Upon enrolling in the programme, you will select your track either within Fundamental Mathematics or Mathematical Modelling. Your MSc programme will consist of a few compulsory courses and a motivated choice of optional courses.
Mandatory courses and electives (75 EC)
- Mandatory courses (15 EC)
- Primary electives (45 EC)
- Secondary electives (15 EC)
Primary electives can be selected either from the local course list or from the Mastermath course list (if approved by the programme director). Secondary electives can be any course from the local course list or any other course at Master’s level (if approved by the programme director). Courses to remedy deficiencies are also counted in this category.
The course list is made up of both local courses as well as courses that are part of the national Mastermath Programme.
- Download the complete overview of the available Mastermath Courses
- Open the list below to see all Local Courses
1. Mathematical ColloquiumThe mathematics department organizes six mathematical colloquia throughout the year: four quarterly colloquia, the Monna lecture and the Kan memorial lecture.The master students of Mathematical Sciences are invited to join these lectures. For first year master students it is mandatory to attend four of these six lectures in the first year.Each student has to upload a short summary (around 200 words, half a A4) of the lecture that he/she has attended on Blackboard.
The lectures will be announced via Blackboard or via the news letter (Wisper).
2. Computational Affine GeometryMaterial
- Ideals, varieties and algorithms (4th edition) by David Cox, available through the UU Library
- Numerical methods and optimization by Eric Walter, available through the UU library
- Knowledge of the notion of affine varieties in terms of ideals, monomial orderings, division algorithm for multivariate polynomials, Dickson's Lemma, Gröbner basis, Hilbert's basis theorem, Buchberger's criterion
- Modelling a robotic arm, and knowledge of the terms forward/inverse kinematic problem, kinematic singularity, configuration space
- Numerical methods for solving systems of non-linear equations including Newton-like methods and fixed-point iterations
- Implement various methods in computer algebra system SAGE
3. History: from ellipses to elliptic curvesThe topic we present is the 18th and 19th century history leading up to ellipticcurves. The subject is accessible from bachelor level, shows how central topics of the20th century are rooted in the 19th (and earlier), and may increase the student’s awareness thatmathematics is a developing and man-made discipline.FormatWe will paint a broad general picture in the lectures, provide recommended reading (primary or secondary sources) and let the students work outthe details themselves. After one week they hand in a “textbook-style” paper of the previous lecture. There is ample room to follow your own interests. Work can be done individually or in pairs.Learning GoalsStudents give proof that they:
- understand the original motivations and contexts to create elliptic curves and related concepts;
- see the similarities and dissimilarities between contemporary mathematics and mathematics of earlier times;
- experience mathematics as a dynamically developing discipline made by real people;
- are able to write a coherent and intelligible text discussing mathematics in a historic context.
report robotica 50%report history of mathematics
50%colloquia lectures 0%orient him/herself in contemporary research in fundamental/applied mathematics xhas knowledge of the notion of affine varieties in terms of ideals, monomial orderings, division algorithm for multivariate polynomials, Dickson's Lemma, Gröbner basis, Hilbert's basis theorem, Buchberger's criterionx is able to model a robotic arm, and has knowledge of the terms forward/inverse kinematic problem, kinematic singularity, configuration spacex is able to apply numerical methods for solving systems of non-linear equations including Newton-like methods and fixed-point iterationsx is able to implement various methods in computer algebra system SAGEx understands the original motivations and contexts to create elliptic curves and related concepts x
Mathematics for Industry
* Course organisation
The course consists of 6 meetings, simulating the well-known Mathematics with Industry
Study Week format.
On the first meeting of the course, real-world industrial problems of a
mathematical nature will be presented by representatives with various industrial
backgrounds, such as software companies, banks, online shops, governmental
research institutes. The participating students will organise themselves
in groups of 8-10 students according to the problem of their interest
and will query the problem presenter in detail about all aspects of the problem,
trying to formulate it in a precise mathematical way.
