# Courses

MAST31003 Topology II (10 cr.)

Fundamentals of general topology, including: topological spaces and bases, connectedness, compactness, separation and countability axioms, metrization and extension theorems.

Prerequisites: Topology I

MAST31002 Functional Analysis (10 cr.)

Introduction to linear functional analysis including Banach and Hilbert spaces and linear operators between them; topology of normed spaces; examples of Banach spaces including sequence and function spaces; three basic principles; Fourier-series; Sobolev spaces; applications to differential equations.

Prerequisites: Analysis I & II, Linear Algebra I & II, Topology I

MAST31001 Real Analysis I (5 cr.)

Basics on real analysis, like L^p spaces, convolution, covering theorems, Lebesgue's differentiation theorem, BV- and absolutely continuous functions.

Prerequisites: Measure and Integral

MAST31004 Real Analysis II (10 cr.)

Hausdorff measure and dimension, self similar fractals, differentiation of Radon measures, Radon-Nikodym derivative, signed measures.

Prerequisites: Measure and Integral, Real Analysis I

MAST31013 Geometric Measure Theory (10 cr.)

The actual content may vary, e.g. fractals, rectifiability, tangent spaces and measures.

Prerequisites: Measure and Integral, Real Analysis I, Real Analysis II

MAST31017 Introduction to Differential Geometry (10 cr.)

Basics on smooth manifolds and mappings between manifolds, tangent and cotangent space, tensors, differential forms. Stokes theorem.

Prerequisites: Topology II

MAST31026 Riemannian Geometry (10 cr.)

Riemannian metric and connection, geodesics and exponential map, curvature, curvature and topology, and comparison theorems.

Prerequisites: Introduction to Differential Geometry

MAST31028 Metric Geometry (10 cr.)

The actual content may vary, including eg. notions of curvature in metric spaces.

Prerequisites:

MAST31027 Metric Surfaces (10 cr.)

The actual content may vary, e.g. minimal surfaces, varifolds, and currents.

Prerequisites:

MAST31018 Spectral Theory (10 cr.)

Unbounded operators in Hilbert spaces; closed, symmetric and self-adjoint operators; spectral theorem; perturbation theory; applications to elliptic PDE

Prerequisites: B.Sc.-level mathematics, Functional Analysis
Recommended: Sobolev space theory, Fourier analysis, theory of distributions

MAST31012 Elliptic Partial Differential Equations (10 cr.)

Advanced topics in spectral theory: max-min-principle, asymptotic analysis, Floquet-Bloch-Gelfand theory for periodic problems, boundary singularities, properties of eigenfunctions

Prerequisites: Functional Analysis, Spectral Theory

MAST31006 Complex Analysis I (10 cr.)

Complex arithmetic, differentiation, power series, contour integration, Mobius transforms

Prerequisites:

MAST31007 Complex Analysis II (10 cr.)

Laurent series, zeros and poles, residues, the argument principle, trigonometric and indefinite integrals, integrals of multi-valued functions

Prerequisites:
Recommended: Complex Analysis I

MAST31016 Sobolev Spaces (10 cr.)

Hölder spaces, Classical Sobolev spaces,Sobolev embeddings, Fractional Sobolev spaces

Prerequisites: Measure and Integration, Functional Analysis

MAST31019 Hyperbolic Metric and Geometric Function Theory (10 cr.)

The hyperbolic metric, various forms of the Schwarz-Pick Lemma, results in geometric function theory

Prerequisites:
Recommended: Complex Analysis I

MAST31014 Harmonic Analysis I (10 cr.)

Maximal operators, Calderon-Zygmund decomposition, BMO, interpolation

Prerequisites:

MAST31015 Harmonic Analysis II (10 cr,)

Singular Integrals, Calderon -Zygmund operators.

Prerequisites:
Recommended: Harmonic Analysis I

MAST31008 Fourier Analysis I (10 cr.)

