Introduction
Students from the master’s programme in Applied and Computational Mathematics will become skilled applied mathematicians, well-prepared for advanced industrial positions, or continuing graduate studies. The programme offers four tracks: Computational Mathematics, Financial Mathematics, Optimisation, and Systems Theory, and Mathematics of Data Science.
Applied and Computational Mathematics at KTH
The programme consists of foundation courses that are mandatory for all students, and once the individual specialisation track is chosen, there are relevant mandatory courses within that area as well. The programme offers four tracks: Computational Mathematics, Financial Mathematics, Optimisation, and Systems Theory, and Mathematics of Data Science.
Regardless of which track students attend, the final term consists of a degree project that may be carried out in an academic or industrial environment in Sweden or abroad. Students are welcome to discuss project ideas with the staff of the Department of Mathematics but are also encouraged to seek other contracts, in the academic world and in industry, to identify suitable projects. The result of the degree project is provided as a written report and as a presentation at a seminar.
Computational Mathematics track
The field of computer simulations is of great importance for high-tech industry and scientific/engineering research, for example, virtual processing, climate studies, fluid dynamics, and advanced materials. Thus, computational science and engineering is an enabling technology for scientific discovery and engineering design. It involves mathematical modelling, numerical analysis, computer science, high-performance computing, and visualisation. The remarkable development of large scale computing in the last decades has turned computational science and engineering into the "third pillar" of science, complementing theory and experiment.
The Computational Mathematics track is mainly concerned with the mathematical foundations of computational science and engineering. However, in this track, we will also discuss issues of high-performance computing. Given the interdisciplinarity, the final curriculum may vary greatly depending on your interests. The Computational Mathematics track contains courses providing knowledge of design, analysis, and application of numerical methods for mathematical modelling, usable in computer simulations catering to both research and prototyping.
Financial mathematics track
Financial mathematics is applied mathematics used to analyse and solve problems related to financial markets. Any informed market participant would exploit an opportunity to make a profit without any risk of loss. This fact is the basis of the theory of arbitrage-free pricing of derivative instruments. Arbitrage opportunities exist but are rare. Typically both potential losses and gains need to be considered. Hedging and diversification aim at reducing risk. Speculative actions on financial markets aim at making profits. Market participants have different views of the future market prices and combine their views with current market prices to take actions that aim at managing risk while creating opportunities for profits. Portfolio theory and quantitative risk management present theory and methods that form the theoretical basis of market participants’ decision making.
Financial mathematics has received lots of attention from academics and practitioners over recent decades and the level of mathematical sophistication has risen substantially. However, a mathematical model is at best a simplification of the real-world phenomenon that is being modelled, and mathematical sophistication can never replace common sense and knowledge of the limitations of mathematical modelling.
Optimisation and Systems Theory track
Optimisation and Systems Theory is a discipline in applied mathematics primarily devoted to methods of optimisation, including mathematical programming and optimal control, and systems theoretic aspects of control and signal processing. The discipline is also closely related to mathematical economics and applied problems in operations research, systems engineering, and control engineering. The master’s education in Optimisation and Systems Theory provides knowledge and competence to handle various optimisation problems, both linear and nonlinear, to build up and analyse mathematical models for various engineering systems, and to design optimal algorithms, feedback control, and filters and estimators for such systems.
Optimisation and Systems Theory has wide applications in both industry and research. Examples of applications include the aerospace industry, engineering industry, radiation therapy, robotics, telecommunications, and vehicles. Furthermore, many new areas in biology, medicine, energy and environment, and information and communications technology require an understanding of both optimisation and system integration.
Mathematics of Data Science track
Statistics is the science of learning from data. Classical statistics is trying to understand data by determining a plausible model for data, and testing whether the data fits the model. Modern learning is concerned with computational statistics and automated methods for extracting information from data. The technological progress and the increased availability of information contributes to the emergence of massive and complex data sets. A variety of scientific fields are contributing to the analysis of such data at the interface of mathematics, statistics, optimisation, and computational methods for learning. Optimal decision-making under uncertainty based in such circumstances requires modelling and discovering relevant features in data, optimisation of decision policies and model parameters, dimension reduction, and large scale computations. Data science based on applied mathematics has the potential for a transformative impact on natural sciences, business, and social sciences.
This is a two-year programme (120 ECTS credits) given in English. Graduates are awarded the degree of Master of Science. The programme is given mainly at KTH Campus in Stockholm by the School of Engineering Sciences (at KTH).
