Study tracks

Mathematics and Statistics Master's Programme has four study tracks: mathematics, applied mathematics, statistics and social statistics. Mathematics study track includes specializations analysis, geometry, algebra and topology, matrhematical logic, matrhematical physics, and applied mathematics study track includes specializations applied analysis, mathematical modelling, mathematics of imaging, stochastics, insurance and financial marthematics.

Courses on functional analysis, real analysis, Fourier analysis, complex analysis and partial differential equations, for example, are available in the analysis study track.

Requirements in analysis:

  1. Core courses and background studies (10-20 credits)
    Mandatory courses: Real analysis I, 5 cr., and Functional analysis, 10 cr.
  2. Optional and specialization courses
     

    Suitable courses for (1) and (2) include the following courses (10 credits each): 

    Fourier analysis, Complex analysis I, Complex analysis II, Real analysis II/Geometric measure theory, Harmonic analysis, Partial Differential Equations I, Partial Differential Equations II/Sobolev spaces, Introduction to differential geometry/Riemannian geometry, Spectral theory, as well as other specialized advanced courses in analysis

  3. Pro gradu (30 credits) and pro gradu seminar (5 credits).
  4. Other courses in mathematics and statistics: Vector analysis II, 5 cr, and/or Differential equations II, 5 cr.
    (Including these courses in the Master degree needs to be agreed with a faculty member).

 

Courses on functional analysis, Fourier analysis, inverse problems, integral equations and partial differential equations, for example, are available in the applied analysis study track.

Requirements in applied analysis:

  1. Core courses and background studies (10-20 credits)
    Recommended course: Functional analysis, 10 cr.
    Other recommended courses: Real analysis, 5 cr.
  2. Optional courses and specialization courses

    Recommended courses: Partial differential equations 1, Integral equations, Inverse Problems

  3. Pro gradu (30 credits) and pro gradu seminar (5 credits).

  4. Other courses in mathematics and statistics: Differential equations II or Vector analysis II or Introduction to wavelets and Fourier analysis 5 cr (Including these courses in the Master degree needs to be agreed with a faculty member).

Courses on algebra, topology, algebraic topology and differential geometry, for example, are available in the geometry, algebra and topology study track.

Requirements in Geometry, Algebra and Topology:

  1. Core courses and background studies
    Mandatory: Topology II (10 cr.) 
    Recommended: Algebra II (10 cr.), compex analysis (10cr.)
  2. Optional courses and specialization courses
    Strongly recommended: Introduction to Algebraic topology or Introduction to differential geometry (10cr.)

    Other recommended courses: A topics course in topology (e.g. Homotopy theory, de Rham cohomology) or a topics course in geometry (Riemannian geometry, metric geometry) 

  3. Pro gradu and pro gradu seminar

  4. Other courses in mathematics and statistics: Topology I (5+5 cr., Algebra I (5+5 cr.),   Vector analysis II (5 cr.)

Courses on mathematical logic, axiomatic set theory, model theory, recursion theory and dependence logic, for example, are available in the mathematical logic study track.

Requirements in Mathematical logic:

  1. Core courses and background studies (10-20 credits)
    Mandatory courses: Mathematical logic, 10 cr., and one of Topology 2, Algebra 2, and Real Analysis, 10 cr.
  2. Optional courses and specialization courses 
    Classification theory (10 credits), non-elementary model theory (5 credits), descriptive set theory (5 credits), large cardinals (5 credits), strong logics (5 credits)
    Suitable courses include the following (10 credits each): Axiomatic set theory, model theory, computability theory, dependence logic, finite model theory
  3. Pro gradu (30 credits) and pro gradu seminar (5 credits).
  4. Other courses in mathematics and statistics: Introduction to logic I (Johdatus logiikkaan I), 5 cr, and Introduction to logic II (Johdatus logiikkaan II), 5 cr . Elements of set theory 10cr
    (Including these courses in the Master degree needs to be agreed with a faculty member).

We offer a strongly science-based Master programme especially suitable for research-oriented MSc students of applied mathematics, as well as for all students interested in using mathematical models of real-life phenomena. We focus on biological applications, especially on ecology and evolution, because advanced modelling skills can be acquired through this without time-consuming specialization into a particular field of applications.

Our courses include mathematical modelling, evolutionary game theory, adaptive dynamics, stochastic population dynamics, spatial models, and mathematical epidemiology.

Requirements in Mathematical modeling: 

  1. Core courses (10-20 credits)
    At least one of the following courses is mandatory: Mathematical modelling (10 cr) or Introduction to mathematical biology (10 cr).
  2. Optional courses and specialization courses
    A minimum of 40 credits in mathematics MSc courses other than the above two core courses. At least 20 credits of these must be from the specialization courses of Mathematical modelling.
  3. Master thesis (30 credits) and Master thesis seminar (5 credits).
  4. Other courses in mathematics and statistics: Differential equations (5 + 5 credits) recommended if not taken earlier.

Courses on mathematical physics, quantum dynamics and kinetic theory, for example, are available in the mathematical physics study track.

Requirements in mathematical physics:

  1. Core courses and background studies (10-20 credits):  At least 15cr from the following A-module courses: Functional analysis, 10 cr., Probability Theory I, 5cr., Probability Theory II, 5cr., Real analysis, 5 cr.
  2. Mandatory requirement for courses in Physics: At least 20 cr. from Physics courses, other than Mathematical Methods for Physicists I, II, and III (FYMM I-III).  Courses included in the Bachelor Degree can be counted towards the total 20 cr.  Suggested topics: Analytical Mechanics, Quantum Mechanics, Statistical Mechanics, Quantum Statistics.

    Formal requirements: 10-60 cr. from optional math courses and 0-35 cr. from physics courses.

    Recommendations for other -studies:
                Fourier Analysis (10cr)
                Either FYMM I (10cr) or Complex Analysis I (10cr)
                Either FYMM II (10cr) or Partial Differential Equations I (10cr)
                At least one course in numerical simulations

    Suggestions for optional optional courses: any of the courses listed above under A1, Partial differential equations II (10 cr), Spectral theory (10cr), Mathematical Modelling (10 cr), Topology II (10 cr), Stochastic Analysis I & II (each 5cr)

    Specialization courses (10–20 credits): At least 10 cr from courses listed under the code "Course in Mathematical Physics"

  3. Pro gradu (30 credits) and pro gradu seminar (5 credits).

  4. Other courses in mathematics and statistics: Vector analysis II, 5cr.

Courses on inverse problems and Bayesian inversion, for example, are available in the mathematics of imaging study track.

Requirements for mathematics of imaging:

  1. Core courses and background studies (10-20 credits)
    Mandatory course in study track: Functional analysis, 10 cr.

  2. Optional courses and specialization courses
    Recommended courses:  Inverse Problems

  3. Pro gradu (30 credits) and pro gradu seminar (5 credits).

  4. Other courses in mathematics and statistics: Numerical linear algebra (Including these courses in the Master degree needs to be agreed with a faculty member).

Courses on probability theory, stochastic analysis and stochastic processes, for example, are available in the stochastics study track.

Requirements in stochastics: 

  1. Core courses and background studies (10-25 credits):
    Mandatory courses: Probability Theory I, 5cr., Probability Theory II, 5cr.
    Recommended courses: Real analysis, 5 cr., Functional analysis 10cr., Statistical inference III 5cr.
  2. Optional courses: master level courses in Mathematics, Statistics
    and/or minor subjects (40-60 cr., of which no more than 30 cr. in minor subjects)
    Recommended courses: Stochastic processes, Stochastic analysis, Large deviations, Malliavin calculus, Mathematical finance, Risk theory, Study track courses in Mathematical Physics, Extreme events, other advanced courses in mathematics or statistics

    Specialization courses (10–20 credits). It is strongly recommended that to take at least 10 cr. from the following courses: Stochastic processes, Stochastic analysis, Malliavin calculus, Mathematical finance, Risk theory. It is also possible to take a course under the code "Study track Course in Mathematical Physics" if the topic is appropriate (to be agreed with a faculty member).

  3. Pro gradu (30 credits) and pro gradu seminar (5 credits).
  4. Other courses in mathematics and statistics: (0-20cr), to be agreed with a faculty member: Vector analysis II, 5cr. Differential equations II 5cr., Statistical inference II 10cr

Requirements in the Financial and Insurance Mathematics

  1. The core courses Probability theory I and II, are mandatory.
  2. It is strongly recommended to take at least one of these two courses,  stochastic methods and/or stochastic analysis.

  3. At least 20 cr. from other specialization courses  like Mathematical finance or Risk theory or Life insurance mathematics, depending whether the plan is focused on insurance or on mathematical finance.

  4. Apart from these requirements the student is free to choose courses from all other specializations in mathematics and statistics, and also courses from different master programs like economics or computer science with mathematical content.

  5. It is also recommended to to choose courses in statistics. Generalized linear models, survival analysis, time series analysis, econometrics and computational statistics are most useful for a mathematician working in the insurance/financial industry.

Degree requirements in the statistics study track

  • The advanced module is 85-120 cr.
    • Computational statistics (5 cr), mandatory course
    • Master´s thesis and seminar (30 cr + 5 cr), mandatory studies
    • Courses (45-80 cr)
      Computational statistics II (5 cr)
      Statistical inference III (5 cr)
      Advanced course in Bayesian statistics (5 cr)
      Spatial modelling and Bayesian inference (5 cr)
      High dimensional statistics (5 cr)
      Time series analysis I (5 cr)
      Time series analysis II (5 cr)
      Nonparametric and robust methods (5 cr)
      Robust regression (5 cr)
      Bayesian inference with OpenBugs (5 cr)
      Event-history analysis (5 cr)
      Statistical methods in public health (5 cr)
      Genome-wide association studies ( 5 cr)
      Phylogenetic inference and data analysis (5 cr)
      Modelling molecular evolution (5 cr)
      Statistical population genetics (5 cr)
      Trends in biostatistics and bionformatics (5 cr)
      Courses from social statistics, mathematics, applied mathematics study tracks.
      Courses from Computer science, Datascience and Life Science Informatics master´s programmes (or some other programme), as approved in personal study plan.

Degree requirements in the social statistics study track

  • The advanced module module is 85-120 cr.
    • Generalized linear models II (5 cr), mandatory course
    • Master´s thesis and seminar (30 cr + 5 cr), mandatory studies
    • Courses (45-80 cr)
      Advanced statistical inference (5 cr)
      Nonparametric and robust methods (5 cr)
      Robust regression (5 cr)
      Structural equation models (5 cr)
      Introduction to register-based research (5 cr)
      Data-analysis with SAS (5 cr)
      Demographic analysis (5 cr)
      Register-based data analysis (5 cr)
      Small area estimation (5 cr)
      Analysis of complex surveys (5 cr)
      Applied logistic regression (5 cr)
      Topics in social statistics (5 cr)
      Survey sampling (5 cr)
      Survey-methodology and European statistical system (5 cr)
      courses from statistics, mathematics and applied study tacks
      courses from other Master´s programmes as approved in personal study plan
      • Students can specialize in official statistics and receive the European Master in Official Statistics (EMOS) certificate.