Fourier series, their properties such as convergence and applications

Prerequisites: Measure and Integration, Functional Analysis

MAST31009 Fourier Analysis II (10 cr.)

Continuous Fourier transform on L^p-spaces and on tempered distributions

Prerequisites: Fourier Analysis I, Real Analysis I

MAST31010 Partial Differential Equations I (10 cr.)

Linear transport equation, Laplace's equation, heat equation and wave equation.

Prerequisites: Differential Equations I & II, Vector Analysis I & II

MAST31011 Partial Differential Equations II (10 cr.)

Sobolev spaces and De Giorgi-Nash-Moser theory.

Prerequisites: Measure and Integration

MAST31403 Integral Equations (10 cr.)

Volterra equations, Compact operators in Hilbert spaces, Fredholm's alternative, applications to PDE's

Prerequisites: Measure and integration, Differential Equations II, Introduction to Functional Analysis

MAST31401 Inverse Problems (15 cr.)

Measurement models leading to ill-posed inverse problems. Computational solution using truncated singular value decmposition and variational regularization. Practical project work with X-ray tomography data collected in the Industrial Mathematics Laboratory.

Prerequisites: Linear Algebra

MAST31404 Introduction to Wavelets and Fourier Analysis (5 cr.)

Fourier transform, windowed Fourier transform, Haar wavelet transform. Introduction to multiresolution analysis. Matlab programming with image processing examples.

Prerequisites: Linear Algebra

MAST31402 Bayesian Inversion (15 cr.)

Bayesian approach to ill-posed inverse problems. Computational methods for exploring the posterior distribution. Theory of well-posedness and stability properties of Bayesian inversion. Practical project work in the Industrial Mathematics Laboratory.

Prerequisites: Probability Theory, Measure and Integration theory

MAST31301 Introduction to Mathematical Physics (10 cr.)

Depends on topics covered.

Prerequisites: Depends on topics covered.

MAST31303 Quantum Dynamics (10 cr.)

Schrödinger equation with various potentials and boundary conditions, Wigner function, unbounded operators, self-adjointness, tensor product spaces, multi-particle systems and their creation operator formalism.

Prerequisites: Measure and Integration

MAST31302 Hamiltonian Dynamics (10 cr.)

Definition of dynamical system and Hamiltonian dynamical system, Hamiltonian Formalism, Linear Hamiltonian systems, Variational principles, Canonical Transformations, Hamilton–Jacobi Theory and Action-Angle Variables, Poisson Brackets and Constants of Motion,Canonical Perturbation Theory, Small divisors and KAM theorem,Transition to Chaos in Hamiltonian Systems, Criteria for Local and Global Chaos.

Prerequisites:

MAST31005 Algebra II (10 cr.)

Advanced topics in the theory of groups, rings, fields and modules.

Prerequisites: Algebraic Structures I & II

MAST31023 Introduction to Algebraic Topology (10 cr.)

Basic notions related to homotopy: fundamental group, examples, applications,  covering space theory;  basic notions related to homology:  chain complexes,  singular homology groups, Eilenberg-Steenrod axioms,  examples, applications.

Prerequisites: Topology I & II, Algebra I

MAST31020 Homotopy theory ( 5 cr.)

Higher homotopy groups; fibrations and cofibrations; exact homotopy sequences.

Prerequisites: Topology I & II, Algebra I, Introduction to Algebraic Topology

MAST31021 Homotopy and Vector Bundles (10 cr.)

Homotopy theory and the fundamental group; basic theory of differentiable manifolds, tangent bundles and more generally vector bundles;   homotopy classification of vector bundles.

Prerequisites: Linear algebra I & II, Algebra I, Vector analysis, Topology I & II

MAST31022 Topological Transformation Groups (10 cr.)

Topological groups, topological properties of actions of compact and locally compact groups, Haar integral, slices and twisted products, fundamental sets.

Prerequisites: Linear algebra I & II, Topology I & II, Algebra I

MAST31024 Advanced Algebraic Topology (10 cr.)

