Courses

The course part of this programme consists of;

  • 15 EC compulsory courses
  • 45 EC primary electives
  • 15 EC secondary electives

This Master's is part of the national Mastermath Programme, which offers you an exceptional list of mathematical courses to choose from. You can choose both local courses (at Utrecht University), as well as courses offered by another Dutch University. The local courses are listed below. For an overview of all courses offered in the Mastermath programme: please visit the Mastermath website.

Local courses (Utrecht)

Mathematics Colloquium (compulsory)

There is no content available for this course.

Mathematics for Industry (compulsory)

There is no content available for this course.

Seminar Selected Topics in the Calculus of Variations

The calculus of variations is an active area of current research with important applications in science and technology (e.g. physics, materials science, image processing,...),
as well as interesting connections with other disciplines of mathematics such as the theory of differential equations,optimal control or geometry. We will start the course with a brief introduction into this field,which is about minimization (or maximization) of functionals, i.e. real-valued maps defined on(infinite-dimensional) function spaces.The classical or indirect approach to variational problems has its roots in the 18th century. Itis concerned with deriving necessary and sucient conditions for extremal functions, assumingtheir existence. A famous counterexample established by Weierstrass in 1850, however, made themathematics community at the time realize that solutions to variational problems may not existin general. This fundamental observation marked the birth of a new branch called the modernor direct methods in the calculus of variations, which we will focus on in this seminar course.The modern methods help to find criteria for the existence of minimizers, but also to establishproperties of minimizing sequences if minimizers do not exist. On a technical level, functionalanalytical tools as well as convex analysis play a key role in addressing these issues.Another interesting and more recent direction of study regards sequences of variational problemsand their asymptotic behavior. The appropriate concept for limit processes in this context wasintroduced by De Giorgi and Franzoni in 1975 and is based on a special notion of convergence forfunctionals, called gamma-convergence. Over the last decades, this framework has turned out to be aversatile and powerful tool for tackling problems that arise from multiscale modeling.
After the introductory part, we will go more into depth on selected topics, chosen according tothe specic interests of the participants. Here are some suggestions:
Existence of minimizers and the direct methods

  • generalized notions of convexity
  • lower semicontinuity of integral functionals
  • relaxation of variational problems with non-convex integrands
  • ntroduction to BV functions
  • variational problems with linear growth
  • Young measures and relaxation

Gamma-convergence and its applications

  • gamma-convergence of integral functionals
  • periodic homogenization via gamma-convergence
  • variational theory of phase transitions
  • dimension reduction via variational convergence
  • discrete to continuum limit passages
  • evolutionary gamma-convergence

MaterialSome general literature:[1] H. Attouch, G. Buttazzo and G. Michaille, Variational analysis in Sobolev and BV spaces -Applications to PDEs and optimization. SIAM, Philadelphia, 2014.[2] A. Braides, Gamma-convergence for beginners. Oxford University Press, Oxford, 2002.[3] B. Dacorogna, Direct methods in the calculus of variations. Springer, New York, 2008.[4] G. Dal Maso, An introduction to Gamma-convergence. Birkhauser Boston, 1993.[5] L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI,1998.[6] I. Fonseca and G. Leoni, Modern methods in the calculus of variations: Lp spaces. Springer,New York, 2007.
 These references provide material for background reading and serve as a starting point for a moredetailed literature search once the topics have been assigned.
PrerequisitesBasics of analysis, functional analysis and measure theory are expected. Familiarity with Sobolevspaces, the concept of weak convergence, and partial differential equations will be helpful, but isnot required. A brief crash-course during the introductory lectures will give the necessary background.
FormatThis seminar is aimed at master students with a background and interest in pure and appliedanalysis.
ECTS7.5
 OrganizationInterested students should contact Carolin Kreisbeck by email (c.kreisbeck@uu.nl). Please outlineyour general interest in the seminar, your familiarity with prerequisites, and specic preferencesregarding topics. Students are asked to make an appointment at least two weeks before the scheduledpresentation to discuss and settle relevant issues, e.g. structure of the talk, choice of literature,questions about the content, etc.
 EvaluationEvery participant is expected to give two seminar talks of 90 minutes each, possibly more, dependingon the number of students. This includes collecting and studying relevant material beforehand,giving a blackboard presentation, preparing a hand-out, and designing homework exercisesfor the other students. Note that hand-in exercises should be approved beforehand by the seminar organizer.
The speaker is responsible for grading the homework assignments. In case of discussionabout the solutions, the final decision will be with the seminar organizer. Attendance at all seminartalks is mandatory. The final grade for the seminar is the weighted average of the homeworkgrades (20%) and the grade for the presentation (80%). The latter takes also the hand-out andthe design of exercises into account.
 Learning goalsAfter completion of the course, the student is able to:

