Department of Mathematics

Syllabus for Master Programme in Mathematics

Masterprogram i matematik

A later update of this programme syllabus has been published.

  • 120 credits
  • Programme code: TMA2M
  • Established: 2009-05-06
  • Established by: The Educational Board of Science
  • Revised: 2010-11-11
  • Revised by: The Educational Board of Science
  • Syllabus applies from: Autumn 2011
  • Responsible faculty: Faculty of Science and Technology

Entry Requirements

Pure Mathematics: Bachelor of Science including 90 ECTS credits in mathematics. Proficiency in English.

Financial Mathematics: Bachelor of Science including 90 ECTS credits in mathematics with at least 20 ECTS credits in probability theory, programming and/or numerical analysis. Proficiency in English.

Mathematical Statistics: Bachelor of Science including 90 ECTS credits in mathematics with at least 30 credits in mathematical statistics. Proficiency in English.

Applied Logic: Bachelor of Science including 60 ECTS credits in computer science and 60 ECTS credits in mathematics with at least 15 credits in basic logic and discrete mathematics. Proficiency in English.

Applied Mathematics: Bachelor of Science including, either 90 ECTS credits in mathematics with at least 20 credits in numerical analysis, programming and/or probability, or 60 credits in mathematics and 60 credits in Physics, Technology or Computer Science. Proficiency in English.

Decisions and Guidelines

1. Decision to offer two-year Master programmes
According to a decision taken by the Vice Chancellor 2006-09-16, Uppsala University will offer two-year Master programmes in biology, computer science, physics, earth science, sustainable development, chemistry, mathematics, computational science and applied biotechnology, all from 2007-07-01. Furthermore according to a decision taken by the Vice Chancellor 2008-06-07 a Master programme in bioinformatics from 2009-07-01.

2. Objectives for a Degree

2.1 Objectives for a Degree of Master (One Year) (Magisterexamen)

Objectives for a Degree of Master (One Year) according to the Higher Education Ordinance, System of Qualifications.

Knowledge and understanding
For a Degree of Master (One Year) students must
- demonstrate knowledge and understanding in their main field of study, including both a broad command of the field and deeper knowledge of certain parts of the field, together with insight into current research and development work; and
- demonstrate deeper methodological knowledge in their main field of study.

Skills and abilities
For a Degree of Master (One Year) students must
- demonstrate an ability to integrate knowledge and to analyse, assess and deal with complex phenomena, issues and situations, even when limited information is available;
- demonstrate an ability to independently identify and formulate issues and to plan and, using appropriate methods, carry out advanced tasks within specified time limits;
- demonstrate an ability to clearly present and discuss their conclusions and the knowledge and arguments behind them, in dialogue with different groups, orally and in writing; and
- demonstrate the skill required to participate in research and development work or to work in other advanced contexts.

Judgement and approach
For a Degree of Master (One Year) students must
- demonstrate an ability to make assessments in their main field of study, taking into account relevant scientific, social and ethical aspects, and demonstrate an awareness of ethical aspects of research and development work;
- demonstrate insight into the potential and limitations of science, its role in society and people’s responsibility for how it is used; and
- demonstrate an ability to identify their need of further knowledge and to take responsibility for developing their knowledge.

2.2 Objectives for a Degree of Master (Two Years) (Masterexamen)
Objectives for a Degree of Master (Two Years) according to the Higher Education Ordinance, System of Qualifications.

Knowledge and understanding
For a Degree of Master (Two Years) students must
- demonstrate knowledge and understanding in their main field of study, including both broad knowledge in the field and substantially deeper knowledge of certain parts of the field, together with deeper insight into current research and development work; and
- demonstrate deeper methodological knowledge in their main field of study.

Skills and abilities
For a Degree of Master (Two Years) students must
- demonstrate an ability to critically and systematically integrate knowledge and to analyse, assess and deal with complex phenomena, issues and situations, even when limited information is available;
- demonstrate an ability to critically, independently and creatively identify and formulate issues and to plan and, using appropriate methods, carry out advanced tasks within specified time limits, so as to contribute to the development of knowledge and to evaluate this work;
- demonstrate an ability to clearly present and discuss their conclusions and the knowledge and arguments behind them, in dialogue with different groups, orally and in writing, in national and international contexts; and - demonstrate the skill required to participate in research and development work or to work independently in other advanced contexts.

Judgement and approach
For a Degree of Master (Two Years) students must
- demonstrate an ability to make assessments in their main field of study, taking into account relevant scientific, social and ethical aspects, and demonstrate an awareness of ethical aspects of research and development work;
- demonstrate insight into the potential and limitations of science, its role in society and people’s responsibility for how it is used; and
- demonstrate an ability to identify their need of further knowledge and to take responsibility for developing their knowledge.

Layout of the Programme

4.7.1 Description of the programme
All courses inside the programme are advanced level courses. After completion of the programme the student graduates with a Diploma of Master of Science in Mathematics. The programme is formed in such a way that after one year of studies the student may graduate with a Diploma of Magister of Sciences in Mathematics.

The Programme has the following five specialisations:

• Mathematics
• Financial Mathematics
• Statistics
• Applied Mathematics
• Applied Logics

This division into specialisations partially reflects the traditional division of mathematics into "pure" and "applied" mathematics. Specialisation "Mathematics" contains mostly studies in pure mathematics, while the other specialisations are devoted to the studies in different branches of applied mathematics: financial mathematics, statistics, "classical" applied mathematics with methods and tools, which mostly rely on integral- and differential calculus, and applied logics, which means mathematical logics with applications to computer sciences.

Progression in the education is based on the inner logical structure of the subject "mathematics". Mathematical notions and results form a complicated logical structure, where different mathematical theories build upon each other.

During the first term a special course is given for each specialisation except the specialisation "financial mathematics". This course either reviews and presents the basic notions necessary from the basic level education in a more general context (courses Topology and Mathematical Statistics respectively) or presents a diverse collection of methods and techniques inside the specialisation (courses Applied Mathematics and Applied Logic respectively). This forms a solid ground for further special courses which either successively extend previous knowledge or lead to an essential widening of some area.

In the specialisation "financial mathematics" progression in the education is based primarily on the progression of courses Finances I, Financial Mathematics II and Financial Mathematics III. These course are obligatory for the Master's Thesis, which completes the studies. Analogously the course Financial Mathematics II is obligatory for Magister Degree.

4.7.2 Comprehensive aims of the education
Specialisation "Mathematics" is primarily designed for those who intend to continue with a Ph.D. study in mathematics or those who would like to get advanced knowledge in the subject, which after an appropriate extension may lead to professional activities outside the university.

Specialisation "Financial Mathematics" should make students well prepared for activities in the financial sector, where use of advanced mathematical, statistical and numerical methods and theories play an important role. The program should also give students the qualification to be accepted for Ph.D. study in mathematics and, in case of an appropriate choice of additional courses, to Ph.D. study in national economy or mathematical statistics.

Specialisation "Mathematical Statistics" is designed so that it should satisfy industrial needs and needs of the society for qualified mathematical statisticians and at the same time prepare for a Ph.D. study in mathematical statistics.

The specialisations "Applied Mathematics" and "Applied Logic" prepare the student for professional activity in those areas where mathematical modelling and numerical computations play a central role. Moreover, depending on the choice of courses, they qualify for Ph.D. studies in one of the mathematical subareas, such as pure mathematics, mathematical logics, numerical analysis, automatic control, system engineering, or computer sciences.

The specialisation “Applied Mathematics” equips the student with knowledge in mathematical modelling, in particular modelling of systems that display non-linear, stochastic, or chaotic behaviour. These are common in diverse contexts, as for example in biology. The applications often involve various areas of mathematics, and numerical methods are frequently applied.

The specialisation in "Applied Logic" equips the student with knowledge in mathematical logic, emphasizing skills in the application and development of formal methods in various contexts, as for instance in verification and synthesis of algorithms, programs and systems.

4.7.3 Aims as expected results of the study
Within the objectives given in the Higher Education Ordinance, the student of the programme must

• have wide and deep knowledge in mathematics
• be able to independently formulate complicated problems in mathematical form
• be able to use and develop theoretical models
• be able to apply results from different areas and carry out extensive computations and simulations to find useful solutions.

For specialisation "mathematics" the student additionally must

• be able to give an account on central notions and important results in the area
• in general be able to describe how the results are connected to each other
• be able to prove most important theorems in the area after a short preparation time.

For specialisation "Financial Mathematics" the student additionally must

• have advanced knowledge about the mathematical theories of the modern financial instruments and be able to use this knowledge to construct pricing models for advanced financial instruments
• have advanced knowledge about stochastic modelling and parabolic differential equations and be able to give an account on this on financial models
• be able to carry out extensive computations and simulations on financial models
• have advanced knowledge in macro and micro economic theory

For specialisation "Mathematical Statistics" the student additionally must

• have acquired advanced tools for stochastic modelling of random phenomena, primarily in natural sciences, medicine, technology and economy
• have deep knowledge in one of the sub areas of mathematical statistics and an ability to apply this knowledge
• be able to analyse and evaluate computer data in relevant areas.

For specialisation "Applied Mathematics" the student additionally must

• have deep knowledge of methods and tools of applied mathematics and be able to apply this knowledge in some applied area
• have extensive training in mathematical modelling and in managing of the complete chain of the following problems: mathematical model, mathematical theory, numerical methods, interpretation, validation and evaluation.

For specialisation "Applied Logic" the student additionally must

• have deep knowledge in mathematical logics and its applications in computer sciences
• have acquired the ability to specify and verify systems and programs with the help of methods, based on logics.

4.7.4 Programme outline
The courses in the programme can be seen from the outline. The extent of the courses is given in higher education credits (hp). Courses, up to 30 credits, also can be chosen from other main fields.

Degree

3.1 Regulations according to Higher Education Ordinance
A Degree of Master (One Year) is obtained after the student has completed course requirements of 60 higher education credits with a certain area of specialisation determined by each higher education institution itself, including at least 30 higher education credits with in-depth studies in the main field of study. In addition, the student must hold a Degree of Bachelor, a Degree of Bachelor of Arts in…, a professional degree worth at least 180 higher education credits or an equivalent foreign qualification.

Exceptions may be made to the requirement of a previous qualification for a student who has been admitted to the educational programme without having had basic eligibility in the form of a qualification. However, this does not apply if in the admissions process an exception has been made under Chapter 7, Section 28, second paragraph on the grounds that there has been insufficient time to issue a qualification certificate.

A Degree of Master (Two Years) is obtained after the student has completed course requirements of 120 higher education credits with a certain area of specialisation determined by each higher education institution itself, including at least 60 higher education credits with in-depth studies in the main field of study. In addition, the student must hold a Degree of Bachelor, a Degree of Bachelor of Arts in…, a professional degree worth at least 180 higher education credits or an equivalent foreign qualification.

Exceptions may be made to the requirement of a previous qualification for a student who has been admitted to the educational programme without having had basic eligibility in the form of a qualification. However, this does not apply if in the admissions process an exception has been made under Chapter 7, Section 28, second paragraph on the grounds that there has been insufficient time to issue a qualification certificate.

Independent project (degree project)
For a Degree of Master (One Year) students must have completed an independent project (degree project) worth at least 15 higher education credits in their main field of study, within the framework of the course requirements.

For a Degree of Master (Two Years) students must have completed an independent project (degree project) worth at least 30 higher education credits in their main field of study, within the framework of the course requirements. The independent project may comprise less than 30 higher education credits, but not less than 15 higher education credits, if the student has already completed an independent project at the second level worth at least 15 higher education credits in their main field of study, or an equivalent project in a foreign educational programme.

3.2 Local regulations
Main fields for a Degree of Master (One Year) and a Degree of Master (Two Years) at the Faculty of Science and Technology are Biology, Computer Science, Physics, Earth Science, Sustainable Development, Chemistry, Mathematics, Computational Science and Applied Biotechnology.

A Degree of Master (one year ) may, except for courses on advanced level, contain one or several courses on basic level comprising not more than 15 higher education credits. A degree of Master ( two years ) may, except for courses on advanced level, contain one or several courses on basic level comprising not more than 30 higher education credits. The course or the courses are meant to provide such additional competence as is needed for in-depth studies in the main field of study and cannot be included in the student’s basic degree.

For a Degree of Master (Two Years) students must have completed an independent project (degree project) worth at least 30 higher education credits

Other Directives

Students who, outside the programme, have acquired equivalent qualifications corresponding to at least 15 credits on advanced level in addition to the degree at Bachelor’s level, may apply to be accepted to a later part of the programme. The application deadline is for the autumn term May 1 and for the spring term December 1.

4.7.7 Grade and examination
Unless otherwise prescribed in the course syllabus, a grade is to be awarded on completion of a course. A student who has taken two examinations in a course or a part of a course without obtaining a pass grade is entitled to have another examiner appointed, unless there are special reasons to the contrary.

4.7.8 Courses together in a degree
Some courses cannot be considered in a degree together. Which courses this concern will be pointed out in each course syllabus.

4.7.9 Qualification and diploma
Upon request, a student who has received a pass grade in a course is to receive a course certificate from the higher education institution. Upon request, a student who meets the requirements for a qualification is to receive a diploma from the higher education institution.

A Degree of Master (One Year) is obtained after the student has completed course requirements of 60 higher education credits with a certain area of specialisation determined by each higher education institution itself, including at least 30 higher education credits with in-depth studies in Mathematics. For a Degree of Master (One Year) students must have completed an independent project (degree project) worth at least 15 higher education credits in Mathematics, within the framework of the course requirements.

A Degree of Master (Two Years) is obtained after the student has completed course requirements of 120 higher education credits with a certain area of specialisation determined by each higher education institution itself, including at least 60 higher education credits with in-depth studies in Mathematics. For a Degree of Master (Two Years) students must have completed an independent project (degree project) worth at least 30 higher education credits in Mathematics, within the framework of the course requirements. A degree of Master (Two Years) may, except for courses on advanced level, contain one or several courses on basic level comprising not more than 30 higher education credits.