Depends on the topic of the course.

Prerequisites: Depends on the topic of the course

MAST31025 De Rham Theory (10 cr.)

Exterior algebra, differential forms, smooth manifolds, de Rham cohomology, integration on manifolds.

Prerequisites: Topology II, Real Analysis I
Recommended: Introduction to differential geometry, Introduction to Algebraic Topology

MAST31201 Mathematical Logic (10 cr.)

The main topics of the course are the completeness theorems of propositional and predicate logic and Gödel's incompleteness theorems. Methods and topics needed for these are formal deduction, definability, primitive recursive and recursive functions.

Prerequisites: Mathematical maturity acquired during B.Sc. level mathematics courses. Introduction to logic I&II help, but are not strictly necessary.

MAST31202 Axiomatic Set Theory (10 cr.)

Ordinals, cardinals, forcing, consistency results.

Prerequisites: Introduction to Logic II

MAST31203 Model Theory (10 cr.)

Ultraproducts, elimination of quantifiers, saturation, atomicity, Ehrenfeucht-Fraisse game.

Prerequisites: Mathematical maturity.

MAST31204 Recursion Theory (10 cr.)

Prerequisites:

MAST31205 Finite model Theory (10 cr.)

Ehrenfeucht–Fraïssé game and the pebble game, finite automata, second-order logic, fixed-point logics, infinitary logic, Turing machines and complexity classes, zero-one laws.

Prerequisites: Introduction to Logic II

MAST31206 Dependence Logic (5 cr.)

Team semantics, dependence logic and its variants.

Prerequisites: Introduction to Logic II

MAST31207 Classification Theory (10 cr.)

Forking, primary models, decompositions.

Prerequisites: Model Theory

MAST31208 Non-Elementary Model Theory (5 cr.)

Abstract elementary classes, tameness, excellence, homogeneity.

Prerequisites: Mathematical maturity.

MAST31209 Descriptive Set Theory (5 cr.)

Borel sets, analytic sets, measure, determinacy.

Prerequisites: Mathematical maturity.

MAST31210 Large Cardinals (5 cr.)

Large cardinals, trees, compactness properties.

Prerequisites: Set Theory

MAST31211 Strong Logics (5 cr.)

Strong logics, completeness theorems.

Prerequisites: Set Theory, Model Theory

MAST31212 History of Logic (5 cr.)

First and second order logic, completeness and incompleteness theorem.

Prerequisites: Basic Logic

MAST31213 Complexity Theory (10 cr.)

Turing machines, basic complexity classes, hierarchy theorems, reductions and completeness.

Prerequisites: Mathematical maturity.

MAST31215 Introduction to Continuous Logic (5 cr.)

Metric structures, formulas and statements of continuous logic, metric ultraproducts, compactness.

Prerequisites: Basic course in logic.

MAST31701 Probability Theory I (5 cr.)

Measure theoretic foundations of probability, independence, laws of large numbers, characteristic functions and the central limit theorm, Gaussian measures, recurrence/transience of random walks.

Prerequisites: Analysis I-II, Topology I, Vector Analysis I, Measure and Integration

MAST31702 Probability Theory II (5 cr.)

Discrete time Markov chains, Poisson process, conditional expectation, martingales.

Prerequisites: Probability Theory I and its prerequisites.

MAST31704 Topics in Probability I (5 cr.)

Some of the following: general aspects of stochastic processes, Markov chains and processes, Brownian motion, ergodic theory, large deviations and concentration of measure, geometric probability, integrable probability, point processes and determinantal processes, random graphs.

Prerequisites: Probability theory I,II, and their prerequisites.

MAST31705 Topics in Probability II (5 cr.)

Some of the following: general aspects of stochastic processes, Markov chains and processes, Brownian motion, ergodic theory, large deviations and concentration of measure, geometric probability, integrable probability, point processes and determinantal processes, random graphs.

Prerequisites: Probability Theory I,II, and their prerequisites.

