Master's Programme in Mathematics and Operations Research
Curriculum 2020–2022
Director of degree programme: Professor Antti Hannukainen
Degree: Master of Science (Technology)
Extent: 120 ECTS
Abbreviation: MathOR
The degree of Master of Science in Technology consists of a major, an optional minor, master's thesis, and elective studies. The Master's Programme in Mathematics and Operations Research offers the following majors:
- Applied Mathematics
- Mathematics
- Systems and Operations Research
The majors in Applied Mathematics and Mathematics are long or compact. 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.
Long major
Major (55–65 ECTS)
Master's thesis (30 ECTS)
Elective studies (25–35 ECTS)
Compact major
Major (40–45 ECTS)
Master's thesis (30 ECTS)
Minor (20–25 ECTS)
Elective studies (25–30 ECTS)
Each student is assigned an academic advisor who guides the student through the programme. The advisor helps the student in writing a personal study plan, and can be consulted in various types of study-related matters, such as selecting a master's thesis supervisor.
The objective of the Master's Programme in Mathematics and Operations Research is to train 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, and to analyze them using advanced methods and algorithms. 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 the Master's Programme in 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. Thus, a large share of the students continue their studies to a doctoral degree.
Applied Mathematics major
Professor in charge: Nuutti Hyvönen
Extent: 55–65 ECTS (long major) or 40–45 ECTS (compact major)
Abbreviation: AM
Code: SCI3053
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.
The major in Applied Mathematics can be taken either as a long major (55–65 ECTS) or a compact major (40–45 ECTS). The student's personal academic advisor provides assistance in planning the curriculum details.
Mandatory studies (15 ECTS)
| Code | Course name | ECTS credits | Period | Recommended year |
|---|---|---|---|---|
| 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 |
Core studies (25–45 ECTS)
Select 25–45 credits for a long major or 25–30 credits for a compact major. In both cases, select at least one course from each of the three categories below.
Computational mathematics
| Code | Course name | ECTS credits | Period | Recommended year |
|---|---|---|---|---|
| MS-E1142 | Computational Algebraic Geometry | 5 |
III (2020–2021) V (2021–2022) |
1 or 2 |
| MS-E1150 | Matrix Theory | 5 | II (2020–2021) | Fall 2020 |
| MS-E1461 | Hilbert Spaces | 5 | I | 1 or 2 |
| MS-E1652 | Computational Methods for Differential Equations | 5 | II | 1 or 2 |
| MS-E1653 | Finite Element Method | 5 | III–IV | 1 or 2 |
| MS-E1654 | Computational Inverse Problems | 5 | IV | 1 or 2 |
Optimization
| Code | Course name | ECTS credits | Period | Recommended year |
|---|---|---|---|---|
| MS-E2121 | Linear Optimization | 5 | III-IV | 1 |
| MS-E2134 | Decision Making and Problem Solving | 5 | I-II | 1 or 2 |
| MS-E2148 | Dynamic Optimization | 5 | III | 1 or 2 |
Probability and statistics
| Code | Course name | ECTS credits | Period | Recommended year |
|---|---|---|---|---|
| MS-E1600 | Probability Theory | 5 | III | 1 |
| MS-E1602 | Large Random Systems | 5 | IV (2020–2021) | Spring 2021 |
| MS-E1603 | Random Graphs and Network Statistics | 5 | I | 1 or 2 |
| MS-E1621 | Algebraic Statistics | 5 | I–II (2020–2021) | Fall 2020 |
| CS-E5710 | Bayesian Data Analysis | 5 | I–II | 1 or 2 |
Students taking a long major select in addition sufficiently many courses in mathematics or other mathematical sciences to obtain a total of 55–65 credits. Courses not in the below list of recommended courses may also be included with the consent of the professor in charge.
