Computer Science 1 (TÖL101G)

Note: Only one course of either TÖL101G Tölvunarfræði 1 or TÖL105G Tölvunarfræði 1a can count towards the BS degree.

The Java programming language is used to introduce basic concepts in computer programming: Expressions and statements, textual and numeric data types, conditions and loops, arrays, methods, classes and objects, input and output. Programming and debugging skills are practiced in quizzes and projects throughout the semester.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Macroeconomics I (HAG103G)

The course aims to give the students an insight into the main theories, concepts, topics, and principles of macroeconomics and macroeconomic activity. The course stresses both the analytical content and applied usefulness of the topics covered and how they relate to various current economic issues at home and abroad. A sound knowledge of macroeconomics prepares students for various other economics courses, and for life.

Introduction to Mathematics (STÆ110G)

The course covers the language of mathematics and the fundamentals of logic and set theory.
The treatment of logic and set theory is naive but sufficiently precise to serve as a foundation for the general use of logic and mathematics in further mathematical studies. Emphasis is placed on basic concepts such as quantifiers, implications, sets, mappings, injective and surjective functions. Training is provided in formulating simple proofs. The course is taught once a week, three class hours at a time. A written final exam will be held in teaching week 12. Students complete assignments during the semester that count for 30% of the final grade.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Linear Algebra (STÆ107G)

Basics of linear algebra over the reals.  

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullk space.  The dot product, length and angle measures.  Volumes in higher dimension and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Microeconomics II (HAG201G)

Intermediate microeconomic theory. Basic factors of price theory, uncertainty, including analysis of demand, costs of production and supply relationships, and price and output determination under various market structures, market failures and public choice.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis IIA (STÆ207G)

Emphasis is laid on the theoretical aspects of the material. The aim is that the students acquire understanding of fundamental concepts and are able to use them, both in theoretical consideration and in calculations. Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobian matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flows. Cylindrical and spherical coordinates. Taylor polynomials. Extrema and classification of stationary points. Extrema with constraints. Implicit functions and local inverses. Line integrals and potential functions. Proper and improper multiple integrals. Change of variables in multiple integrals. Simply connected regions. Integration on surfaces. Theorems of Green, Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Theoretical Statistics (STÆ313M)

Likelihood, Sufficient Statistic, Sufficiency Principle, Nuisance Parameter, Conditioning Principle, Invariance Principle, Likelihood Theory. Hypothesis Testing, Simple and Composite Hypothesis, The Neyman-Pearson Lemma, Power, UMP-Test, Invariant Tests. Permutation Tests, Rank Tests. Interval Estimation, Confidence Interval, Confidence, Confidence Region. Point Estimation, Bias, Mean Square Error. Assignments constitute 30% of the final grade.

Complex Analysis I (STÆ301G)

Complex numbers and the topology of the complex plane. Sequences and series of complex numbers. Differentiable and holomorphic functions. Sequences and series of functions; power series. Path integration and primitives. The exponential function and related functions. Winding numbers. The Cauchy theorem, the integral formula of Cauchy and consequences. The identity theorem, the open mapping theorem and the maximum principle. Laurent series, isolated singularities and their classification. The theorem of residues and residue calculus. The argument principle and Rouché's theorem. Connections with real analysis: The Cauchy-Riemann equations, harmonic functions and the integral formulas of Poisson and Schwarz. Holomorphic functions defined by integrals (e. g. the Laplace transformation). Conformal mapping and the Riemann mapping theorem.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Mathematical Analysis IIIA (STÆ304G)

The course is an introduction to three important tools of applied mathematics, namely ordinary differential equations, Fourier-series and partial differential equations.  Some basic theoretical properties are proved and solution methods presented. 

Ordinary differential equations: linear differential equations of order n, the Cauchy problem, Picard's existence theorem, solution by power series and equations with singular points.  Fourier series: convergence point-wise, uniformly and in the mean-square, Parseval's equation. 

Partial differential equations: the heat equation and the wave equation solved on a finite interval by separation of variables and Fourier series and their solutions compared, the Dirichlet problem for the Laplace equation on the rectangle and the disc, the Poisson integral formula.

Linear algebra II (STÆ401M)

Modules and linear maps. Free modules and matrices. Quotient modules and short exact sequences. Dual modules. Finitely generated modules over a principal ideal domain. Linear operators on finite dimensional vector spaces.