MAST31801 Mathematical Finance I (5 cr.)

Arbitrage pricing theory and Market completeness. Pricing in incomplete markets. The necessary concepts from convex analysis will be also introduced.

Prerequisites: Probability Theory I & II
Recommended: Stochastic Analysis

MAST31805 Mathematical Finance II (5 cr.)

Financial markets in continuous time. Black and Scholes formula. Incomplete market models. Interest rate models.

Prerequisites: Probability theory I & II, and their prerequisites
Recommended: Stochastic Analysis

MAST31802 Risk Theory (10 cr.)

Number of claims, Total claim amount, Reinsurance, Outstanding claims, Ruin theory.

Prerequisites: Probability Theory
Recommended: Stochastic Analysis I & II

MAST31806 Advanced Risk Theory (5 cr.)

Prerequisites: Probability Theory, Risk Theory
Recommended: Stochastic Analysis I & II

MAST31803 Life Insurance Mathematics (10 cr.)

Deterministic interest rates, Mortality, Prices, Reserves, Multi state models.

Prerequisites: Probability Theory
Recommended: Stochastic Analysis I & II

MAST31804 Tariff Theory (5 cr.)

Deductibles, Collective premium, Crediblility theory, Exponential smoothing, Bonus systems, Premiums and solvency.

Prerequisites: Probability Theory, Risk Theory
Recommended: Stochastic Analysis I & II

MAST31706 Stochastic Analysis I (5 cr.)

The  course is focused on stochastic integration theory, with respect to martingales and processes with finite variations, including continuous martingales and processes with jumps.

Prerequisites: Probability theory I & II, and their prerequisites

MAST31710 Stochastic Analysis II (5 cr.)

Stochastic differential equations and applications.

Prerequisites: Probability theory I & II, and their prerequisites

MAST31707 Malliavin Calculus (10 cr.)

The purpose of this course is to empower the students with the modern tools of Malliavin calculus. Among the applications of Malliavin calculus is the construction of anticipative stochastic integrals, and the Ito-Clark-Ocone representation formula of Brownian functionals. The Malliavin calculus with respect to the Poisson process will be also discussed.

Prerequisites: Probability Theory I & II
Recommended: Stochastic Analysis I &II

MAST31501 Mathematical Modelling (10 cr.)

This course focuses on how to construct and analyse mathematical models of population behavior, rigorously derived from the mechanics of the underlying processes on the level of the individuals, and the analysis of these models to obtain results relevant to the application field. Many examples are taken from ecology but provide methods that are transferable to other fields as well. Topics include mono- and bimolecular reactions, reaction networks, the principle of mass action, predator-prey models, competition models, diffusion and taxis, pattern formation, structured populations and developmental delays.

Prerequisites: BSc courses on differential equations, linear algebra, probability theory.

LSI Introduction to Mathematical Biology (10 cr.)

This course is intended as a first course in mathematical biology, with emphasis on ecology. Subject areas include the dynamics of populations and of interacting species, natural selection and evolution, with some examples from other fields such as enzyme kinetics and regulation. An advantage of approaching mathematical biology via ecology is that the necessary biological background is minimal. The methods used for the construction and analysis of the models are transferable to other fields.

Prerequisites: BSc courses on differential equations, linear algebra, probability theory.

Adaptive dynamics is a modern mathematical framework to model evolution by natural selection, where selection derives from (possibly complex) ecological interactions between the individuals. The course contains the methods and theorems of adaptive dynamics as well as a number of applications to concrete biological problems.

Prerequisites: Ordinary Differential Equations

MAST31504 Stochastic Population Models (10 cr.)

This is a course about population models that cannot be properly described or analysed in a purely deterministic way because of the presence of noise. We consider two kinds of noise depending on its origin: The noise may be exogenous, i.e., due to autonomous processes external to the population itself and affecting it by causing population parameters to fluctuate in time. Especially in small populations, the noise may also be endogenous, i.e., due to stochastic demographic in the number of births and deaths within any given interval of time.