Recommended courses
| Code | Course name | ECTS credits | Period | Recommended year |
|---|---|---|---|---|
| MS-E1050 | Graph Theory | 5 | I | 1 or 2 |
| MS-E1052 | Combinatorial Network Analysis | 5 | II (2021–2022) | Fall 2021 |
| MS-E1110 | Number Theory | 5 | II | 1 or 2 |
| MS-E1111 | Galois Theory | 5 | IV (2020–2021) | Spring 2021 |
| MS-E1687 | Advanced Topics in Cryptography | 5 | III–IV | 1 or 2 |
| CS-E3190 | Principles of Algorithmic Techniques | 5 | I–II | 1 or 2 |
| CS-E4710 | Machine Learning: Supervised Methods | 5 | I-II | 1 or 2. |
| CS-E4500 | Advanced Course in Algorithms | 5 | III–IV | 1 or 2 |
| CS-E4510 | Distributed Algorithms | 5 | I–II | 1 or 2 |
| CS-E4340 | Cryptography | 5 | I-II | 1 or 2 |
| CS-E4530 | Computational Complexity Theory | 5 | IV-V | 1 or 2 |
| CS-E4555 | Combinatorics | 5 | V | 1 or 2 |
| CS-E4580 | Programming Parallel Computers | 5 | V | 1 or 2 |
| CS-E5740 | Complex Networks | 5 | I–II | 1 or 2 |
| CS-E5745 | Mathematical Methods for Network Science | 5 | III | 1 or 2 |
| PHYS-E0412 | Computational Physics | 5 | III–V | 1 or 2 |
Mathematics major
Professor in charge: Juha Kinnunen
Credits: 55–65 ECTS (long major) or 40–45 ECTS (compact major)
Abbreviation: MA
Code: SCI3054
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 projects 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, mathematical physics, matrix theory, partial differential equations, probability theory, and real analysis.
Mathematics is a versatile major: the student has the opportunity to choose her/his field of mathematics with no pre-assigned restrictions. The course content of the major is planned under the supervision of the student's personal academic advisor, with emphasis on the mathematics courses lectured at the Department of Mathematics and Systems Analysis. You have the opportunity to include a minor in your studies. Depending on whether a minor is included or not, the extent of the major is 40–45 credits or 55–65 credits.
Recommended courses include:
| Code | Course name | ECTS credits | Period | Recommended year |
|---|---|---|---|---|
| MS-E1050 | Graph Theory | 5 | I | 1 |
| MS-E1052 | Combinatorial Network Analysis | 5 | II (2021-2022) | Fall 2021 |
| MS-E1110 | Number Theory | 5 | II | 1 |
| MS-E1111 | Galois Theory | 5 | IV (2020–2021) | Spring 2021 |
| MS-E1142 | Computational Algebraic Geometry | 5 | III (2021), V (2022) |
1 or 2 |
| MS-E1143 | Algebraic Geometry 1 | 5 | I (2021–2022) | Fall 2021 |
| MS-E1144 | Algebraic Geometry 2 | 5 | II (2021–2022) | Fall 2021 |
| MS-E1150 | Matrix Theory | 5 | II (2020-2021) | Fall 2020 |
| MS-E1200 | Lie Groups and Lie Algebras | 5 | IV (2021–2022) | Spring 2022 |
| MS-E1280 | Measure and Integral | 5 | II | 1 |
| MS-E1281 | Real Analysis | 5 | IV (2021–2022) | Spring 2022 |
| MS-E1461 | Hilbert Spaces | 5 | I | 1 |
| MS-E1462 | Banach Spaces | 5 | II (2020-2021) | Fall 2020 |
| MS-E1531 | Differential Geometry | 5 | III (2021-2022) | Spring 2022 |
| MS-E1600 | Probability Theory | 5 | III | 1 |
| MS-E1602 | Large Random Systems | 5 | IV (2020–2021) | Spring 2021 |
| MS-E1687 | Advanced Topics in Cryptography | 5 | III–IV | 1 or 2 |
| CS-E4555 | Combinatorics | 5 | V | 1 or 2 |
In addition, the student can take courses in applied mathematics, operations research, or other mathematical sciences.
Examples of possible orientations
Algebra and discrete mathematics
- MS-E1050 Graph Theory
- MS-E1110 Number Theory
- MS-E1111 Galois Theory
- MS-E1142 Computational Algebraic Geometry
- MS-E1143 Algebraic Geometry 1
- MS-E1144 Algebraic Geometry 2
- MS-E1200 Lie Groups and Lie Algebras
- other courses, and an optional minor chosen under the guidance of the personal academic advisor.