Galois theory (STÆ403M)

Selected topics in commutative algebra. 

Subject matter: Fields and field extensions.  Algeraic extensions, normal extensions and seperable extensions.  Galois theory.  Applications.  
Noetherian rings.  Hilbert's basis theorem and Hilbert's Nullstellensatz.

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Measure and Integration Theory (STÆ402M)

Measure spaces, measures, outer measures. The Lebesgue measure on Rn. Measurable functions, the monotone convergence theorem, Fatou’s Lemma. Integrable functions, Lebesgue’s  dominated convergence theorem and applications. Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces. Fourier series. Product of measure spaces, theorems of Tonelli and Fubini. Complex measures. Jordan decomposition and Lebesgue decomposition of measures, Radon-Níkodým theorem. Continuous linear functionals on Lp-spaces. Image measures, transformation formula for the Lebesgue measure on Rn.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Discrete mathematics (TÖL104G)

Propositions, predicates, inference rules. Set operations and Boolean algebra. Induction and recursion. Basic methods of analysis of algorithms and counting. Simple algorithms in number theory. Relations, their properties and representations. Trees and graphs and related algorithms. Strings, examples of languages, finite automata and grammars.

Innovation, IP protection, and application (VON101M)

Minimum number of students registered for the course to be taught: 10

This course will cover basics in intellectual property rights with emphasis on patents. What is intellectual property and when and how can they best be protected? How does intellectual property rights work (e.g. patents), and how can they be used and applied in innovation and industry. The system‘s organization will be covered, as well as the patent process, making a financial plan relating to patents, and database searches.

Operations Research 2 (IÐN508M)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover integer programming and modeling with stochastic programming. The student will become familiar with building mathematical models using Python.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbours, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course, the students work on a practical machine learning project.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Research Project (STÆ262L)

Research Project

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Introduction to Logic (STÆ528M)

Logical deductions and proofs. Propositional calculus, connectives, truth functions and tautologies. Formal languages, axioms, inference rules. Quantifiers. First-order logic. Interpretations. The compactness theorem. The Lövenheim-Skolem theorem. Computability, recursive functions. Gödel's theorem.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Combinatorics (STÆ533M)

This course is aimed at second and third year undergraduate mathematics students. The purpose is to introduce the student to several combinatorial structures, methods of their enumeration and useful properties. Particular emphasis will be placed on the systematic use of generating functions in enumeration.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Formal Languages and Computability (TÖL301G)

Finite state machines, regular languages and grammars, push-down automata, context-free languages and grammars, Turing machines, general languages and grammars, and their basic properties. Recursive and recursively enumerable languages, reduction between languages, connection to decision problems and proving unsolvability of such problems. The complexity classes P and NP, and NP-completeness. Examples of various models of computation.

Algorithms in Bioinformatics (TÖL504M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Algorithms in Bioinformatics (TÖL604M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Introduction to data science (REI202G)

The course provides an introduction to the methods at the heart of data science and introduces widely used software tools such as numpy, pandas, matplotlib and scikit-learn.

The course consists of 6 modules:

1. Introduction to the Python programming language.
2. Data wrangling and data preprocessing.
3. Exploratory data analysis and visualization.
4. Optimization.
5. Clustering and dimensionality reduction.
6. Regression and classification.

Each module concludes with a student project.

Note that there is an academic overlap with REI201G Mathematics and Scientific Computing and both courses cannot be valid for the same degree.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Statistical Consulting (MAS201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Applied data analysis (MAS202M)

The course focuses on statistical analysis using the R environment. It is assumed that students have basic knowledge of statistics and the statistical software R. Students will learn to apply a broad range of statistical methods in R (such as classification methods, resampling methods, linear model selection and tree-based methods). The course on 12 weeks and will be on "flipped" form. This means that no lectures will be given but students will read some material and watch videos before attending classes. Students will then work on assignments during the classes.

Research Project (STÆ262L)

Research Project

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Representation Theory of Finite groups (STÆ626M)

The definition of a linear representation of a group.  Schur's lemma. Group characters. The group algebra. Representation theory of the symmetric group. Applications of representation theory to group theory: Theorems of Frobenius and Burnside.