Prerequisites: differential equations, probability theory

MAST31503 Spatial Models in Ecology and Evolution (10 cr.)

This course will explore how to model the dynamics and evolution of populations with spatial movement, spatial constraints and spatial interactions between organisms. We study diffusion, travelling waves, pattern formation and Turing instability, stochastic patch occupancy models, structured metapopulation models, probabilistic cellular automata and coupled map lattices. We also discuss topical issues of evolutionary biology where spatial structure plays a crucial role, e.g. the evolution of mobility (dispersal), specialisation to different environments, and the evolution of altruistic behaviour. This is a course in applied mathematics. Instead of choosing the problem to suit a method, we emphasize the use of versatile techniques. We introduce/review methods for ordinary differential equations and difference equations, partial differential equations, Fourier analysis, stochastic processes, pair approximation methods, game theory and adaptive dynamics. When necessary, we turn to numerical analysis.

Prerequisites: BSc courses on differential equations, linear algebra, probability theory; basic computer programming for project work
Recommended: Mathematical modelling or Introduction to mathematical biology

LSI Evolution and the Theory of Games (5 cr.)

This course is about modelling animal behavior using the theory of games. Loosely speaking, a game is a mathematical model of a conflict of interest where the costs and benefits to individual players do not only depend on their own behavior but also on the behavior of the others. This makes a game essentially different from a mere optimization problem. After a short introduction to various basic notions and techniques from general game theory, we rapidly focus on games as a means to understand the evolution of animal behavior such as aggression, cooperation, bluff and others.

Prerequisites:

LSI Mathematics of Infectious Diseases (10 cr.)

This course is an introduction to mathematical modelling of the dynamics of infectious diseases in human and other populations. The topics include the basic models of epidemics (e.g. SIR); the basic reproduction number (R0); vaccination; the final size of an epidemic; persistence; the evolution of pathogens; diseases in small communities; time to extinction; epidemics in structured host populations; multi-level mixing (households); epidemics on networks. The course is given as a book-reading course based on a textbook that approaches much of the material via problem-solving. Lectures and exercise classes are combined; next to traditional lectures, also students present sections of the book and discuss the solutions of the problems.

Prerequisites: BSc courses on differential equations, linear algebra, probability theory

MAST32001 Computational Statistics I (5 cr.)

Most important numerical methods and principles for statistics. Theory and practice of methods for sampling from probability distributions including rejection sampling, importance sampling, generic Markov chain Monte Carlo and Hamiltonian Monte Carlo. Overview of methods for approximate inference including Laplace approximation and variational inference. The computer projects can be implemented in Python or R.

Prerequisites: BSc courses on linear algebra, probability theory, statistical inference; basic programming skills
Recommended: Fundamentals of differential equations

MAST32002 Computational Statistics II (5 cr.)

A larger project work in one of the topics covered in Computational Statistics I.

Prerequisites: Computational Statistics I

MAST32005 Spatial Modelling and Bayesian Inference (5 cr.)

Gaussian process, its properties and use in spatial statistics. Inference and prediction with hierarchical Gaussian process models. Maximum a posterior and Markov chain Monte Carlo approaches methods for inference.

Prerequisites:

MAST32004 Advanced Course in Bayesian Statistics (5 cr.)

Foundations of Bayesian statistics, model comparison and validation, decision analysis and experimental design.

Prerequisites: Bachelor’s studies
Recommended: Computational Statistics

MAST32006 High Dimensional Statistics (5 cr.)

Statistical inference when either the number of data units is large and/or each unit has been measured on a large number of variables. Includes techniques for variable selection, dimension reduction and large-scale inference. Applications across modern data science.

Prerequisites: Bachelor’s studies
Recommended: Computational Statistics

MAST33009 Structural Equation Models (5 cr.)

Basics of structural equation models, introduction to Mplus software, single-group analyses (first-order confirmatory factor analysis, full structural equation model), multiple-group analyses (analysis of covariance structures, full structural equation model).

