Applied and Computational Mathematics

Applied and Computational Mathematics at KTH Royal Institute of Technology is a comprehensive master's program designed to equip students with advanced knowledge and practical skills in mathematical modeling, computational methods, and their applications across various scientific and engineering disciplines. The program emphasizes the development of analytical thinking, problem-solving capabilities, and proficiency in modern computational tools, preparing graduates for careers in research, industry, and academia. Throughout the program, students engage with a broad curriculum covering numerical analysis, differential equations, optimization, stochastic processes, data analysis, and machine learning, among other topics. The program is structured to combine theoretical foundations with hands-on projects, fostering an environment where students can apply mathematical concepts to real-world challenges. Students benefit from state-of-the-art laboratories, collaboration with industry partners, and opportunities to participate in cutting-edge research projects. The program also includes opportunities for international exchange, internships, and specialization within areas such as computational science, applied mathematics, or mathematical modeling. Graduates of Applied and Computational Mathematics are well-equipped to contribute to technological innovation, data-driven decision-making, and scientific discovery, opening pathways to careers in software development, data analysis, scientific research, finance, and engineering. The program is taught in English and welcomes applicants from diverse backgrounds with a strong foundation in mathematics and computing. With a strong emphasis on interdisciplinary learning and practical application, Applied and Computational Mathematics at KTH prepares students to meet the complex mathematical and computational challenges of today’s digital and data-driven world. Upon graduation, students will possess a robust skill set and a global perspective, enabling them to thrive in a variety of professional environments and contribute meaningfully to societal advancement through scientific and technological innovation.

Courses for all specialisations

Year 1

The tracks of the Master’s program Applied and Computational Mathematics are closely connected and a skilled applied mathematician has knowledge and skills from several of the fields of applied mathematics presented below.

Mandatory courses

• Theory and Methodology of Science with Applications (Natural and Technological Science) 7.5 credits
• Applied Numerical Methods 7.5 credits
• Probability Theory 7.5 credits

Conditionally elective courses

• Applied Linear Optimization 7.5 credits
• Mathematical Systems Theory 7.5 credits
• Systems Engineering 7.5 credits

Optional courses

• Molecular Modeling 7.5 credits
• Computational Chemistry 7.5 credits
• Bioinformatics and Biostatistics 7.0 credits
• Visualization 7.5 credits
• Advanced Computation in Fluid Mechanics 7.5 credits
• Machine Learning 6.0 credits
• Mathematical Modelling of Biological Systems 9.0 credits
• Introduction to High Performance Computing 7.5 credits
• Optimization 6.0 credits
• Computational Methods for Stochastic Differential Equations 7.5 credits
• Topics in Scientific Computings 3.0 credits
• Program Construction in C++ for Scientific Computing 7.5 credits
• Advanced Individual Course in Scientific Computing 6.0 credits
• Project Course in Scientific Computing 7.5 credits
• Financial Mathematics, Basic Course 7.5 credits
• Applied Nonlinear Optimization 7.5 credits
• Geometric Control Theory 7.5 credits
• Optimal Control Theory 7.5 credits
• Applied Systems Engineering 7.5 credits
• Regression Analysis 7.5 credits
• Modern Methods of Statistical Learning 7.5 credits
• Portfolio Theory and Risk Management 7.5 credits
• Time Series Analysis 7.5 credits
• Computer Intensive Methods in Mathematical Statistics 7.5 credits
• Martingales and Stochastic Integrals 6.0 credits
• Game Theory 7.5 credits
• Financial Derivatives 7.5 credits
• Risk Management 7.5 credits
• Computational Fluid Dynamics 7.5 credits
• Applied Computational Fluid Dynamics 5.0 credits

Year 2

Conditionally elective courses

• Mathematical Systems Theory 7.5 credits
• Systems Engineering 7.5 credits

Optional courses

• Molecular Modeling 7.5 credits
• Computational Chemistry 7.5 credits
• Bioinformatics and Biostatistics 7.0 credits
• Machine Learning 6.0 credits
• Mathematical Modelling of Biological Systems 9.0 credits
• Introduction to High Performance Computing 7.5 credits
• Optimization 6.0 credits
• Applied Systems Engineering 7.5 credits
• Modern Methods of Statistical Learning 7.5 credits
• Portfolio Theory and Risk Management 7.5 credits
• Risk Management 7.5 credits