Analysis of Algorithms (TÖL403G)

Methodology for the design of algorithms and the analysis of their time conplexity. Analysis of algorithms for sorting, searching, graph theory and matrix computations. Intractable problems, heuristics, and randomized algorithms.

Computer Science 1 (TÖL101G)

Note: Only one course of either TÖL101G Tölvunarfræði 1 or TÖL105G Tölvunarfræði 1a can count towards the BS degree.

The Java programming language is used to introduce basic concepts in computer programming: Expressions and statements, textual and numeric data types, conditions and loops, arrays, methods, classes and objects, input and output. Programming and debugging skills are practiced in quizzes and projects throughout the semester.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Introduction to Mathematics (STÆ110G)

The course covers the language of mathematics and the fundamentals of logic and set theory.
The treatment of logic and set theory is naive but sufficiently precise to serve as a foundation for the general use of logic and mathematics in further mathematical studies. Emphasis is placed on basic concepts such as quantifiers, implications, sets, mappings, injective and surjective functions. Training is provided in formulating simple proofs. The course is taught once a week, three class hours at a time. A written final exam will be held in teaching week 12. Students complete assignments during the semester that count for 30% of the final grade.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Linear Algebra (STÆ107G)

Basics of linear algebra over the reals.  

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullk space.  The dot product, length and angle measures.  Volumes in higher dimension and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis IIA (STÆ207G)

Emphasis is laid on the theoretical aspects of the material. The aim is that the students acquire understanding of fundamental concepts and are able to use them, both in theoretical consideration and in calculations. Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobian matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flows. Cylindrical and spherical coordinates. Taylor polynomials. Extrema and classification of stationary points. Extrema with constraints. Implicit functions and local inverses. Line integrals and potential functions. Proper and improper multiple integrals. Change of variables in multiple integrals. Simply connected regions. Integration on surfaces. Theorems of Green, Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Complex Analysis I (STÆ301G)

Complex numbers and the topology of the complex plane. Sequences and series of complex numbers. Differentiable and holomorphic functions. Sequences and series of functions; power series. Path integration and primitives. The exponential function and related functions. Winding numbers. The Cauchy theorem, the integral formula of Cauchy and consequences. The identity theorem, the open mapping theorem and the maximum principle. Laurent series, isolated singularities and their classification. The theorem of residues and residue calculus. The argument principle and Rouché's theorem. Connections with real analysis: The Cauchy-Riemann equations, harmonic functions and the integral formulas of Poisson and Schwarz. Holomorphic functions defined by integrals (e. g. the Laplace transformation). Conformal mapping and the Riemann mapping theorem.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Mathematical Analysis IIIA (STÆ304G)

The course is an introduction to three important tools of applied mathematics, namely ordinary differential equations, Fourier-series and partial differential equations.  Some basic theoretical properties are proved and solution methods presented. 

Ordinary differential equations: linear differential equations of order n, the Cauchy problem, Picard's existence theorem, solution by power series and equations with singular points.  Fourier series: convergence point-wise, uniformly and in the mean-square, Parseval's equation. 

Partial differential equations: the heat equation and the wave equation solved on a finite interval by separation of variables and Fourier series and their solutions compared, the Dirichlet problem for the Laplace equation on the rectangle and the disc, the Poisson integral formula.

Linear algebra II (STÆ401M)

Modules and linear maps. Free modules and matrices. Quotient modules and short exact sequences. Dual modules. Finitely generated modules over a principal ideal domain. Linear operators on finite dimensional vector spaces.

Galois theory (STÆ403M)

Selected topics in commutative algebra. 

Subject matter: Fields and field extensions.  Algeraic extensions, normal extensions and seperable extensions.  Galois theory.  Applications.  
Noetherian rings.  Hilbert's basis theorem and Hilbert's Nullstellensatz.

Measure and Integration Theory (STÆ402M)

Measure spaces, measures, outer measures. The Lebesgue measure on Rn. Measurable functions, the monotone convergence theorem, Fatou’s Lemma. Integrable functions, Lebesgue’s  dominated convergence theorem and applications. Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces. Fourier series. Product of measure spaces, theorems of Tonelli and Fubini. Complex measures. Jordan decomposition and Lebesgue decomposition of measures, Radon-Níkodým theorem. Continuous linear functionals on Lp-spaces. Image measures, transformation formula for the Lebesgue measure on Rn.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Discrete mathematics (TÖL104G)

Propositions, predicates, inference rules. Set operations and Boolean algebra. Induction and recursion. Basic methods of analysis of algorithms and counting. Simple algorithms in number theory. Relations, their properties and representations. Trees and graphs and related algorithms. Strings, examples of languages, finite automata and grammars.

