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 MSc in Mathematics
MSc in Mathematics
The master's degree in Mathematics aims to train mathematicians capable of understanding, framing, formulating and solving problems in research, development and innovation, in the context of public administration, banking, trade, industry and services. This master program provides a wide range of mathematical knowledge, from advanced fundamental theory to a wide range of real applications. The use of computational tools is also present whenever necessary and appropriate.
Specialization areas
Applied Analysis and Computation: offers an integrated training in scientific computing, more precisely in finding solutions of quantitative problems using computational programming and simulation, mathematical models, and a diversity of analytical, statistical, and numerical techniques.
The Master in Mathematics in this area of specialization can be certified by the European Consortium for Mathematics in Industry (ECMI) as a master program in Tecnomathematics, as long as certain requirements are met.
Statistics, Optimization and Mathematics in Finance: provides training to solve problems involving treatment and analysis of data and support to decision making. The main focus goes to ways of dealing with the unknown, the uncertainty, the risk, the scarceness and the efficiency, in particular in the context of applications in Finance.
Pure Mathematics: provides solid and advanced training in Algebra, Analysis and Geometry, and stimulates the beginning of research activities within these areas.
PROFESSONAL GOALS  PROGRAM  ADMISSION REQUIREMENTS  CONTACTS 

Professional Goals
The MSc in Mathematics Program offers a broadband advanced training, that helps developing abstract thinking and stimulates autonomy and creativity. Furthermore, it provides a large spectrum of theoretical, practical, and computational knowledge, with enormous applicability potential in the most diverse analytical and quantitative problems where Mathematics plays a relevant role. Thus, the career opportunities for holders of the MSc in Mathematics are vast and varied.
These career opportunities range from research to other public and private sectors of productive activity, such as:
 banking, management, and insurance;
 administration, service industry, commerce, information technology (IT), and telecommunications;
 industry and technological development.

Program
The MSc in Mathematics Program offered by FCTUC is a 2year course divided in 4 semesters that confers the Degree of Master of Science. There are three areas of specialization:
 Applied Analysis and Computation or Applied Analysis and Computation – Tecnomathematics (ECMI)
 Statistics, Optimization and Mathematics in Finance
 Pure Mathematics

Applied Analysis and Computation
The MSc in Mathematics offered by FCTUC is a 2year program divided in 4 semesters that confers the Degree of Master of Science. The first year of the specialization in Applied Analysis and Computing is composed of 10 courses of 6 ECTS each. In this area of specialization,
There are two alternative ways to complete the second year:
 a seminar course, a research project and the preparation of a thesis;
 a course chosen from the other two areas of specialization, a course on Mathematical Modelling, a Seminar Course on Mathematical Modelling, and the preparation of a thesis (Applied Analysis and Computation  ECMI's Tecnomathematics model).
The study program of the MSc in Mathematics  specialization in Applied Analysis and Computation is the following:
1st semester
 Advanced Programming (6 ECTS)
 Mathematical Methods in Physics and Biology (6 ECTS)
 Parallel Computing (6 ECTS)
 Stochastic Processes and Calculus (option A, 6 ECTS)
 Time Series (option A, 6 ECTS)
 Commutative Algebra (option B, 6 ECTS)
 Differentiable Manifolds (option B, 6 ECTS)
 Game Theory (option B, 6 ECTS)
 Linear Programming and Combinatorial Optimization (option B, 6 ECTS)
 Optimization Methods in Finance (option B, 6 ECTS)
 Real And Functional Analysis (option B, 6 ECTS)
Note: The student must choose one course marked as option A and one course marked as option B.
Note: The choices of the optional courses depend on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
2nd semester
 Computational Visualization (6 ECTS)
 Numerical Methods for Partial Differential Equations (6 ECTS)
 Numerical Optimization (6 ECTS)
 Algebraic Geometry (option C, 6 ECTS)
 Algebraic Topology (option C, 6 ECTS)
 Category Theory (option C, 6 ECTS)
 Groups and their representations (option C, 6 ECTS)
 Mathematics in Finance (option C, 6 ECTS)
 Partial Differential Equations (option C, 6 ECTS)
 Risk Theory (option C, 6 ECTS)
 Sampling and Surveys (option C, 6 ECTS)
Note: The student must choose two courses marked with option C.
Note: The choices of the optional courses depend on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
3rd semester
 Dissertation in Applied Analysis and Computation (annual, 15 ECTS)
 Research Project in Applied Analysis and Computation (9 ECTS)
 Seminar in Applied Analysis and Computation (6 ECTS)
4th semester
 Dissertation in Applied Analysis and Computation (annual, 30 ECTS)