Prerequisites: Bachelor’s studies

MAST33003 Nonparametric and Robust Methods (5 cr.)

Sign and rank estimates, tests and confidence intervals. Hodges-Lehmann -type methods.

Prerequisites: Probability II and Statistical Inference II
Recommended: Bachelor’s studies

MAST33004 Robust Regression (5 cr.)

LS-, LAD-,  M-, GM-, S-, LTS-, MM- and R-estimation. Measures of robustness.  Hypothesis testing.

Prerequisites: Probability II and Statistical Inference II
Recommended: Bachelor’s studies

LSI Trends in Biostatistics and Bioinformatics (LSI)

LSI Phylogenetic Inference and Data Analysis (LSI)

LSI Statistical Population Genetics (LSI)

MAST32007 Time Series Analysis I (5 cr.)

The basic concepts in strationary time series analysis. Empirically most relevant autoregressive moving average (ARMA) model: its properties, model selection, estimation, testing and forecasting. The basic models for conditional heteroskedasticity particularly applied in empirical finance are studied.

Prerequisites: Bachelor’s studies in statistics

MAST32008 Time Series Analysis II (5 cr.)

The estimation and testing theory of the vector autoregressive (VAR) model and the vector error correction (VEC) form of the VAR model

Prerequisites: Bachelor’s studies in statistics

MAST33005 Linear Mixed Models

for nonstationary trending time series particularly applied in economic applications.

Prerequisites: Subject level statistical inference

MAST33006 Generalized Linear Mixed Models

MAST33007 Discrete Markov Processes

MAST33008 Statistical Demography

MAST32003 Statistical Inference III (5 Cr.)

Delta method and convergence in probability and distribution. Large sample properties of the logarithmic likelihood function and maximum likelihood estimator. Asymptotic test statistics and confidence bounds.

Prerequisites: Bachelor’s studies.

(Bachelor’s Level) Generalized Linear Models I (5 Cr.)

Continuous, dichotomous,  polytomic  and count response.  Log-linear models.  Comparison of the models. Deviance.

Prerequisites: Probability II and Statistical Inference II
Recommended: Bachelor’s studies.

MAST33001 Generalized Linear Models II (5 Cr.)

Continuous, dichotomous,  polytomic  and count response.  Log-linear models.  Comparison of the models. Deviance.

Prerequisites: Probability II and Statistical Inference II
Recommended: Bachelor’s studies.

Connections and special aspects of widely used methods of inference.

Prerequisites: Subject level statistical inference

MAST33018 Survey Sampling (5 Cr.)

Course gives an overview on sampling methods and estimation under different sampling designs and the use of the methods in empirical research and statistics production. Methods include simple random sampling, systematic sampling, Bernoulli sampling, Poisson sampling, PPS sampling, stratified sampling and multi-stage sampling and the estimation of finite population parameters under the various sampling designs, including HT estimation, GREG estimation and calibration as well as variance estimation. Special emphasis is in methods to incorporate auxiliary information into sampling design and estimation design. Empirical examples and case studies are given.

Prerequisites:

MAST33019 Survey Methodology And European Statistical System (5 Cr.)

Course covers key materials required for the core module, including essentials on ESS and NSS as well as the necessary statistics quality and production process related topics.

Prerequisites:

MAST33010 Introduction To Register-Based Research (5 Cr.)

This course provides introduction to empirical research using secondary data sources (i.e. data that have not been collected primarily for the purposes of specific research question), most widely used administrative registers, study designs suitable for register-based research and the principles of data preprocessing as seen in published research.

Prerequisites:
Recommended: Bachelor’s studies

MAST33011 Data Analysis With SAS (5 Cr.)

The course covers the essential data handling and data analysis tools implemented in selected SAS software packages (BASE, STAT, GRAPH, IML). Data analysis by SAS tools.

Prerequisites:

MAST33012 Demographic Analysis (5 Cr.)