Track; Computational Mathematics (COMA)

The field of computer simulations is of great importance for high-tech industry and scientific/engineering research, e.g. virtual processing, climate studies, fluid dynamics, advanced materials, etc. Thus, Computational Science and Engineering (CSE) is an enabling technology for scientific discovery and engineering design. CSE involves mathematical modeling, numerical analysis, computer science, high-performance computing and visualization. The remarkable development of large scale computing in the last decades has turned CSE into the "third pillar" of science, complementing theory and experiment. 

The track Computational Mathematics (COMA) is mainly concerned with the mathematical foundations of CSE. However, in this track we will also discuss issues of high-performance computing. Given the interdisciplinarity, your final curriculum may vary greatly depending on your interests.

Year 1

Mandatory courses

• Numerical Solutions of Differential Equations 7.5 credits
• Parallel Computations for Large- Scale Problems 7.5 credits

Year 2

Mandatory courses

• Matrix Computations for Large-scale Systems 7.5 credits
• The Finite Element Method 7.5 credits

Track; Financial Mathematics (FMIA)

Financial mathematics is applied mathematics used to analyze 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 the last 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 modeled, and mathematical sophistication can never replace common sense and knowledge of the limitations of mathematical modeling. 

Year 1

Mandatory courses

• Financial Mathematics, Basic Course 7.5 credits

Conditionally elective courses

• Regression Analysis 7.5 credits
• Time Series Analysis 7.5 credits

Year 2

Mandatory courses

• Portfolio Theory and Risk Management 7.5 credits

Conditionally elective courses

• Financial Derivatives 7.5 credits
• Risk Management 7.5 credits

Track; Mathematical Statistics (MASA)

Statistics is the science of learning from data. Decision making based on only partial information and incomplete data is a necessity and statistics is an essential tool in such circumstances. The world is full of stochastic phenomena and randomness appear in many ways. Probability theory is the framework for describing stochastic phenomena. 

Mathematical statistics is a structured approach to statistics based on probability theory. Mathematical statistics typically aims at determining a plausible model for a stochastic phenomenon or a set of observations, and to use this model to make predictions about the future or to design optimal strategies for decision making. Computational statistics is a rapidly developing field of mathematical statistics that includes theory and methods for efficient stochastic simulation. Computational statistics have become indispensable in a variety of applications. 

Year 1

Conditionally elective courses

• Regression Analysis 7.5 credits
• Modern Methods of Statistical Learning 7.5 credits
• Time Series Analysis 7.5 credits
• Computer Intensive Methods in Mathematical Statistics 7.5 credits
• Martingales and Stochastic Integrals 6.0 credits

Year 2

Conditionally elective courses

• Modern Methods of Statistical Learning 7.5 credits
• Risk Management 7.5 credits

Track; Optimization and Systems Theory (OPSA)

Optimization and Systems Theory is a discipline in applied mathematics primarily devoted to methods of optimization, 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. 

Master’s education in Optimization and Systems Theory provides knowledge and competence to handle various optimization problems, both linear and nonlinear, to build up and analyze mathematical models for various engineering systems, and to design optimal algorithms, feedback control, and filters and estimators for such systems.

Optimization and Systems Theory has wide applications in both industry and research. Examples of applications include 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 understanding of both optimization and system integration. 

Year 1

Conditionally elective courses

• Applied Linear Optimization 7.5 credits
• Applied Nonlinear Optimization 7.5 credits
• Mathematical Systems Theory 7.5 credits
• Geometric Control Theory 7.5 credits
• Optimal Control Theory 7.5 credits
• Systems Engineering 7.5 credits
• Applied Systems Engineering 7.5 credits

Year 2

Conditionally elective courses

• Mathematical Systems Theory 7.5 credits
• Systems Engineering 7.5 credits
• Applied Systems Engineering 7.5 credits

Degree project and thesis

The project may be carried out in an academic or industrial environment in Sweden or abroad. Students are welcome to discuss project ideas with the staff at the mathematics department, but are also encouraged to seek other contacts, in the academic world and in industry, to identify suitable projects. The result of this thesis work is provided as a written report and as a presentation at a seminar.

