Study tracks and specializations
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, mathematical logic, mathematical physics, and applied mathematics study track includes specializations applied analysis, mathematical modelling, mathematics of imaging, stochastics, insurance and financial mathematics and probabilistic modelling.
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:
 Core courses and background studies (1020 credits)
Mandatory courses: Real analysis I, 5 cr., and Functional analysis, 10 cr.  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
 Pro gradu (30 credits) and pro gradu seminar (5 credits).

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:
 Core courses and background studies (1020 credits)
Recommended course: Functional analysis, 10 cr.
Other recommended courses: Real analysis, 5 cr.  Optional courses and specialization courses
Recommended courses: Partial differential equations 1, Integral equations, Inverse Problems

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

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:
 Core courses and background studies
Mandatory: Topology II (10 cr.)
Recommended: Algebra II (10 cr.), compex analysis (10cr.)  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)

Master´s thesis and seminar

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:
 Core courses and background studies (1020 credits)
Mandatory courses: Mathematical logic, 10 cr., and one of Topology 2, Algebra 2, and Real Analysis, 10 cr.  Optional courses and specialization courses
Classification theory (10 credits), nonelementary 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  Pro gradu (30 credits) and pro gradu seminar (5 credits).
 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).
Mathematical modelling
We offer a strongly sciencebased Master programme especially suitable for researchoriented MSc students of applied mathematics, as well as for all students interested in using mathematical models of reallife phenomena. We focus on biological applications, especially on ecology and evolution, because advanced modelling skills can be acquired through this without timeconsuming specialization into a particular field of applications.
Requirements in Mathematical modelling:
1. Core courses (1020 credits)
At least one of the following courses is mandatory:
MAST31501 Mathematical modelling (10 cr) OR
LSI33001 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. The following specialization courses are offered on a regular basis:
MAST31505 Adaptive dynamics (10 cr)
LSI33002 Evolution and the theory of games (5 cr)
MAST31504 Stochastic population models (10 cr)
MAST31503 Spatial models in ecology and evolution (10 cr)
LSI33003 Mathematics of infectious diseases (10 cr)
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:
 Core courses and background studies (1020 credits): At least 15cr from the following Amodule courses: Functional analysis, 10 cr., Probability Theory I, 5cr., Probability Theory II, 5cr., Real analysis, 5 cr.
 Mandatory requirement for courses in Physics: At least 20 cr. from Physics courses, other than Mathematical Methods for Physicists I, II, and III (FYMM IIII). 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: 1060 cr. from optional math courses and 035 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 simulationsSuggestions 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"

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

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:

Core courses and background studies (1020 credits)
Mandatory course in study track: Functional analysis, 10 cr. 
Optional courses and specialization courses
Recommended courses: Inverse Problems 
Pro gradu (30 credits) and pro gradu seminar (5 credits).

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:
 Core courses and background studies (1025 credits):
Mandatory courses: Probability Theory I, 5cr., Probability Theory II, 5cr.
Recommended courses: Real analysis, 5 cr., Functional analysis 10cr., Statistical inference III 5cr.  Optional courses: master level courses in Mathematics, Statistics
and/or minor subjects (4060 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 statisticsSpecialization 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).
 Pro gradu (30 credits) and pro gradu seminar (5 credits).
 Other courses in mathematics and statistics: (020cr), 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
 The core courses Probability theory I and II, are mandatory.

It is strongly recommended to take at least one of these two courses, stochastic methods and/or stochastic analysis.

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.

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.

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.
The Probabilistic modelling specialisation constructs a bridge between theoretical mathematics and its applications to practical data analysis tasks. Our courses include core theoretical and practical courses in statistics and machine learning. The probabilistic modelling specialisation is especially suitable for researchoriented MSc students who would like to continue for PhD studies in statistics or machine learning as well as for students who are interested in using mathematics in challenging real world data analysis tasks.
Requirements in the Probabilistic modelling:
1. Core courses. Choose at lest 10 credits.
Recommended courses: Probability theory I, Probability theory II
2. Optional courses and specialization courses
Highly recommended courses: Computational statistics I, Advanced course in Bayesian statistics, Advanced Course in Machine Learning
Other recommended courses: Computational statistics II, Spatial modelling and Bayesian inference, High dimensional statistics, Introduction to Machine Learning, Probabilistic graphical models
3. Master thesis (30 credits) and Master thesis seminar (5 credits).
4. Other courses in mathematics and statistics and other master's programs.
Degree requirements in the statistics study track
The advanced module is 85120 cr
MAST32001 Computational statistics I (5 cr), mandatory course
MAST30001 Master´s thesis seminar (5 cr), mandatory
MAST32000 Master´s thesis (30 cr), mandatory
Courses 4580 cr
MAST32002 Computational statistics II (5 cr)
MAST32004 Advanced course in Bayesian statistics (5 cr)
MAST32005 Spatial modelling and Bayesian inference (5 cr)
MAST32006 High dimensional statistics (5 cr)
MAST32007 Time series analysis I (5 cr)
MAST32008 Time series analysis II (5 cr)
MAST33003 Nonparametric and robust methods (5 cr)
MAST33004 Robust regression (5 cr)
MAST32009 Bayesian inference with OpenBugs (5 cr)
LSI34006 Eventhistory analysis (5 cr)
LSI34005 Statistical methods in public health (5 cr)
LSI34002 Genomewide association studies ( 5 cr)
LSI34003 Phylogeny inference and data analysis (5 cr)
LSI34007 Modelling molecular evolution (5 cr)
LSI34004 Statistical population genetics (5 cr)
LSI34001 Topics in biostatistics and bionformatics (5 cr)
MAST32011 Computer age statistical inference (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 85120 cr
MAST33001 Generalized linear models II (5 cr), mandatory course
MAST30001 Master´s thesis seminar (5 cr), mandatory
MAST33000 Master´s thesis (30 cr), mandatory
Courses 4580 cr:
MAST33013 Nonparametric and robust methods (5 cr)
MAST33004 Robust regression (5 cr)
MAST33009 Structural equation models (5 cr)
MAST33010 Introduction to registerbased research (5 cr)
MAST33011 Dataanalysis with SAS (5 cr)
MAST33012 Demographic analysis (5 cr)
MAST33013 Registerbased data analysis (5 cr)
MAST33014 Small area estimation (5 cr)
MAST33015 Analysis of complex surveys (5 cr)
MAST33018 Survey sampling (5 cr)
MAST33019 Surveymethodology and European statistical system (5 cr)
 courses from statistics, mathematics and applied mathematics 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.