Measurement and modeling of health, life table, multi-state life table, Sullivan's method for health expectancies, stationarity assumptions for the Sullivan method.

Prerequisites: Generalized Linear Models; Linear Algebra I
Recommended: Bachelor’s studies

MAST33013 Register-Based Data Analysis (5 Cr.)

During this course essential aspects of pragmatic register-based data analysis will be covered: how to store and preprocess large data sets, how to combine different data sources, which kind of study designs can be used, how to avoid common biases and how to document the research process in a repeatable manner.

Prerequisites: Introduction to register-based research
Recommended: Bachelor’s studies, event-history analysis, programming

MAST33014 Small Area Estimation (5 Cr.)

The course covers topics in modern statistical methods for the estimation of parameters for population subgroups or domains and small areas (Small Area Estimation, SAE).

Prerequisites: Survey sampling

MAST33015 Analysis Of Complex Surveys (5 Cr.)

The course covers topics in the analysis of complex survey data where the complexity arises from the complex (multi-stage) cluster sampling design. Complex sampling designs involve correlation between observations within clusters. Methods are needed that properly account for the complexities in the analysis phase. Methods covered include variance estimation by linearization, jackknife and bootstrap, design-based Wald tests of independence and homogeneity, linear and logistic regression and ANCOVA, and linear mixed modelling. Methods are applied with SAS and R tools to complex survey data collected by stratified cluster sampling.

Prerequisites: Survey sampling

MAST33016 Applied Logistic Regression (5 Cr.)

The logic of statistical analysis of dichotomous responses and their practical analysis.

Prerequisites: Introductory statistics

MAST31405 Inverse Problems Project Work (5 Cr.)

Measurement and calibration of data, implementation of a computational inversion method.

Prerequisites: Inverse Problems

MAST31506 Dynamics Of Lotka-Volterra Systems (5 Cr.)

This course focuses on how to analyse the dynamics of Lotka-Volterra systems

Prerequisites: BSc courses on ordinary differential equations, linear algebra

MAST31216 Models Of Arithmetic

Basic theory of models of arithmetic including the arithmetized completeness theorem; Scott sets.

Prerequisites:

MAST31029 Clifford Analysis

Theory of Clifford algebras and applications to analysis.

Prerequisites: Analysis I & II Linear algebra, Partial differential equations is good but not necessary.
Recommended: Master’s studies

MAST31304 Topics In Mathematical Physics

The content of the course may vary over time. Topics   will be taken from currently active research areas in mathematical physics.

Prerequisites:

MAST31709 Optimal Stochastic Control With Applications To Finance

Deterministic and stochastic control. Dynamic programming theory. Hamilton-Jacobi-Bellman equation. Filtering theory. Optimal investment with partial information.

Prerequisites: Basic notions of probability theory and stochastic calculus.

MAST31901 History Of Mathematics (5 Cr.)

Number systems, Greek mathematics, algorithms for computing divisions and square roots, problem solving before algebra, algebra cossa, trigonometry, analytical geometry, infinitesimal calculus, features of 18th century mathematics. Early mathematics in Finland.

Prerequisites: Bachelor’s studies

MAST31902 Analytic Number Theory (10 Cr.)

Number theoretic functions, their growth rate. Perron formula. Dirichlet series. Prime number theorem.  Dirichlet theorem on primes in an arithmetic sequence. Basic properties of Riemann zeta function and its connection to the distribution of primes. Other topics will be decided durung the lecture.

Prerequisites: Basic knowledge of elementary number theory and algebra. Complex analysis I, Basic measure and in integration theory
Recommended: Complex Analysis II

MAST30002 Maturity Test

MAST31903 Philosophy Of Science

MAST30004 Work Experience (5 Cr.)

Working in an area of mathematics and statistics relevant to the MAST-programme.

Prerequisites: Bachelor’s studies in mathematics or statistics
Recommended: A core course and some specialization studies of MAST-programme master studies