Master's Programme in Mathematics, 120 Credits

Entry Requirements

A Bachelor's degree or equivalent first-cycle qualification comprising of at least 180 ECTS or a corresponding qualification from an internationally recognised university. Specific entry requirements are at least 90 ECTS in Mathematics, at least 7,5 ECTS in Mathematical Statistics and at least 7,5 ECTS in Programming Methodology, or equivalent. Proficiency in English equivalent to Swedish upper secondary course English B/6.

Degree

After a completed programme of 120 credits, or 60 credits, the student can apply for and obtain a Degree of Master of Science (120 credits) or a Degree of Master of Science (60 credits) in accordance with local qualification descriptor established by the Vice-Chancellor, see https://www.umu.se/student/mina-studier/examen/krav-och-huvudomraden/examensbeskrivningar.
In Swedish, Degree of Master of Science (60 credits) and Degree of Master of Science (120 credits) are called Filosofie magisterexamen and Filosofie masterexamen, respectively. The degree is awarded in the main field of study (Mathematics).

Description of the education for current education cycle

The education is at an advanced level (second cycle). The aims for second-cycle courses and study programmes are set out in the Higher Education Act, Chapter 1 Section 9. Second-cycle courses and study programmes shall involve the acquisition of specialist knowledge, competence and skills in relation to first-cycle courses and study programmes, and in addition to the requirements for first-cycle courses and study programmes shall:

• further develop the ability of students to integrate and make autonomous use of their knowledge,
• develop the students' ability to deal with complex phenomena, issues and situations, and
• develop the students' potential for professional activities that demand considerable autonomy, or for research and development work.

National goals for current degree

Degree of Master of Science (60 credits)

Knowledge and understanding
For a Degree of Master (60 credits) the student shall

• demonstrate knowledge and understanding in the main field of study, including both an overview of the field and specialised knowledge in certain areas of the field as well as insight into current research and development work, and
• demonstrate specialised methodological knowledge in the main field of study.

Competence and skills
For a Degree of Master (60 credits) the student shall

• demonstrate the ability to integrate knowledge and analyse, assess and deal with complex phenomena, issues and situations even with limited information,
• demonstrate the ability to identify and formulate issues autonomously as well as to plan and, using appropriate methods, undertake advanced tasks within predetermined time frames,
• demonstrate the ability in speech and writing to report clearly and discuss his or her conclusions and the knowledge and arguments on which they are based in dialogue with different audiences, and
• demonstrate the skills required for participation in research and development work or employment in some other qualified capacity.

Judgement and approach
For a Degree of Master (60 credits) the student shall

• demonstrate the ability to make assessments in the main field of study informed by relevant disciplinary, social and ethical issues and also to demonstrate awareness of ethical aspects of research and development work,
• demonstrate insight into the possibilities and limitations of research, its role in society and the responsibility of the individual for how it is used, and
• demonstrate the ability to identify the personal need for further knowledge and take responsibility for his or her ongoing learning.

Degree of Master of Science (120 credits)

Knowledge and understanding
For a Degree of Master (120 credits) the student shall

• demonstrate knowledge and understanding in the main field of study, including both broad knowledge of the field and a considerable degree of specialised knowledge in certain areas of the field as well as insight into current research and development work, and
• demonstrate specialised methodological knowledge in the main field of study.

 
Competence and skills
For a Degree of Master (120 credits) the student shall

• demonstrate the ability to critically and systematically integrate knowledge and analyse, assess and deal with complex phenomena, issues and situations even with limited information,
• demonstrate the ability to identify and formulate issues critically, autonomously and creatively as well as to plan and, using appropriate methods, undertake advanced tasks within predetermined time frames and so contribute to the formation of knowledge as well as the ability to evaluate this work
• demonstrate the ability in speech and writing both nationally and internationally to clearly report and discuss his or her conclusions and the knowledge and arguments on which they are based in dialogue with different audiences, and
• demonstrate the skills required for participation in research and development work or autonomous employment in some other qualified capacity.