  • give an overview of modern variational methods and knows how to use them for solving problems.
  • acquire a solid understanding of a specic mathematical topic by reading graduate-level textbooks and scientic papers.
  • organize material from the literature in analysis into a well-structured and comprehensible presentation.
  • choose appropriate means of communicating mathematics both in written and oral form.
  • formulate and correct exercises that keep a balance between relevance and feasibility.
  • explain specic topics from the content list of the seminar and put them into perspective.

Evaluation matrix presentation (80%)
(includes hand-out and
design of exercises)homework (20%)give an overview of modernvariational methods and know how to use them for solvingproblemsxxacquire a solid understandingof a specic mathematical topic by reading graduate-leveltextbooks and scientic papersx organize material from the literaturein analysis into a well-structured and comprehensible presentationx choose appropriate means ofcommunicating mathematics bothin written and oral formxxformulate and correct exercisesthat keep a balance between relevance and feasibilityx explain specic topics from thecontent list of the seminar andput them into perspectivexx    

Seminar Scientific Computing

The seminar course address the theory and techniques of data assimilation, as well as their applications. Thus, by the end of the course the students will know the basics of data assimilation, data-assimilation algorithms, and how they can be employed in the real-world applications.

The seminar course is organized as following: each seminar meeting a scientific paper is presented by a student followed by a discussion between the course participants. Course grades are based on article presentation and active participation during discussions. The course is primarily intended for master students and beginning PhD students in mathematics.
 Seminar Scientific ComputingPresentation 45%Chair 5%Report 50%is able to read and understand a scientific paper (suggested by the tutor) on data assimilation in a limited time periodx  present the content of the paper at a level accessible to other studentsx  chair a session by asking two questions x ask questions, participate in a discussion after a presentation x answer questionsx  implement a data assimilation method from an assigned paper in Matlab (or other language)  xwrite a report based on assigned test cases and a data assimilation method  xdeliver the report at a comparison session  x 

Seminar Applications of Mathematics in Radiation Research

The seminar will start with an introduction to radiation research and an overview of suggestions for topics and papers on these topics. Topics will be assigned to students based on their interests and background. In subsequent lectures students will give presentations on the research papers they have read.

Prerequisites: basic knowledge of calculus, linear algebra and differential equations. Prior knowledge in radiation research is not needed.
Students from other masters than Mathematical sciences or Physics (such as Biology, Computer Science, Medicine) are also welcome, if they do not fear mathematical formulas and have basic knowledge of the prerequisites.

The seminar will be led by Fieke Dekkers from RIVM (the National Institute for Public Health and the Environment). For more information, send an email to s.a.j.dekkers@uu.nl.

Internships at RIVM may be available to students participating in the seminar.
References:

  1. Modeling dose deposition and DNA damage due to low-energy β(-) emitters.
    Alloni D, Cutaia C, Mariotti L, Friedland W, Ottolenghi A.
    Radiat Res. 2014 Sep;182(3):322-30. doi: 10.1667/RR13664.1. Epub 2014 Aug 12.
  2. A two-mutation model of radiation-induced acute myeloid leukemia using historical mouse data. Dekkers F, Bijwaard H, Bouffler S, Ellender M, Huiskamp R, Kowalczuk C, Meijne E, Sutmuller M.
    Radiat Environ Biophys. 2011 Mar;50(1):37-45. doi: 10.1007/s00411-010-0328-7. Epub 2010 Sep 15.
  3. Ionizing radiation and leukemia mortality among Japanese Atomic Bomb Survivors, 1950-2000.
    Richardson D, Sugiyama H, Nishi N, Sakata R, Shimizu Y, Grant EJ, Soda M, Hsu WL, Suyama A, Kodama K, Kasagi F.
    Radiat Res. 2009 Sep;172(3):368-82. doi: 10.1667/RR1801.1.
  4. Radiation exposure and circulatory disease risk: Hiroshima and Nagasaki atomic bomb survivor data, 1950-2003. Shimizu Y, Kodama K, Nishi N, Kasagi F, Suyama A, Soda M, Grant EJ, Sugiyama H, Sakata R, Moriwaki H, Hayashi M, Konda M, Shore RE. BMJ. 2010 Jan 14;340:b5349. doi: 10.1136/bmj.b5349.

presentation, 85%exercises (set by student) 10%take home excercises 5%is able to read and understand a scientific paper (chosen by the student or suggested) on radiation research with applications of mathematics in a limited time periodx is able to identify the mathematics relevant to the paperx is able to present the content of the paper at a level accessible to other studentsx is able to design exercises linked to a presentation he/she has given for other students x is able to answer exercises set by other students on the topic of these students’presentations x

Seminar Algebraic Topology

There is no content available for this course.

Seminar Constructible Sets

Seminar organizer: Jaap van Oosten (UU). TIME/DAY/ROOM OF THIS SEMINAR WILL BE ARRANGED BY JAAP VAN OOSTEN together with the participants of the seminar.
 
This seminar is about a classical model of Set theory, created by Gödel in 1938, proving the consistency of the Generalized Continuum Hypothesis (which implies the Axiom of Choice). This model has been used extensively in later years, in order to settle more intricate consistency problems.
 
Material A basic reference is the book Constructibility by Devlin. This will be supplemented with other material in due course.
 
Prerequisites Bachelor-level mathematics, including the Foundations of Mathematics course.
 
Format This seminar is aimed at master students interested in Logic. There is a maximum number of (active) participants of 10.
 
ECTS: 7.5.
 
Schedule: Second Semester, block 3+4, 2 x 45 minutes per week
 
Language: English
 
Evaluation: Participants are expected to give three presentations of 45 minutes each. Since each session consists of 2x45 minutes, students work in a “team” of 2, and work in collaboration.
Moreover, each team devises 1 or 2 homework exercises, which are solved by the other students, graded by the team. Simultaneously with handing out the homework, the team hands a “model solution” to the teacher.
Students are encouraged to give feedback to each other.
 
Learning goals After completion of the course, the student is able to:

  • convert material from part of graduate-level textbook or a scientific paper into a coherent and comprehensible presentation for fellow students and mathematicians in general.
  • choose appropriate means of communicating theoretical mathematics to fellow students and mathematicians, in written and oral form.
  • Formulate and grade exercises that keep a balance between relevance, interest, and feasibility.
  • Work together as a team, thereby acquiring collaboration skills.
  • Provide reasonable feedback to fellow students.

Evaluation matrix: 
  Presentation(s)Homework Understanding the material2050Effective communication of the material200Formulating and correcting homework
 010
  

Seminar Ergodic Theory

There is no content available for this course.

Introduction to Complex Systems

There is no content available for this course.

Laboratory Class Scientific Computing

Content: This course is part of the Scientific Computing track of the Mathematical Sciences master's programme.
There are two major subjects that are treated in the course:

  • (pseudo)random number generation together with the application of (pseudo)random numbers in Monte Carlo integration for high dimensional integrals
  • Genetic Algorithms for determining a(n approximate) minimiser of a hard NP-complete) optimisation problem (as the traveling salesman problem, minimum-energy charge configuration problem, knapsack problem, etc.).

A review is given of the basic concepts of Probability Theory and Statistics as required for a proper treatment of the course subjects.
The subjects of the course are related: Genetic Algorithms, as well as Monte Carlo rely on random numbers, as is the case for many other algorithms for scientific computing. It is crucial that sequences of these kind of numbers (or of numbers that could pass for being random) are highly efficiently being generated. To achieve this, all resources of a computer have to be exploited. This requires coding in a programming language as C++ that allows to control the use of memory and can exploit the possibilities of the central process unit (CPU). Actually, the primary goal of the course is to teach students to develop efficient computer programs in an object oriented language as C++ for solving large scale Scientific Computing problems.
Achieving high efficiency does not only depend on the efficiency of the coding but also on the choice of the solution methods and parameters for these: they have to be tuned to the Scientific Computing problem at hand. This requires insight in the mathematical background of both the solution method and the Scientific Computing problem.
In practice, a computational scientist works in a team which makes reporting extra important. The students have to write two reports on the results that they obtain with their codes.

