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Master's Programme in Mathematics and Operations Research
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Curriculum 2024–2026
About the programme
Director of degree programme: Professor Lasse Leskelä
Degree: Master of Science (Technology)
Extent: 120 ECTS
Abbreviation: MathOR
The Master’s Programme in Mathematics and Operations Research offers the following majors:
- Mathematics
- Applied Mathematics
- Systems and Operations Research
The majors in Mathematics and Applied Mathematics are compact or long. The major in Systems and Operations Research is compact. Students taking a compact major must also take a minor. Students taking a long major may include a minor in their elective studies.
Master's Programme in Mathematics and Operations Research trains experts with a broad knowledge of mathematical methods and strong problem solving skills for tackling challenging problems in science, industry, and society.
The students learn to think mathematically. They also learn to build mathematical and statistical models, to analyse them using advanced methods and algorithms, and to communicate the results effectively. Through their choice of major, the students can focus on mathematical theory, computational and statistical methods, or modelling and problem solving skills that are needed in practice.
Graduates from Mathematics and Operations Research are in growing demand in many industries and in the public sector. This growth is driven by the ability to collect more data about a great variety of phenomena, which together with advances in mathematical methods and greater computational power makes it possible to apply mathematical skills ever more extensively.
All three majors provide an excellent basis for building a career in scientific research, and many graduates continue their studies in a doctoral programme.
Total
120
ECTS
40–65
ECTS
ECTS
20–25
ECTS
25–30
ECTS
* Compact major 40-45 ECTS. Long major 60-65 ECTS. Systems and Operations Research is only offered as a compact major.
** Contains compulsory language studies.
Major 40–65 ECTS
Extent: 40–45 ECTS (compact major) or 60–65 ECTS (long major)
Professor in charge: Juha Kinnunen
Code: SCI3054
Objectives
The major in Mathematics helps students develop their mathematical thinking so that they understand how mathematical theories are constructed and how mathematical problems are formulated and solved. The education is research-based with all courses taught by mathematicians who lead active research project related to the course topics. This major is partly intended as a preparation for students who plan to become researchers or university teachers in mathematics or related sciences. A master’s degree in mathematics also provides the student with a broad range of skills in problem solving, logical reasoning, and flexible thinking, which are attributes valued throughout the society. The covered mathematical areas include abstract and commutative algebra, algebraic geometry, complex analysis, differential geometry, graph theory, harmonic analysis, mathematical physics, matrix theory, partial differential equations, probability theory, and real analysis.
Structure
Mathematics is a versatile major: students can choose their field of mathematics with no preassigned restrictions. Students may include a minor in their studies. Depending on whether a minor is included or not, the extent of the major is 40–45 credits or 60–65 credits.
Recommended courses
| Code | Course name | ECTS | Period/year |
|---|---|---|---|
| MS-E1050 | Graph Theory | 5 | I/1 |
| MS-E1052 | Combinatorial Network Analysis | 5 | II (2025-2026)/1 or 2 |
| MS-E1082 | Special course: Advanced Topics in Algebra | 5 | IV (2024-2025)/1 or 2 |
| MS-E1110 | Number Theory | 5 | II/1 |
| MS-E1111 | Galois Theory | 5 | IV (2024-2025)/1 or 2 |
| MS-E1142 | Computational Algebraic Geometry | 5 | V (2025-2026)/1 or 2 |
| MS-E1145 | Algebraic Geometry | 5 | I–II (2025-2026)/1 or 2 |
| MS-E1150 | Matrix Theory | 5 | II (2024-2025)/1 or 2 |
| MS-E1200 | Lie Groups and Lie Algebras | 5 | IV (2025-2026)/1 or 2 |
| MS-E1210 | Sobolev Spaces | 5–10 | I–V/1 or 2 |
| MS-E1215 | Elliptic Partial Differential Equations | 5–10 | I–V/1 or 2 |
| MS-E1280 | Measure and Integral | 5 | II/1 (or during BSc) |
| MS-E1281 | Real Analysis | 5 | IV (2025-2026)/1 or 2 |
| MS-E1423 | Fourier Theory | 5 | III/1 |
| MS-E1426 | Harmonic Analysis | 5 | II (2024-2025)/1 or 2 |
| MS-E1461 | Hilbert Spaces | 5 | I/1 (or during BSc) |
| MS-E1462 | Banach Spaces | 5 | II (2024-2025)/1 or 2 |
| MS-E1531 | Differential Geometry | 5 | III (2025-2026)/1 or 2 |
| MS-E1542 | General Topology | 5 | IV /1 or 2 |
| MS-E1600 | Probability Theory | 5 | I/1 |
| MS-E1604 | Brownian Motion and Stochastic Analysis | 5 | IV (2024-2025)/1 or 2 |
| MS-E1688 | Special Course: Advanced Cryptography* | 5 | I–II (2025-2026)/1 or 2 |
| CS-E4340 | Cryptography | 5 | I–II/1 |
| CS-E4370 | Applied Cryptography | 5 | III–IV/1 |
| CS-E4380 | Special Course: Advanced Cryptography* | 5 | I–II (2024-2025)/1 or 2 |
| CS-E4700 | Logic and Hard Computational Problems | 5 | I–II/1 or 2 |
* Can be completed several times as a part of major studies
In addition, the student can take courses in applied mathematics, operations research, or other mathematical sciences.