Applied Analysis and Computation  Tecnomathematics
The MSc in Mathematics offered by FCTUC is a 2year program divided in 4 semesters that confers the Degree of Master of Science. The first year of the specialization in Applied Analysis and Computing – Technomathematics (ECMI) is composed of 10 courses of 6 ECTS. The second year consists of a course chosen from the other two areas of specialization, a course on Mathematical Modelling, a Seminar Course on Mathematical Modelling, all in the first semester, and the preparation of a thesis.
The European Consortium for Mathematics in Industry (ECMI) is a consortium of academic institutions and industrial companies operating on a European scale, with the main purpose of promoting and supporting the use of mathematical models in social or economic relevant activities. For a master's course in Mathematics  specialization in Applied Analysis and Computing to be certified by the ECMI as master in Techonomathematics, the following requirements must be met:
At least one of the four semesters of the master must be abroad, at one ECMI university (or another institution approved by the ECMI educational committee).
The unit Seminar in Mathematical Modelling must correspond to the participation in an international ECMI Modelling Week.
The dissertation must be written in English, mathematics traverse an industrial theme and be submitted for approval by the ECMI education committee.
The study program of the MSc in Mathematics  specilization in Applied Analysis and Computation (Tecnomathematics) is the following:
1st semester
 Advanced Programming (6 ECTS)
 Mathematical Methods in Physics and Biology (6 ECTS)
 Parallel Computing (6 ECTS)
 Stochastic Processes and Calculus (option A, 6 ECTS)
 Time Series (option A, 6 ECTS)
 Commutative Algebra (option B, 6 ECTS)
 Differentiable Manifolds (option B, 6 ECTS)
 Game Theory (option B, 6 ECTS)
 Linear Programming and Combinatorial Optimization (option B, 6 ECTS)
 Optimization Methods in Finance (option B, 6 ECTS)
 Real and Functional Analysis (option B, 6 ECTS)
Note: The student must choose one course marked with option A and one course marked with option B.
Note: The choices of the optional courses depend on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
2nd semester
 Computational Visualization (6 ECTS)
 Numerical Methods for Partial Differential Equations (6 ECTS)
 Numerical Optimization (6 ECTS)
 Algebraic Geometry (option C, 6 ECTS)
 Algebraic Topology (option C, 6 ECTS)
 Category Theory (option C, 6 ECTS)
 Groups and their representations (option C, 6 ECTS)
 Mathematics in Finance (option C, 6 ECTS)
 Partial Differential Equations (option C, 6 ECTS)
 Risk Theory (option C, 6 ECTS)
 Sampling and Surveys (option C, 6 ECTS)
Note: The student must choose two courses marked with option C.
Note: The choice of all optional courses depends on the previous qualification path of the student under supervision of the Coordinator of the Master Program, who should approve and later follow each individual choice.
3rd semester
 Dissertation in Applied Analysis and Computation (annual, 15 ECTS)
 Mathematical Modelling (6 ECTS)
 Seminar course in Mathematical Modelling (3 ECTS)
 Semester option (6 ECTS)
Nota: The student must choose the course Semester option from all optional courses from the first semester that have not yet been attended.
Note: The choice of the optional course depends on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
4th semester
 Dissertation in Applied Analysis and Computation (annual, 30 ECTS)