Topics covered
Optimisation, mathematical systems theory, systems engineering, modeling and simulation, numerical methods and applications, parallel and high-performance computations, big data, machine learning, arbitrage pricing, portfolio theory, and risk management.
Career
Advanced mathematics and computer simulations are present within several important fields, their use has increased dramatically by the rapid development in computer software and hardware. Financial mathematics, medicine, and biology are prevalent areas, but students will be able to bring the usage of mathematics and simulations into a multitude of applications.
The graduates of this programme are in high demand in the labour market as well as in academia. Alumni work in large and smaller companies like Ericsson, ABB, Comsol, SAAB, RaySearch Labs, Modelon, If, Citibank, Brainlab, ÅF, Atlas Copco, Elekta, Process Systems Enterprise, Goldman Sachs, and many others. Another alternative is an academic career in which the programme’s alumni continue with their doctoral studies at KTH, other Swedish universities, or other leading European and US universities.
After graduation
Technology manager, deep learning software engineer, team lead, professor, PhD student, credit risk analyst, CEO, head of marketing and sales, technical director, development engineer.
"During my first year at KTH, I joined a student association called KTH Formula. We designed and built an electric car that we took to international competition in the UK. It was very fun and rewarding."
Dina Faraj, Sweden/Iraq
Sustainable development
Graduates from KTH have the knowledge and tools for moving society in a more sustainable direction, as sustainable development is an integral part of all programmes. The particular strength of mathematics is its high degree of abstraction, with one and the same mathematical model used to describe very different features in many different areas of application. This versatility leads to the effect that once you can quantify phenomena, you will be able to investigate these phenomena independently of their source, for example in science, engineering, society, and the economy. Many of the UN goals of sustainable development are accordingly linked to Applied Mathematics, to name just a few: Good health and well-being, affordable and clean energy, Decent work and economic growth, Industry, innovation and infrastructure, Sustainable cities and communities, Climate action, Life below water, Reduced inequality and others. The master’s programme in Applied and Computational Mathematics provides the student with the knowledge and tools applicable for their successful treatment. You will see examples of how to do this in different courses. It is not uncommon for the final master’s degree project to be devoted to questions related to sustainable development and its various goals. The examples of sustainable development goals addressed by the programme are:
13 Climate Action
3 Good Health and Well-Being
9 Industry, Innovation, and Infrastructure
Examples of master’s degree projects relating to Climate Action are: Efficient computational methods for climate models (collaboration with SMHI); Consequences of climate change for the electric power supply (in collaboration with SWECO); Polynomial chaos expansion for climate economy assessment (in collaboration with Karlsruhe Institute of Technology)
Examples of master’s degree projects relating to Good Health and Well-Being are: Optimal construction of medical equipment for cancer treatment (in collaboration with RaySearch Labs); Simulation of suturing for surgeon training (in collaboration with SenseGraphics); Proton arc therapy optimisation (in collaboration with RaySearch Labs);
Examples of master’s degree projects relating to Industry, Innovation, and Infrastructure are Optimal traffic planning for autonomous vehicles (in collaboration with Volvo Construction Equipment); Optimal energy management for parallel hybrid electric vehicles (in collaboration with Scania); Optimal driving decision based on energy and time costs (in collaboration with Volvo).
Faculty and research
The programme is run by the Department of Mathematics. The Department of Mathematics at KTH hosts some of the strongest Swedish research groups in mathematics. It comprises four units: Mathematics, Mathematical Statistics, Optimisation and Systems Theory, and Numerical Analysis. Jointly, these units perform research in a broad spectrum of mathematical disciplines, ranging from pure to applied mathematics. Some of the current larger research centres hosted at the department are:
Random matrices, sponsored by the Wallenberg foundation
Image processing, sponsored by SSF
PDE, sponsored by the ERC/VR/Gustafsson's foundation
MathDataLab, sponsored by Brummer & Partners
Research carried out at the Division of Optimisation and Systems Theory includes various topics in mathematical systems theory (Xiaoming Hu, Per Enqvist, and Johan Karlsson), with particular emphasis on stochastic systems, filtering, identification, and robust and nonlinear control; mathematical programming (Anders Forsgren and Per Enqvist), with large-scale nonlinear programming, structural optimisation; and a wide range of applications. Examples of applications include radiation therapy (Forsgren), robotics (Hu), and telecommunications (Karlsson).