Examples of orientations
Algebra and discrete mathematics
- MS-E1050 Graph Theory
- MS-E1052 Combinatorial Network Analysis
- MS-E1082 Special Course: Advanced Topics in Algebra
- MS-E1110 Number Theory
- MS-E1111 Galois Theory
- MS-E1142 Computational Algebraic Geometry
- MS-E1145 Algebraic Geometry
- MS-E1200 Lie Groups and Lie Algebras
- MS-E1688/CS-E4380 Special Course: Advanced Cryptography
- CS-E4340 Cryptography
- CS-E4370 Applied Cryptography
- Other courses, and an optional minor
Analysis and partial differential equations
- MS-E1210 Sobolev Spaces
- MS-E1215 Elliptic Partial Differential Equations
- MS-E1280 Measure and Integral
- MS-E1281 Real Analysis
- MS-E1423 Fourier Theory
- MS-E1426 Harmonic Analysis
- MS-E1461 Hilbert Spaces
- MS-E1462 Banach Spaces
- Other courses, and an optional minor
Mathematical physics
- MS-E1145 Algebraic Geometry
- MS-E1200 Lie Groups and Lie Algebras
- MS-E1461 Hilbert Spaces
- MS-E1462 Banach Spaces
- MS-E1531 Differential Geometry
- MS-E1600 Probability Theory
- MS-E1604 Brownian Motion and Stochastic Analysis
- Other courses, and an optional minor
Extent: 40–45 ECTS (compact major) or 60–65 ECTS (long major)
Professor in charge: Lasse Leskelä
Code: SCI3053
Objectives
The major in Applied Mathematics is designed for students interested in mathematics, statistics, and their applications. It is based on a solid mathematical core that gives the student a broad set of skills for working on diverse mathematical and computational problems. The major also includes an elective part that provides flexibility to orientate toward a master's thesis project in a chosen application area. A high proportion of students taking this major continue their studies to a doctoral degree.
The importance of mathematical techniques is increasing in science and engineering as new fields employing sophisticated mathematical and statistical models are constantly emerging. The driving forces for such development are the ever-increasing computational resources, which should be used wisely and to their full power. This requires the education of mathematicians and statisticians who are able to interact and collaborate with experts in application areas. The major in Applied Mathematics responds to this need.
Structure
Applied Mathematics can be taken either as a compact major (40–45 ECTS) or a long major (60–65 ECTS).