 
Judgement and approach
 
For a Degree of Master (120 credits) the student shall

• demonstrate the ability to make assessments in the main field of study informed by relevant disciplinary, social and ethical issues and also to demonstrate awareness of ethical aspects of research and development work,
• demonstrate insight into the possibilities and limitations of research, its role in society and the responsibility of the individual for how it is used, and
• demonstrate the ability to identify the personal need for further knowledge and take responsibility for his or her ongoing learning.

Examination format

Each syllabus sets out the examination formats used in each individual course.

Grades

Each syllabus sets out the grades used in the course.

Transfer of Credits

A student who believes to have gained knowledge from previous relevant studies or professional experience that may be equivalent to a course or part of a course in the programme can apply for transfer of credits. Granting a transfer of credits means that the student will not have to study the parts of the programme included in the decision. Information on transfer of credits is available on Umeå University's website.

General

Requirements for a Degree of Master of Science (60 credits) in Mathematics at Umeå University
Degree is awarded once the student has completed the course requirements of 60 credits, of which at least 45 credits at a second-cycle level. Within the completed courses at a second-cycle level, at least 30 credits shall cover the main field of study, Mathematics. In addition, a Degree of Bachelor of at least 180 credits or equivalent foreign qualification is required. For this qualification, the student shall, within the scope of the course requirements, have completed an independent project (degree project) of at least 15 credits at a second-cycle level in the main field of study, Mathematics. 

Requirements for a Degree of Master of Science (120 credits) in Mathematics at Umeå University
Degree is awarded once the student has completed the course requirements of 60 credits, of which at least 45 credits at a second-cycle level. Within the completed courses at a second-cycle level, at least 60 credits shall cover the main field of study, Mathematics. In addition, a Degree of Bachelor of at least 180 credits or equivalent foreign qualification is required. For this qualification, the student shall, within the scope of the course requirements, have completed an independent project (degree project) of at least 30 credits at a second-cycle level in the main field of study, Mathematics. The degree project may be replaced by a degree project of at least 15 credits if the student has already completed an individual assignment at a second-cycle level of at least 15 credits in the main field of study covered by the qualification and this work also shall be included in the qualification.

Compulsory courses
Compulsory courses are courses that all students enrolled in the programme normally study. A student enrolled in the study programme is guaranteed a seat in all compulsory courses, provided that the entry requirements for the course in question are met. Entry requirements are set out in each respective course syllabus. The list below names compulsory courses in Mathematics that are usually studied within the programme and are included in the qualification. The courses are listed in alphabetical order under each respective category.

Compulsory courses at a second-cycle level in Mathematics
5MA200 Research in the Mathematical Sciences, 7.5 credits
5MS073 Probability Theory, 7.5 credits
5MA184 Numerical Methods for Partial Differential Equations, 7.5 credits
5MA180 Stochastic Differential Equations 7.5 credits 

Degree projects
5MA128 Thesis Project for a Degree of Master of Science (60 credits) in Mathematics, 15 credits
5MA194 Thesis Project for a Degree of Master of Science (120 credits) in Mathematics, 30 credits

Elective courses
Elective courses are a selection of courses that Umeå University offers within the scope of the programme and where the student chooses which courses to enrol in. The student is guaranteed a seat in one of these courses, provided that the entry requirements for the courses in question are met. However, the student is not guaranteed a seat in their first choice courses. Entry requirements are set out in each respective course syllabus.
 
Elective courses at a second-cycle level in Mathematics
5MA201 Extremal and Probabilistic Combinatorics, 7.5 credits
5MA212 Algebraic and Geometric Combinatorics, 7.5 credits
5MA177 Integer Programming, 7.5 credits
5MA176 The Finite Element Method, 7.5 credits
5MA175 Financial Mathematics, 7.5 credits
5MA178 Monte Carlo Methods for Financial Applications, 7.5 credits
5MA179 Enterprise Risk Management, 15 credits
5MA210 Mathematical Modeling of Economic Systems, 7.5 credits
5MA203 Graph Theory, 7,5 credits
5MA202 Transform methods, 7.5 credits

 
Elective courses at a second-cycle level in Mathematical Statistics
5MS084 Statistical Learning with High-Dimensional Data, 7.5 credits
5MS083 Stationary Stochastic Processes, 7.5 credits
5MS085 Time Series Analysis, 7.5 credits
5MS077 Current Research in Mathematical Statistics, 7.5 credits
5MS056 Multivariate Data Analysis

 
Elective courses at a second-cycle level in Computing Science
5DV200 Computational Complexity, 7.5 credits
5DV182 Efficient Algorithms, 7.5 credits
 
Elective courses at a second-cycle level in Physics
5FY167 Computational Fluid Dynamics, 7.5 credits
5FY188 Monte Carlo Simulations of Critical Phenomena in Physics, 7.5 credits

Elective courses at a first-cycle level in Mathematics
5MA171 Continuous Optimization, 7.5 credits

Elective courses at a first-cycle level in Physics
5FY126 Information, Networks and Markets, 7.5 credits
 
Compulsory profile courses
The two-year master's programme has three main tracks or profiles:

• Computational Mathematics
• Financial Mathematics
• Discrete Mathematics.

Each of these tracks correspond to a specialisation and to a suggested package of elective courses.

 
Specialisation in Computational Mathematics:
5MA176 The Finite Element Method, 7.5 credits
5FY167 Computational Fluid Dynamics, 7.5 credits
5MA177 Integer Programming, 7.5 credits
5DV182 Efficient Algorithms, 7.5 credits
5DV200 Computational Complexity, 7.5 credits
5MA171 Continuous Optimization, 7.5 credits
5MS084 Statistical Learning with  High-Dimensional Data Analysis, 7.5 credits
5MA202 Transform Methods 7.5 credits

Specialisation in Financial Mathematics:
5MA175 Financial Mathematics, 7.5 credits
5MA178 Monte Carlo Methods for Financial Applications, 7.5 credits
5MA179 Enterprise Risk Management, 15 credits
5MA210 Mathematical Modelling of Economic Systems, 7.5 credits
5FY126 Information, Networks and Markets, 7.5 credits
5MS056 Multivariate Data Analysis, 7.5 credits
5MA202 Transform Methods 7.5 credits
5MS085 Time Series Analysis, 7.5 credits
5MS084 Statistical Learning with  High-Dimensional Data Analysis, 7.5 credits
5MS083 Stationary Stochastic Processes, 7.5 credits
5MS077 Current Research Topics in Mathematical Statistics

Specialisation in Discrete Mathematics:
5MA201 Extremal and Probabilistic Combinatorics, 7.5 credits
5MA212 Algebraic and Geometric Combinatorics, 7.5 credits
5MA177 Integer Programming, 7.5 credits
5MA203 Graph Theory, 7,5 credits
5FY188 Monte Carlo Simulations of Critical Phenomena in Physics, 7.5 credits
5FY126 Information, Networks and Markets, 7.5 credits
5DV182 Efficient Algorithms, 7.5 credits
5DV200 Computational Complexity, 7.5 credits
5MS083 Stationary Stochastic Processes, 7.5 credits
5MA202 Transform Methods 7.5 credits

Free electives
Free electives within the programme are applied for in open competition. Free electives can be studied at Umeå University or at other higher education institutions in Sweden or abroad.

Programme overview
The courses included in the programme are listed under the heading 'Study Plan' in the order they are studied.

Degree project/independent project
The degree project concludes the programme and may be initiated once the entry requirements in the course syllabus are met. In the degree project, comprising 30 credits for a Degree of Master (120 credits) or 15 credits for a Degree of Master (60 credits), the student shall apply the knowledge acquired during their studies and present the result orally and in a written report/thesis. The work shall include some form of subject-specific specialisation within the field. The degree project is usually completed individually. However, it is also occasionally permitted for two students to cooperate on a degree project. The degree project can advantageously be completed in cooperation with the business world. A client supervisor shall be appointed and act as the student's day-to-day contact and support during the course of the work. A thesis supervisor at the university shall always be appointed and be responsible for ensuring that the required subject specialisation is achieved. The report shall be linguistically and stylistically designed to ensure its quality is equivalent to reports published within the university and the industry. The report shall include an English abstract and an English translation of the title. Alternatively, the entire report may be written in English.

Deferment of studies

Information on deferment of studies is available on Umeå University's website.

Approved leave from studies

Information on approved leave from studies is available on Umeå University's website.

Discontinuation

Information on discontinuation is available on Umeå University's website.