The research in the Division of Numerical Analysis includes numerical methods for stochastic and deterministic differential equations (Anders Szepessy (a member of the Royal Swedish Academy of Sciences), Mattias Sandberg), analysis and computational methods for differential-algebraic equations (Michael Hanke), numerical methods for micro and complex flow (Anna-Karin Tornberg (a member of the Royal Swedish Academy of Sciences and the Royal Swedish Academy of Engineering Sciences), Katarina Gustavsson), multiscale methods (Olof Runborg), finite element methods for multiphase flow (Sara Zahedi). The researchers are working actively in many interdisciplinary cooperative ventures, e.g., the Swedish e-Science Research Centre (SeRC), the Linné FLOW Centre, and Karolinska Institutet. Students will also have access to Sweden’s fastest supercomputers via the PDC Centre for High-Performance Computing.
The Division of Mathematical Statistics hosts active groups in probability theory and statistical theory with applications to finance and risk management (Boualem Djehiche, Henrik Hult, Sigrid Källblad, Camilla Landen), statistical learning (Henrik Hult, Jimmy Olsson, Tatjana Pavlenko, Pierre Nyquist, Joacim Anden), Monte Carlo methods (Henrik Hult, Jimmy Olsson, Pierre Nyquist), computational statistics (Jimmy Olsson, Pierre Nyquist, Tatjana Pavlenko), and high-dimensional models (Kevin Schnelli, Tatjana Pavlenko).
Admission requirements
To be eligible for the programme, you must have been awarded a bachelor's degree, be proficient in English, and meet the programme-specific requirements.
Bachelor's degree
A bachelor's degree, equivalent to a Swedish bachelor's degree, or equivalent academic qualifications from an internationally recognized university, is required. Students who are following longer technical programmes, and have completed courses equivalent to a bachelor's degree, will be considered on a case-by-case basis.
Students in their final year of undergraduate studies may apply and, if qualified, will receive a conditional acceptance. These applicants must include a written statement according to the instructions given by University Admissions. Students in the final year of undergraduate studies at a Swedish university do not have to provide a written statement in order, if qualified, to receive a conditional acceptance. They must, however, have completed 150 ECTS credits in the bachelor’s programme by 1 February.
English proficiency
English language proficiency equivalent to (the Swedish upper secondary school) English course B/6 is required. The requirement can be satisfied through a result equal to, or higher than, those stated in the following internationally recognized English tests:
TOEFL Paper-based: Score of 4.5 (scale 1-6) in written test, a total score of 575.
TOEFL ITP is not accepted.
TOEFL iBT internet-based: Score of 20 (scale 0-30) in written test, a total score of 90
IELTS Academic/IELTS UKVI: A minimum overall mark of 6.5, with no section lower than 5.5
Cambridge ESOL: Cambridge English: Advanced (CAE) Certificate in Advanced English or Cambridge English: Proficiency (CPE) (Certificate of Proficiency in English)
Michigan English Language Assessment Battery (MELAB): Minimum score of 90
The University of Michigan, ECPE (Examination for the Certificate of Proficiency in English)
Pearson PTE Academic: Score of 62 (writing 61)
The language requirement can also be fulfilled through previous university and upper secondary school studies. More information on recognized English tests, previous studies, and required documents is provided by University Admissions.
Specific requirements for the master's programme in Applied and Computational Mathematics
A bachelor's degree corresponding to 180 ECTS credits, or equivalent, with at least 45 ECTS credits in Mathematics.
The students are required to have documented knowledge corresponding to basic university courses in analysis in one and several variables, linear algebra, numerical analysis, ordinary and partial differential equations, and integral transforms, mathematical statistics, and basics of programming in a higher programming language.
Application documents
Your application is not complete without the required supporting documentation. The following general and programme-specific documents must therefore be included in the application in the specified order:
General documents
Certificates and diplomas from previous university studies
Transcript of completed courses and grades included in your degree
Proof of English proficiency
A copy of your passport including personal data and photograph, or other identification documents
Specific documents for the master's programme in Applied and Computational Mathematics
Letter of motivation
Summary sheet *
*In order for your application to be considered complete, you need to fill out the online summary sheet. If you do not include a summary sheet, this will negatively affect your evaluation score. Please be sure to fill out all of the required information before you submit the form.
If you have questions regarding the summary sheet please contact the programme directly.