Analysis and partial differential equations
- MS-E1280 Measure and Integral
- MS-E1281 Real Analysis
- MS-E1461 Hilbert Spaces
- MS-E1462 Banach Spaces
- other courses, and an optional minor chosen under the guidance of the personal academic advisor
Mathematical physics
Systems and Operations Research major
Professors in charge: Ahti Salo
Credits: 40–42 ECTS (compact major)
Abbreviation: OR
Code: SCI3055
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 Science of Better”. 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.
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.
Mandatory courses (30 cr)
| Code | Course name | ECTS credits | Period | Recommended Year |
|---|---|---|---|---|
| MS-E2112 | Multivariate Statistical Analysis | 5 | III-IV | 1. (or earlier) |
| MS-E2121 | Linear Optimization | 5 | III-IV | 1. (or earlier) |
| MS-E2122 | Nonlinear Optimization | 5 | I-II | 2. |
| MS-E2133 | Systems Analysis Laboratory II | 5 | I-II | 2. |
| MS-E2134 | Decision Making and Problem Solving | 5 | I-II | 1. |
| MS-E2148 | Dynamic Optimization | 5 | III | 1. |
Select at least one seminar:
| Code | Course name | ECTS credits | Period | Recommended Year |
|---|---|---|---|---|
| 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. |
If organized, MS-E2142 Seminar on optimization also qualifies.
Select further courses from the following list to complete the required 40-42 credits:
| Code | Course name | ECTS credits | Period | Recommended Year |
|---|---|---|---|---|
| MS-E2108 | Independent Research Projects in Systems Analysis | 5-8 | I, II, III, IV, V | 1. or 2. |
| MS-E2114 | Investment Science | 5 | I–II | 1. or 2. (or earlier) |
| MS-E2117 | Riskianalyysi | 5 | III–IV | 1. (or earlier) |
| MS-E2129 | Systeemien identifiointi | 5 | I–II (every other year) | 1. or 2. |
| MS-E2130 | Matemaattinen malliajattelu | 5 | I-II | 1. or 2. |
| MS-E2170 | Simulation | 5 | III | 1. (or earlier) |
| CS-E4710 | Machine Learning: Supervised Methods | 5 | I-II | 1. or 2. |
All other MS-E2XXX courses also qualify.
Minor (20–25 cr)
Students taking a compact major must also take a minor (20–25 ECTS). Students taking a long major are encouraged to include a minor in their elective studies. The student's academic advisor can provide assistance in selecting a suitable minor. 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 on Aalto University’s minors:
Elective studies 25–30 cr
Students choose 25–35 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 conset of the programme director, other Finnish universities through Flexible Study Right (JOO).
Also, studies completed abroad during student exchange can be included in the elective studies (exchange studies can also form an international minor or be included in the major) with the consent of the programme director. Work experience (at most 10 credits) completed during an internship in Finland or abroad can also be included in the elective studies.
More information about studies abroad and internships:
In Aalto School of Science, students have compulsory language studies as a part of the bachelor’s degree. If these language studies have not been completed before entering the master’s programme, they must be included in the master’s degree as explained below:
A student whose language of school education is Finnish or Swedish must take 2 credits in the other national language. In addition, all students must demonstrate both oral (o) and written (w) proficiency in one foreign language (3 credits).
Students with excellent command of English (e.g. English as a first language) may apply for the exemption from the compulsory foreign language requirement and take 3 ECTS of Finnish courses instead. In this case, according to the Degree Regulations, the student has not demonstrated the requisite written and oral language requirement in a foreign language, which is reflected in the appendix of the degree certificate. Students may apply for an exemption in the beginning of each term (deadlines 15 September and 15 January) with an application form.
The language studies are included in the elective studies.
More information about language courses can be found on the Language Centre’s website.
Master's thesis 30 cr
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.
More information about the aims and requirements of the master's thesis project: