Schriftelijk open boek examen in de examenperiode. De opdracht telt mee voor 5 van de 20 punten. De score voor de opdracht wordt overgedragen naar de tweede zittijd.

Inhoud

De cursus overloopt de chronologische ontwikkeling van de wiskunde, maar met speciale aandacht voor historiografische discussies en terugkerende thema's. Zo wordt de vraag naar transformatie en revolutie binnen het wiskundig denken aan de orde gesteld, alsook de veranderende positie van de wiskunde in de tijds- en plaatsgebonden culturele context. Ook komen institutionele factoren (onderwijs, professionalisering, gemeenschapsvorming) aan bod. Voor elke periode wordt een algemene karakterisering gegeven die de oriëntatie van het wiskundig denken in die periode verheldert.

• Wiskunde in de Oudheid (de opvattingen van Pythagoras en Plato, de kenmerken van de Griekse wiskunde, belangrijke wiskundigen: Eudoxus, Euclides, Archimedes, Apollonius, Pappus)
• Wiskunde in de Europese Middeleeuwen (Boethius, Leonardo Fibonacci, introductie van Arabische wetenschap, universiteiten)
• De Renaissance van de wiskunde (ontwikkeling van algebra, trigonometrie, invloed van humanisme en de herontdekking van de Griekse wiskunde, wiskundige elementen in kunst en techniek)
• De ontwikkeling van de infinitesimaalrekening (Descartes, Fermat, Newton, Leibniz)
• Het oplossen van wiskundige problemen in de achttiende eeuw (Bernoulli’s, d’Alembert, Euler)
• Verdieping in de negentiende eeuw: het streven naar strengheid en het zoeken naar onderliggende structuren in algebra en meetkunde (Gauss, Cauchy, Abel, Galois, Riemann, Plücker, Klein, Weierstrass,…)
• De ontwikkeling van de statistiek (Pascal, Huygens, Bernoulli, De Moivre, Bayes, Laplace, Quetelet, Poisson, Galton, Pearson)
• De grondslagen van de wiskunde. Arithmetisering (Cantor, Dedekind) en axiomatisering (Frege, Peano, Russell).
• Recente ontwikkelingen: topics uit de wiskunde van de 20ste eeuw.

Studiemateriaal

Het studiemateriaal wordt via Toledo beschikbaar gesteld en bestaat uit

• slides die tijdens de lessen besproken worden;
• teksten uit handboeken en artikels;
• oefeningen
• ...

Content

Review of basic arithmetics: Euler function, congruences of Euler and Wilson, Chinese Remainder Theorem.
Structure of the unit group of Zn.
Solubility of congruences: Lemma Hensel-Rychlik.
Quadratic reciprocity laws of Gauss and Jacobi.
Fast algorithms for congruences and primality testing.
The field of p-adic numbers.
p-adic numbers and the Hilbert symbol.
Rational points on a conic. The Hasse principle.
Quadratic rings.
Whole points on conic sections.
Applications in cryptography.
Prime numbers and the Riemann zeta function (introductory).
Elliptic curves

Course material

Syllabus

Format: more information

Lectures

Content

Same as lectures.

Course material

Same as lectures + Toledo.

Format: more information

Exercises.

Explanation

The evaluation consists of:

- a written exam during the examination period (with grade E).
- 3 (non-obligatory) assignments during the semester (with grade T).

The final grade is calculated according to the formule max{E,(3E+T)/4}.

It is not possible to retake the assignments for the second examination attempt, but the previously submitted assignments do count for the final grade.