| Code | Course name | ECTS | Period/year |
|---|---|---|---|
Compulsory studies (20 ECTS)These core courses are compulsory unless already included in previous studies. Students who have studied similar content at another university should contact the responsible professor of the major to discuss their personal study plan. |
|||
| MS-E1600 | Probability Theory | 5 | I/1 |
| MS-E1651 | Numerical Matrix Computations | 5 | I/1 |
| MS-E2112 | Multivariate Statistical Analysis | 5 | III–IV/1 |
| MS-E2122 | Nonlinear Optimization | 5 | I–II/2 |
Optional studies (20–45 ECTS)Select sufficiently many courses from the following categories to complete a total of 40–45 credits for a compact major or 60–65 credits for a long major. |
|||
Computational mathematics |
|||
| MS-E1423 | Fourier Theory | 5 | III/1 or 2 |
| MS-E1461 | Hilbert Spaces | 5 | I/1 (or during BSc) |
| MS-E1652 | Computational Methods for Differential Equations | 5 | II (2025-2026)/1 or 2 |
| MS-E1653 | Finite Element Method | 5 | III–IV/1 |
| MS-E1654 | Computational Inverse Problems | 5 | IV/1 |
| MS-E2121 | Linear Optimization | 5 | III–IV/1 |
| MS-E2148 | Dynamic Optimization | 5 | III/1 |
| CS-E5755 | Nonlinear Dynamics and Chaos | 5 | III–IV/1 |
| CS-E5795 | Computational Methods in Stochastics | 5 | I–II/2 |
Statistics and probability |
|||
| MS-C2128 | Prediction and Time Series Analysis | 5 | II/1 (or during BSc) |
| MS-E1603 | Random Graphs and Network Statistics | 5 | V (2025-2026)/1 or 2 |
| MS-E1604 | Brownian Motion and Stochastic Analysis | 5 | IV (2024-2025)/1 or 2 |
| MS-E1623 | How to Lie with Statistics? | 5 | II/1 or 2 |
| MS-E1624 | High-Dimensional Statistics | 5 | IV/1 |
| MS-E1622 | Algebraic Methods in Data Science | 5 | III–IV (2024-2025)/1 or 2 |
| MS-E1280 | Measure and Integral | 5 | II/1 (or during BSc) |
| CS-E5710 | Bayesian Data Analysis | 5 | I–II/2 |
| CS-E4825 | Probabilistic Machine Learning | 5 | III–IV/1 |
| CS-E5745 | Mathematical Methods for Network Science | 5 | III/1 or 2 |
Discrete applied mathematics |
|||
| MS-E1050 | Graph Theory | 5 | I/1 or 2 |
| MS-E1052 | Combinatorial Network Analysis | 5 | II (2025-2026)/1 or 2 |
| MS-E1142 | Computational Algebraic Geometry | 5 | V (2025-2026)/1 or 2 |
| MS-E1150 | Matrix Theory | 5 | II (2024-2025)/1 or 2 |
| MS-E2145 | Combinatorial Optimization | 5 | III–IV/1 |
| CS-E4565 | Combinatorics of Computation | 5 | V/1 |
| CS-E4720 | Computational Geometry | 5 | III–IV/1 |
Extent: 40–45 ECTS (compact major)
Professor in charge: Fabricio Oliveira
Code: SCI3055
Objectives
The objective of the major in Systems and Operations Research is to educate experts who master quantitative methods and have professional skills for solving decision problems arising in the planning and management of complex systems. Core topics in the major are optimization, statistics, forecasting, dynamic systems, simulation, and decision and risk analysis. The major includes hands-on laboratory exercises and project assignments on real problems posed by external organizations.
Overall, the major is founded on a balanced engineering-economic approach which builds on structured problem solving, systems thinking, and mathematical modelling. This approach allows operations research (OR) professionals to optimize the use of limited resources, to improve the efficiency of production and services, and to support decisions in recognition of risks and multiple objectives, for instance.
Methods of systems and operations research are used extensively in application areas such as finance, logistics, manufacturing, services, defence, energy, environment, and healthcare. OR is widely known as the scientific discipline for making “Smarter Decisions for a Better World”. The rapidly expanding field of analytics employs OR methods to provide data-driven decision support.
Graduates in Systems and Operations Research are in growing demand by firms, consultancies, research institutes, and the public sector. Many start their careers as analysts and then advance to more senior positions and even to top management. The major provides an excellent basis for continued studies and graduates often choose to pursue a doctoral degree.
Structure
All courses can be taken in the first or second year of studies, and many even during bachelor studies. The recommended years for the core courses are shown below.