Innovation, IP protection, and application (VON101M)

Minimum number of students registered for the course to be taught: 10

This course will cover basics in intellectual property rights with emphasis on patents. What is intellectual property and when and how can they best be protected? How does intellectual property rights work (e.g. patents), and how can they be used and applied in innovation and industry. The system‘s organization will be covered, as well as the patent process, making a financial plan relating to patents, and database searches.

Operations Research 2 (IÐN508M)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover integer programming and modeling with stochastic programming. The student will become familiar with building mathematical models using Python.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbours, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course, the students work on a practical machine learning project.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Research Project (STÆ262L)

Research Project

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Introduction to Logic (STÆ528M)

Logical deductions and proofs. Propositional calculus, connectives, truth functions and tautologies. Formal languages, axioms, inference rules. Quantifiers. First-order logic. Interpretations. The compactness theorem. The Lövenheim-Skolem theorem. Computability, recursive functions. Gödel's theorem.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Combinatorics (STÆ533M)

This course is aimed at second and third year undergraduate mathematics students. The purpose is to introduce the student to several combinatorial structures, methods of their enumeration and useful properties. Particular emphasis will be placed on the systematic use of generating functions in enumeration.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Formal Languages and Computability (TÖL301G)

Finite state machines, regular languages and grammars, push-down automata, context-free languages and grammars, Turing machines, general languages and grammars, and their basic properties. Recursive and recursively enumerable languages, reduction between languages, connection to decision problems and proving unsolvability of such problems. The complexity classes P and NP, and NP-completeness. Examples of various models of computation.

Algorithms in Bioinformatics (TÖL504M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Algorithms in Bioinformatics (TÖL604M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Introduction to data science (REI202G)

The course provides an introduction to the methods at the heart of data science and introduces widely used software tools such as numpy, pandas, matplotlib and scikit-learn.

The course consists of 6 modules:

1. Introduction to the Python programming language.
2. Data wrangling and data preprocessing.
3. Exploratory data analysis and visualization.
4. Optimization.
5. Clustering and dimensionality reduction.
6. Regression and classification.

Each module concludes with a student project.

Note that there is an academic overlap with REI201G Mathematics and Scientific Computing and both courses cannot be valid for the same degree.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Statistical Consulting (MAS201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Applied data analysis (MAS202M)

The course focuses on statistical analysis using the R environment. It is assumed that students have basic knowledge of statistics and the statistical software R. Students will learn to apply a broad range of statistical methods in R (such as classification methods, resampling methods, linear model selection and tree-based methods). The course on 12 weeks and will be on "flipped" form. This means that no lectures will be given but students will read some material and watch videos before attending classes. Students will then work on assignments during the classes.

Research Project (STÆ262L)

Research Project

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Representation Theory of Finite groups (STÆ626M)

The definition of a linear representation of a group.  Schur's lemma. Group characters. The group algebra. Representation theory of the symmetric group. Applications of representation theory to group theory: Theorems of Frobenius and Burnside.

Analysis of Algorithms (TÖL403G)

Methodology for the design of algorithms and the analysis of their time conplexity. Analysis of algorithms for sorting, searching, graph theory and matrix computations. Intractable problems, heuristics, and randomized algorithms.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Computer Science 1 (TÖL101G)

Note: Only one course of either TÖL101G Tölvunarfræði 1 or TÖL105G Tölvunarfræði 1a can count towards the BS degree.

The Java programming language is used to introduce basic concepts in computer programming: Expressions and statements, textual and numeric data types, conditions and loops, arrays, methods, classes and objects, input and output. Programming and debugging skills are practiced in quizzes and projects throughout the semester.

Physics 1 R Lab (EÐL108G)

There are 4 lab sessions with experiments mainly from mechanics, with emphasis on teaching students methods of data collection and data processing. Student hand in a lab report on each experiment. They also hand in a final report from one of these that is intended to look more like a journal article.