Organisation: The course consists of weekly, four hour long sessions of which the first hour is used by the teacher to discuss the theory from the lecture notes and the remaining time is used by the students to work on their assignments and reports.
For both subjects as mentioned above there is an existing C++ code base, from which almost all essential parts are missing, that the students need to explore and complete. This exploration is guided by assignments from the lecture notes and course website, of which a number have to be handed in for grading. After the students have completed the assignments and turned the code base into a working program, they need to investigate the theoretical results from the lecture notes by performing experiments with the programs they have written, which results in a report. Two such reports have to be written (in English and formatted in LaTeX): one concerning (pseudo)random number generation and Monte Carlo, and one concerning Genetic Algorithms. For the final report the students choose either the traveling salesman problem, the minimum-energy charge configuration, or the knapsack problem to test the effectiveness of the Genetic Algorithm software they have developed.

Examination: The final grade is based on two reports and a few hand-in assignments. Each report makes up 40% of the student's grade. Together, the two reports account for 80% of the student's grade. The remaining 20% is determined by the average grade of the hand-in assignments.

Note: The student is not allowed to use (partial) texts in their reports written by others and use them within their own reports without proper quotation and citation. This includes slightly adapting the source texts.

Prerequisites: Basic knowledge of programming in an imperative language (C, C++, Java); also knowledge of numerical methods (particularly numerical errors, basic integration techniques, and error estimates). two reports
80%hand in assignment
20%is familiar with the basic concepts of Probability Theory (and Statistics), (LC)random number generators (RNGs), Monte-Carlo methods, Genetic Algorithms (GAs), and some local search methodsxxis able to obtain insight on how to judge on quality of RNGs and the applicability of the other methodsxxis able to implement advanced algorithms in C or C++, to work with object-oriented classes, and to correctly apply concepts such as inheritance and polymorphismxxis able to use external libraries written by third parties, and will know how to write accessible codexxis able to write a coherent and concise reportsx

Algorithms in Finance

There is no content available for this course.

PhD Research Training

There is no content available for this course.

Research Project: Proposal

There is no content available for this course.

Research Project: Thesis

At the end of a master project, the student is able to

  1. Study relevant literature and gain in-depth knowledge in a certain mathematical topic;
  2. Conduct research in the field of mathematical sciences and report on it in a manner that meets customary standards of the discipline;
  3. Work together on a research team (e.g., in a hierarchical team of supervisor and junior member(s), together with peers or as a trainee in a company's research team or unit);
  4. Communicate conclusions both written and orally as well as the underlying knowledge, grounds and considerations to various audiences in English (e.g. the research team, fellow researchers in the same area and master students in the same general area of mathematics);
  5. Judge and evaluate mathematical research and publications;
  6. Independently perform literature searches;
  7. Enroll in a Ph.D. programme in mathematics or begin a career as a professional mathematician.

The final mark for a master project is built out of three marks which measure how well the student performed at achieving the learning goals. A detailed document with guidelines for the evaluation of the master project is published on the students website.
process 30%thesis 50%defence 20%is able to study relevant literature and gain in-depth knowledge in a certain mathematical topicxxxis able to conduct research in the field of mathematical sciences and report on it in a manner that meets customary standards of the disciplinexxxcan work together on a research team (e.g., in a hierarchical team of supervisor and junior member(s), together with peers or as a trainee in a company's research team or unit)x xcan communicate conclusions both written and orally as well as the underlying knowledge, grounds and considerations to various audiences in English (e.g. the research team, fellow researchers in the same area and master students in the same general area of mathematics)xxxis able to judge and evaluate mathematical research and publicationsxx can independently perform literature searchesx can enroll in a Ph.D. programme in mathematics or begin a career as a professional mathematicianxxx
The average mark is the weighted average of the three marks for process, thesis and defence. The final mark is the minimum of the average mark and the mark for the thesis. Accordingly, in order for the student to pass, both the mark for the thesis and the average mark should be at least six.

Orientation on Mathematical Research (compulsory)

There is no content available for this course.