Statistics, Optimization and Mathematics in Finance
The MSc in Mathematics offered by FCTUC is a 2year program divided in 4 semesters that confers the Degree of Master of Science. The first year of the specialization in Statistics, Optimization, and Mathematics in Finance, is composed of 10 courses of 6 ECTS each. The second year is devoted to a seminar course and to the preparation of a thesis.
The study program of the MSc in Mathematics  specialization in Statistics, Optimization and Mathematics in Finance is the following:
1st semester
 Linear Programming and Combinatorial Optimization (6 ECTS)
 Optimization Methods in Finance (6 ECTS)
 Stochastic Processes and Calculus (6 ECTS)
 Time Series (6 ECTS)
 Commutative Algebra (option A, 6 ECTS)
 Computational Visualization (option A, 6 ECTS)
 Differentiable Manifolds (option A, 6 ECTS)
 Game Theory (option A, 6 ECTS)
 Mathematical Methods in Physics and Biology (option A, 6 ECTS)
 Parallel Computing (option A, 6 ECTS)
 Real and Functional Analysis (option A, 6 ECTS)
Note: The student must choose one course marked with option A.
Note: The choice of the optional course depends on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
2nd semester
 Mathematics in Finance (6 ECTS)
 Numerical Optimization (6 ECTS)
 Sampling and Surveys (6 ECTS)
 Advanced Programming (option B, 6 ECTS)
 Algebraic Geometry (option B, 6 ECTS)
 Algebraic Topology (option B, 6 ECTS)
 Category Theory (option B, 6 ECTS)
 Groups and their representations (option B, 6 ECTS)
 Numerical Methods for Partial Differential Equations (option B, 6 ECTS)
 Partial Differential Equations (option B, 6 ECTS)
 Risk Theory (option B, 6 ECTS)
Note: The student must choose two courses marked with option B.
Note: The choices of the optional courses depend on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
3rd semester
 Dissertation in Statistics, Optimization and Mathematics in Finance (annual, 24 ECTS)
 Seminar in Statistics, Optimization and Mathematics in Finance (6 ECTS)
4th semester
 Dissertation in Statistics, Optimization and Mathematics in Finance (annual, 30 ECTS)

Pure Mathematics
The MSc in Mathematics offered by FCTUC is a 2year program divided in 4 semesters that confers the Degree of Master of Science. The first year of the specialization in Pure Mathematics is composed of 10 courses of 6 ECTS each. The second year is devoted to a seminar course and to the preparation of a thesis.
The study program of the MSc in Mathematics  specialization in Pure Mathematics is the following:
1st semester
 Commutative Algebra (6 ECTS)
 Differentiable Manifolds (6 ECTS)
 Real and Functional Analysis (6 ECTS)
 Stochastic Processes and Calculus (6 ECTS)
 Computational Visualization (option A, 6 ECTS)
 Game Theory (option A, 6 ECTS)
 Linear Programming and Combinatorial Optimization (option A, 6 ECTS)
 Mathematical Methods in Physics and Biology (option A, 6 ECTS)
 Optimization Methods in Finance (option A, 6 ECTS)
 Parallel Computing (option A, 6 ECTS)
 Time Series (option A, 6 ECTS)
Note: The student must choose one course marked with option A.
Note: The choice of the optional course depends on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
2nd semester
 Partial Differential Equations (6 ECTS)
 Category Theory (option B, 6 ECTS)
 Groups and their representations (option B, 6 ECTS)
 Algebraic Geometry (option C, 6 ECTS
 Algebraic Topology (option C, 6 ECTS)
 Advanced Programming (option D, 6 ECTS)
 Mathematics in Finance (option D, 6 ECTS)
 Numerical Methods for Partial Differential Equations (option D, 6 ECTS)
 Numerical Optimization (option D, 6 ECTS)
 Risk Theory (option D, 6 ECTS)
 Sampling and Surveys (option D, 6 ECTS)
Note: The student must choose at least one course marked with option B and at least one course marked with option C. The total number of 30 ECTS for the semester can be completed with courses marked with option D.
Note: The choices of the optional courses depend on the previous qualification path of the student and should follow the advise of the Program Coordinator, who is also responsible for its approval.
3rd semester
 Dissertation in Pure Mathematics (annual, 24 ECTS)
 Seminar in Pure Mathematics (6 ECTS)
4th semester
 Dissertation in Pure Mathematics (annual, 30 ECTS)

Admission requirements
One of the following requirements must be met by the applicants for this Master program:
 Hold a BSc degree or legal equivalent;
 Hold a foreign academic degree after completing 1st cycle of studies organized according to the principles of the Bologna Process;
 Hold a foreign academic degree that is recognized as meeting the objectives of the Bachelor’s degree by the Scientific Council of the Faculty of Science and Technology;
 Hold an academic, scientific or professional curriculum that is recognized by Scientific Council of Faculty of Science and Technology to attest the capacity to complete this cycle of studies.
Candidates should check the admission requirements available on the university's site, in addition to the information provided here.

Contacts
Departamento de Matemática da FCTUC
Apartado 3008
EC Santa Cruz
3001 – 501 Coimbra
Telefone: +351 239 791 150
Fax: +351 239 793 069
Email: secmat@mat.uc.pt
Program coordination:
Carlos Tenreiro (coordinator, DMUC)
DMUC’s representative at ECMI’s educational committee:
 Marta Pascoal (DMUC)