Requirements

1. A completed Bachelor's degree, corresponding to a Swedish Bachelor's degree (180 ECTS credits), or equivalent academic qualifications from an internationally recognised university.
2. Students in their final year of undergraduate education may apply to KTH and, if qualified, receive conditional acceptance. If you have not yet completed your studies, please include a written statement issued by the degree awarding university. This statement must be certified and stamped by the Academic Registrar's Office, the Examinations Office or equivalent of the institution. Statements from other staff members, such as faculty members, will not be accepted.
3. Students who are following longer technical programmes, and have completed courses equivalent to a Bachelor´s degree (180 ECTS credits), will be considered on a case-by-case basis.
4. Cover sheet (generated from the web-based application). However, if you have a Swedish personal ID number or if you choose to upload your documents, the cover sheet is not required.
5. Certificates and diplomas from previous education at an internationally recognised university.
6. Transcripts of records (including course list). All courses taken and grades must be included. Sort them in reverse chronological order, i.e. put the last received document on top.
7. Proof of English proficiency.
8. A copy of your passport or some other document of identification. If you are from an EU/EEA country or Switzerland and are required to document your citizenship status in order to be considered exempt from paying application and tuition fees, your passport copy must be certified. If you are not a citizen of an EU/EEA country or Switzerland, certification of your passport copy is not required.
9. Completed summary sheet
10. IELTS A minimum overall mark of 6.5, with no section lower than 5.5 (only Academic Training accepted). 
11. TOEFL Paper-based test: total result of 575 (written test, minimum grade 4.5)
12. TOEFL  Internet-based test: total result of 90 (written test, minimum grade 20)

• A Bachelor of Science corresponding to 180 ECTS, 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, differential equations and transforms, mathematical statistics, and basics of programming in a higher programming language.

  The specific requirements may be assessed as not fulfilled if

• the grade point average below 75% of the scale maximum
• the degree awarding institution is not considered to meet acceptable quality standards by the authorities of the country in which the institution is located
• the degree does not qualify for admission to equivalent Master’s level in the country where the degree is awarded

Scholarships

• KTH Scholarship
• Russian Presidency Scholarship for Abroad Studies

The Master's Programme in Applied and Computational Mathematics at KTH Royal Institute of Technology is a comprehensive and rigorous program designed to equip students with advanced skills in mathematical modeling, numerical analysis, and computational techniques. The program aims to prepare students for both research and industry by emphasizing practical applications of mathematics in various fields such as engineering, finance, data science, and technology. The curriculum typically includes foundational coursework in calculus, linear algebra, differential equations, programming, and algorithms, alongside specialized subjects like computational methods, optimization, stochastic processes, and data analysis. Students often have the opportunity to participate in projects, internships, and collaborations with industry partners, providing real-world experience and fostering innovation. The program is structured to develop both theoretical understanding and practical problem-solving abilities, ensuring graduates are capable of tackling complex mathematical challenges. Teaching methods include lectures, seminars, group work, and computer labs, encouraging active learning and teamwork. KTH emphasizes interdisciplinary approaches, often integrating recent advances in computational science and technology to solve contemporary problems. Graduates from the program are well-positioned for careers in academia, research institutions, and various industries requiring advanced mathematical expertise. The program staff includes experienced professors and researchers committed to high-quality education and research. International students are welcome, and the university promotes diversity and inclusion within its academic community. The program's location in Stockholm provides access to a vibrant tech and innovation ecosystem, supporting students' professional growth and networking opportunities. No specific language requirements are outlined, but courses are typically conducted in English to accommodate international students. Overall, the Applied and Computational Mathematics program at KTH offers a challenging and rewarding education for students passionate about mathematics and its applications in modern technology and science.