In the next 4 meetings, each group will work on its problem and try to solve it,
where necessary contacting the problem owner for further input.
In the final meeting, the solution found by the group is presented in an
oral presentation, in the presence of the industrial representatives,
and a concise (10-20 page) report is handed in with a description of the
solution intended for the problem owner, including a 1-page management summary.
* Course coordinators
Prof. dr. Rob Bisseling (Mathematical Institute, Utrecht University,
Scientific Computing) and Dr. Fieke Dekkers (National Institute for
Public Health and the Environment, RIVM, and Mathematical Institute, Utrecht University).
*** Content ***
* Learning goals
After the completion of the course, the student is able to:
- translate a possibly ill-posed industrial problem into a mathematical problem
that captures the essence of the original problem
- solve this problem within a given limited amount of time, possibly
in approximated form or with additional assumptions on the input
- work together in a team with diverse backgrounds, towards a common goal
- present the solution orally in a form understandable to the original problem poser
- present the solution in a written report, which is concise but still contains
the most important insights.
The aim of the course is to provide students with industrial experience
and actual problem solving skills in an actual industrial context,
as a preparation for a future career where mathematicians contribute
their part in interdisciplinary teams working on real-life problems.
*** Entry requirements ***
Mathematical maturity in a diversity of subfields of mathematics, at the level
of having finished a bachelor degree in mathematics or equivalent.
*** Required materials ***
Bring your own laptop, for internet access, data analysis,
and possibly for running/developing software.
*** Instructional formats ***
Presentations (by problem posers and students), attendance required.
Full-day working sessions for solving problems, writing the final report, and preparing
the final presentation.
* Language: English
Final written team report 50%, final team presentation (by at most 3 team members) 20%,
active individual participation 30% (judged on the basis of attendance, activity,
and an individual log of work done).
*Evaluation matrix: report 50 %presentation 20%personal log 30%is able to translate a possibly ill-posed industrial problem into a mathematical problem
that captures the essence of the original problemx is able to solve this problem within a given limited amount of time, possibly
in approximated form or with additional assumptions on the inputx is able to recognize and describe his/her personal contribution to group work xis able to present the solution orally/in slides in a form understandable to the original problem poser x is able to present the solution in a written report, which is concise but still contains the most important insightsx
Orientation on Mathematical Research
The aim of this course is to orient a student in contemporary mathematical research by doing to projects in specialized research areas of their choice; possible specializations include algebraic geometry, number theory, differential geometry, algebraic topology, logic, scientific computing, applied analysis, probability, statistics and complex systems.
The course starts with a presentation on how to write and present research and how to work efficiently in groups.
The students choose their favorite topics from a list of possible projects proposed by staff members of the institute that function as project supervisors. The course coordinators make the final assignment of projects to students, taking into account that each project should have a group of 4-6 students working on it. A first project is assigned to be finished and graded by the middle of the semester, and a second project by the end of the semester. The student groups work on their project in consultation with the project supervisor, towards giving a presentation for their fellow students and handing in a written report. Presentation are held a bit before the (strict) hand-in deadline of the written reports.
The keep the number of required attendings reasonable, depending on the number of students, students are divided into two independent subgroups at the start of the process, or treated as one group.
Admission to the master Mathematical Sciences.
For each of the projects, the students give a (collective) presentation of 30 minutes (including questions) and write a report. Attendance at presentations is mandatory and will be checked by sign-in lists. Not every student needs to be involved in the presentation of each project, but every student needs to have presented at least one of the projects (s)he was involved in.
After the project supervisors have been consulted (at least on the process and the written thesis), the presentation (30%), process (20%) and written thesis (50%) is graded by the course coordinators. The students will receive detailed written feedback on all aspects of the project. The final grade is the average of the two project grades.
Although individual feedback can be given in the reports, both the presentation as well as the written thesis grade is collective, i.e., the same for all group members: the output is the responsibility of the entire group. Possible problems in a group need to be addressed with the course coordinators as soon as possible, and discussions a posteriori make little sense. Grades on process (involvement, etc.) can be given individually in consultation with the supervisor, and the group if necessary.
“toetsmatrijs'' written report
(process and product) 70%presentation 30%work efficiently with a team of 4-6 people on on a topic of interest in contemporary research in mathematicsxxunderstand in depth at least two contemporary research topics in mathematicsxxexplain those topics understandably and engagingly to fellow students and mathematicians in written formx the same in oral form x
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: H. Attouch, G. Buttazzo and G. Michaille, Variational analysis in Sobolev and BV spaces -Applications to PDEs and optimization. SIAM, Philadelphia, 2014. A. Braides, Gamma-convergence for beginners. Oxford University Press, Oxford, 2002. B. Dacorogna, Direct methods in the calculus of variations. Springer, New York, 2008. G. Dal Maso, An introduction to Gamma-convergence. Birkhauser Boston, 1993. L.C. Evans, Partial differential equations. American Mathematical Society, Providence, RI,1998. 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.
OrganizationInterested students should contact Carolin Kreisbeck by email (firstname.lastname@example.org). 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 email@example.com.
Internships at RIVM may be available to students participating in the seminar.
- 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.
- 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.
- 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.
- 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
The exact choice of topics to cover will depend on the participants, but will likely include the following:
- Rational homotopy groups of spheres and related spaces
- Rationalization of spaces
- Sullivan’s polynomial differential forms
- Minimal models
- Differential graded Lie algebras
- Formality, in particular of Kähler manifolds
- Elliptic and hyperbolic spaces
Berglund, Rational homotopy theory.
Félix, Halperin, and Thomas, Rational homotopy theory.
Hess, Rational homotopy theory: a brief introduction.
Students should be familiar with the contents of a first course in algebraic topology, covering singular (co)homology and CW complexes, and with basic commutative algebra (rings and modules).
The seminar is aimed at MSc and graduate students with some background in topology. Talks will be prepared and given by the participants, in close coordination with the organizers. Along with their talk speakers will also provide a handout with homework exercises for all participants, to be approved by the organizers. The speaker is responsible for grading these homeworks.
Participants are expected to give two seminar talks (each of which is a 2x45 min presentation), possibly more or less depending on the number of participants. Participants are expected to attend every seminar meeting. The final grade for the seminar is based on your talks and handouts (40%) and on your homework grades (60%).
After completion of the course, the student is able to:
- convert material from part of a 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 correct exercises that keep a balance between relevance, interest, and feasibility,
- explain specific topics from the content list of the seminar to fellow students, and put them in perspective as far as their relevance to wider mathematics is concerned.
Evaluation matrix PresentationsHomeworkUnderstanding the material20%0%Effective communication of the material20%0%Formulating and correcting homework0%10%Homework grades0%50%
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.
Schedule: Second Semester, block 3+4, 2 x 45 minutes per week
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.
Presentation(s)Homework Understanding the material2050Effective communication of the material200Formulating and correcting homework
Seminar Ergodic Theory
Content: In this seminar the following concepts will be represented:
(1) The notion of measure preserving (stationarity), Several interpretations examples and the Poincare Recurrence Theorem.
(2) The notion of ergodicity (which is a weak notion of independence), and its characterization.
(3) Ergodic Theorems (generalizations of the Strong Law of Large Numbers) such as Birkhoff and Von Neumann’s Ergodic Theorems.
(4) Some consequences of the Ergodic Theorems and the notions of weakly and strongly mixing, and isomorphism between measure preserving systems.
(5) Notion of entropy, the Shannon-Mcmillan Breiman Theorem, and Lochs Theorem.
(6) Some applications: like transformations generating several representations of numbers like continued fractions and continued fractions, irrational rotations and skew products of irrational rotations. Also some applications in percolation.
(7) Construction of invariant and ergodic measures for continuous transformations, unique ergodicity, uniform distribution and Benford’s Law.
- An Introduction to Ergodic Theory by Peter Walters, Springer Verlag.
- Ergodic Theory by Karl Petersen, Cambridge University Press.
- Parry, Topics in Ergodic Theory, Cambridge University Press.
- Ergodic theory of numbers by K. Dajani and C. Kraaikamp, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC 2002.
- Introduction to dynamical systems by M. Brin and Stuck G. Cambridge University Press, Cambridge, 2002. xii+240 pp. ISBN: 0-521-80841-3
- Ergodic Theory: with a view towards Number Theory by Manfred Einsiedler and Thomas Ward.
- Invitation to ergodic theory by Cesar Silva.Student Mathematical Library, 42. American Mathematical Society, Providence, RI, 2008. x+262 pp. ISBN: 978-0-8218-4420-5
- .Lecture Notes on Ergodic Theory by Omri Sarig.
- Some additional lecture notes written by me will be available on Blackboard and on my website.
Prerequisites: Basic knowledge of measure theory and an exposure to Functional Analysis.
Format: This seminar is aimed at master students with a background in measure theory, and some elementary knowledge of undergraduate Functional Analysis is useful in some parts. The maximum number of (active) participants of the seminar is 10.
Evaluation: Participants are expected to give two seminar talks (i.e., two times 2 x 45 minutes presentation), possibly more depending on the number of participants. They will study the material beforehand, hold a blackboard presentation about it, and make a handout. They will pose a hand-in exercise to the other seminar participants (to be handed in at the next lecture), that should be approved beforehand by the seminar organizers. The speaker is responsible for grading this hand-in exercise. In case of discussion about the solutions, the seminar organizers decide. It is obligatory to be present at all talks in this seminar (unless force majeure). The final grade for the seminar is based on the average grade that the seminar organizers give to your talks and handouts (60%), and on your homework grades (40%).
- 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 correct exercises that keep a balance between relevance, interest, and feasibility.
- explain specific topics from the content list of the seminar to fellow students, and put them in perspective as far as their relevance to wider mathematics is concerned.
PresentationsHomeworkUnderstanding the material300Effective communication300Formulating and correcting Homework010Homework Grades030
Introduction to Complex Systems
Topics. Emergence, synchronization, entropy, large deviations, resilience of complex systems, critical transitions. Applications in biology, climate science, economics, sociology, innovation science.
Prerequisites. Familiarity with elementary concepts from linear algebra (eigen values), dynamical systems and probability. Programming in mathematical software (Matlab, Mathematica).
Format. Lectures will alternate between introductions to concepts and theory of complex systems and guest lectures by researchers from other disciplines. Each student will submit two research reports involving computer simulations. There will be a final exam.
Learning goals with assessment weighting:
- read and demonstrate (in class discussions) understanding of multidisciplinary literature (20%)
- understand and be able to apply mathematical analysis and methods to carry out and write two project reports involving computer simulation (50%)
- demonstrate understanding of theoryetical concepts on final exam (30%)
in class discussion
20%project reports 50%final exam 30%is able to read and demonstrate (in class discussions) understanding of multidisciplinary literaturex understands and is able to apply mathematical analysis and methods to carry out and write two project reports involving computer simulation x is able to demonstrate understanding of theoretical concepts (emergence, synchronization, entropy, large deviations, resilience of complex systems, critical transitions) on final exam x
Laboratory Class Scientific Computation
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
Applying Mathematics in Finance
With the rise of high frequency trading on stock markets during the past decade, algorithmic trading plays an increasingly important role on stock markets, for example to exploit mispricing. In the course students will learn to build financial algorithms based on clear and feasible valuation logic. During the sessions students will learn what valuation techniques can be applied to a range of financial products. Moreover, the role of financial algorithms in modern finance and fintech will be discussed. This course is given with the support of an internationally leading firm in the field of market making. This firm trades a wide range of products: listed derivatives, cash equities, exchange–traded funds, bonds and foreign currencies. This cooperation enables students to test their algorithms by applying them to actual price data of a wide range of financial products.
The programming of algorithms will be done in Python, a programming language that is used more and more in firms that work with big data and comprehending it is therefore helpful for your future career. The students are expected to have some basic knowledge of Python; a small document discussing the expected level will be distributed via Blackboard prior to the start of the course.
The first two weeks consist of two sessions per week where we discuss the valuation of stocks, bonds, and derivatives. In addition, there are two Python workshops which are optional to join. From course week three onwards, each course week will consist of a lecture and a workshop. The lectures discuss arbitrage setups used in financial markets, such as simple pairs trading using stocks, but also delta hedging using options. The workshops will be geared towards the development of algorithms to be built to exploit mispricing in stocks or derivatives.
Lectures and workshops
To be announced
To be announced on Blackboard
PhD Research Training
- Lecture: how to write an academic cv
- Writing your own academic cv
- Study templates, possible structure
- collecting/commenting on good and bad aspects of academic cv’s (one session)
- writing your own academic cv, collecting feedback (one)
- Lecture: how to write a research proposal
- Writing your own research proposal:
- Study templates, possible structure
- Discussions with potential research project supervisor
- Collecting in-depth and state-of-the-art information on the research topic, using online tools and databases
- collecting/commenting on good and bad aspects of research projects (one)
- writing your own research proposal, presenting it, collecting feedback (at least three sessions)
Material Some references:
Prerequisites Finished first year of Mathematical Sciences Masters; admitted to UGC.
Format Homework; a few lectures; some group discussions sessions; private discussions with research project supervisor.
Schedule: Variable, to be agreed upon, in the Springer Room.
Evaluation: Participants are expected to actively study cv’s and research plan, approach supervisors, collect information, and write and present their own cv and research plan. Evaluation is Pass/Fail only and based on participation in lectures and sessions (20%), written cv (20%) and written research project (60%).
Learning goals After completion of the course, the student is able to:
- Assess academic cv’s.
- Write a convincing academic cv.
- Identify good research problems within a well-chosen topic in geometry; collect data on these; understand state-of-the-art of this topic; find a supervisor and interact with her/him.
- Write a good research proposal for PhD research.
- Communicate in written and oral form about the research proposal.
Discussions and Presentation(s)Hand-insEffective communication of and reflection on own research skills (oral)100Effective communication of and reflection on own research skills (written cv)020Effective communication of and reflection on own research plans (oral)10 Finding, identifying and formulation state-of-the-art research question, using research literature, understanding context030Effective communication of and reflection on a state-of-the-art research problem (written research proposal)030
Research project: proposal
go/no go evaluation, 100%is able to explain to his/her supervisor (in English) the research question(s) he/she is going to work on in the thesis project (research plan)xcan place the research question(s) in the context of other research/literaturexis able to explain to his/her supervisor (in English) the approach of the research plan and place this plan in the context of other methods/approaches to investigate the research question(s) at hand.x
Research project: thesis
At the end of a master project, the student is able to
- Study relevant literature and gain in-depth knowledge in a certain mathematical topic;
- Conduct research in the field of mathematical sciences and report on it in a manner that meets customary standards of the discipline;
- 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);
- 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);
- Judge and evaluate mathematical research and publications;
- Independently perform literature searches;
- 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.
Research project/Master's thesis (45 EC)
The research part of the Master’s programme consists of the thesis research project of 45 ECTS. At the end of this project you will write a Master's Thesis. This project can be carried out in the Mathematical Institute of Utrecht University or in a company in the Netherlands or abroad. You will plan and conduct your own research in one of the specialisations under the supervision of a staff member, and possibly with the help of an external advisor.