| Code | Course | ECTS | Period/year |
|---|---|---|---|
Mandatory courses (20 ECTS) |
|||
| MS-E2112 | Multivariate Statistical Analysis | 5 | III–IV/1 (or during BSc) |
| MS-E2121 | Linear Optimization | 5 | III–IV/1 (or during BSc) |
| MS-E2132 | Laboratory Assignments in Operations Research II | 5 | I–II/2 |
| MS-E2135 | Decision Analysis | 5 | I–II/1 |
Seminars (5–12 ECTS)Include at least one or both seminars in your major studies: |
|||
| MS-E2177 | Seminar on Case Studies in Operations Research | 5–7 | III–V/1 or 2 |
| MS-E2191 | Graduate Seminar on Operations Research | 5 | I–II/2 |
Specialization studiesChoose sufficiently many courses from the following lists to complete the required 40–45 credits for the major. The second list contains special courses planned for 2024–2026 that might be offered at different periods than indicated below. All other MS-E2XXX courses also qualify, and additional courses may be offered. Check courses.aalto.fi for an updated list. |
|||
| MS-E2122 | Nonlinear Optimization | 5 | I–II/1 or 2 |
| MS-E2145 | Combinatorial Optimization | 5 | III–IV/1 or 2 |
| MS-E2148 | Dynamic Optimization | 5 | III/1 or 2 |
| MS-E2114 | Investment Science | 5 | I–II/1 or 2 (or during BSc) |
| MS-E2117 | Riskianalyysi** | 5 | III–IV/1 (or during BSc) |
| MS-E2130 | Mathematical Modelling* | 5 | I–II/1 or 2 |
| MS-E2150 | Discrete Event Simulation | 5 | IV (2025-2026)/1 (or during BSc) |
| MS-E2155 | System Dynamics Simulation | 5 | IV (2024-2025)/1 (or during BSc) |
| MS-E2107 | Independent Research Projects in Systems and Operations Research Analysis | 5–8 | I, II, III, IV, V/1 or 2 |
| CS-E4715 | Supervised Machine Learning | 5 | I-II/1 or 2 |
Courses on special topics: |
|||
| MS-E2120 | Suomalaisen yhteiskunnan resilienssi** | 1–5 | I-II/1 or 2 |
| MS-E2147 | Optimization in Public Transport | 5 | I–II/2 |
| MS-E2160 | Stochastic Programming and Robust Optimization | 5 | I–II/2 |
*Teaching language is English. Course can be completed in Finnish or English.
**Lectures are given in Finnish.
Master's thesis 30 ECTS
Students are required to write a master’s thesis, which is an individual research project with a workload of 30 credits. The topic of the thesis is usually related to the student’s major, or in some special cases to a minor. The thesis work must have one supervisor and may have one or two advisors. The supervisor is a professor at Aalto University who ensures that the thesis meets all aims and requirements set by the School of Science. The advisor is usually from an organization for which the thesis is written. The thesis advisor shall hold at least a master’s degree. The advisor is an expert in the field of the thesis, who can give advice on content and writing of a thesis. The duties of the advisor are agreed on by the student, supervisor, and advisor.
Master’s thesis work also includes a maturity essay, and a seminar presentation or an equivalent presentation.
For more information about the master's thesis process, please see MyCourses >>
Minor 20–25 ECTS
Students taking a compact major must also take a minor (20–25 ECTS). A minor can also be included in elective studies. Students can choose a minor from minors offered by Aalto University or, with the consent of the programme director, from other Finnish universities or universities abroad. The minor is confirmed in the Personal Study Plan.
More information:
Elective studies 25–30 ECTS
Students choose 25–30 credits of elective studies. As elective studies, students can complete a minor and/or take individual courses. Individual elective courses can also be taken from other programmes at Aalto University or, with the consent of the programme director, other Finnish universities through Flexible Study Right (JOO).
Elective studies must be university-level studies that fulfil the degree requirements and, in general, studies that are offered as degree studies at the university in question. Universities also offer courses that are targeted for a larger audience. The suitability of these studies is evaluated taking into consideration the learning outcomes of the degree that the courses are planned to be included in. Elective studies may not overlap with student's other studies. For this reason, the programme may restrict the choices in elective studies.
Also, studies completed abroad during student exchange can be included in the elective studies (exchange studies can also form an international minor) with the consent of the programme director. Work experience (1–10 ECTS) completed during an internship in Finland or abroad can also be included in the elective studies. If the course JOIN-A0003 Contributing in Community (3 ECTS) is included in the Master’s degree, only 7 credits of practical training can be accepted in the degree.
More information :
If you have compulsory language studies in your master's degree, they are included in the elective studies.
- If you have completed your bachelor's degree in Finland (in Aalto or in another higher education institute), you have fulfilled the compulsory language requirements in the respective degree or received the exemption. You do not need to complete language studies in the master's degree.
- If your language of education is Finnish or Swedish and you have completed your bachelor’s degree outside of Finland, you must
- demonstrate proficiency in national languages by writing the maturity test in your language of education (Finnish or Swedish) and complete the language proficiency tests (2 ECTS) in the other national language. Read more about the language of education here. You may also apply for an exemption of demonstrating proficiency in national languages.
- complete 3 ECTS in one foreign language (including both oral (o) and written (w) proficiency).
- If you have completed your bachelor’s degree outside of Finland, you are required to complete only 3 ECTS in one foreign language (including both oral (o) and written (w) proficiency). Students, whose language of education is not Finnish or Swedish, may alternatively complete an elementary course in Finnish or in Swedish (3 ECTS). The courses in national languages can be at any level on CEFR scale.
Language studies are included in students’ elective studies and are agreed in the personal study plan (HOPS). Language center offers the language studies.
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