Physics 1 R (EÐL107G)

Introduce students to methods and fundamental laws of mechanics, waves and thermodynamics, to the extent that they can apply their knowledge to solve problems. 

Concepts, units, scales and dimensions.  Vectors. Kinematics of particles. Particle dynamics, inertia, forces and Newton's laws. Friction. Work and energy, conservation of energy. Momentum, collisions. Systems of particles, center of mass. Rotation of a rigid body.  Angular momentum and moment of inertia. Statics. Gravity. Solids and fluids, Bernoulli's equation. Oscillations: Simple, damped and forced. Waves. Sound.  Temperature. Ideal gas. Heat and the first law of thermodynamics. Kinetic theory of gases. Entropy and the second law of thermodynamics.

Note that the textbook is accessible to students via Canvas free of charge.

Introduction to Mathematics (STÆ110G)

The course covers the language of mathematics and the fundamentals of logic and set theory.
The treatment of logic and set theory is naive but sufficiently precise to serve as a foundation for the general use of logic and mathematics in further mathematical studies. Emphasis is placed on basic concepts such as quantifiers, implications, sets, mappings, injective and surjective functions. Training is provided in formulating simple proofs. The course is taught once a week, three class hours at a time. A written final exam will be held in teaching week 12. Students complete assignments during the semester that count for 30% of the final grade.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Linear Algebra (STÆ107G)

Basics of linear algebra over the reals.  

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullk space.  The dot product, length and angle measures.  Volumes in higher dimension and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Physics 2 R (EÐL206G)

Introduction to electrodynamics in material; from insulators to superconductors.  Charge and electric field. Gauss' law. Electric potential. Capacitors and dielectrics. Electric currents and resistance. Circuits. Magnetic fields. The laws of Ampère and Faraday. Induction. Electric oscillation and alternating currents. Maxwell's equations. Electromagnetic waves. Reflection and refraction. Lenses and mirrors. Wave optics.

Physics 2 R Lab (EÐL207G)

There are four 4 hour lab sessions and two 3 hour sessions, from optics and electromagnetism. Students hand in a lab report on each experiment. They also hand in a final report from one of the 4 hour experiments that is intended to look more like a journal article.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis IIA (STÆ207G)

Emphasis is laid on the theoretical aspects of the material. The aim is that the students acquire understanding of fundamental concepts and are able to use them, both in theoretical consideration and in calculations. Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobian matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flows. Cylindrical and spherical coordinates. Taylor polynomials. Extrema and classification of stationary points. Extrema with constraints. Implicit functions and local inverses. Line integrals and potential functions. Proper and improper multiple integrals. Change of variables in multiple integrals. Simply connected regions. Integration on surfaces. Theorems of Green, Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Geophysical Exploration (JEÐ504M)

A full semester course – 14 weeks.

a) One week field work at the beginning of autumn term.  Several geophysical methods applied to a practical problem.

b) Geophysical exploration methods and their application in the search for energy resources and minerals. Theoretical basis, instruments, measurement procedures, data processing and interpretation. Seismic reflection and refraction, gravity, magnetics, electrical methods, borehole logging. Practical work includes computations, model experiments.  Interpretation and preparation of report on field work done at beginning of course.

Complex Analysis I (STÆ301G)

Complex numbers and the topology of the complex plane. Sequences and series of complex numbers. Differentiable and holomorphic functions. Sequences and series of functions; power series. Path integration and primitives. The exponential function and related functions. Winding numbers. The Cauchy theorem, the integral formula of Cauchy and consequences. The identity theorem, the open mapping theorem and the maximum principle. Laurent series, isolated singularities and their classification. The theorem of residues and residue calculus. The argument principle and Rouché's theorem. Connections with real analysis: The Cauchy-Riemann equations, harmonic functions and the integral formulas of Poisson and Schwarz. Holomorphic functions defined by integrals (e. g. the Laplace transformation). Conformal mapping and the Riemann mapping theorem.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Mathematical Analysis IIIA (STÆ304G)

The course is an introduction to three important tools of applied mathematics, namely ordinary differential equations, Fourier-series and partial differential equations.  Some basic theoretical properties are proved and solution methods presented. 

Ordinary differential equations: linear differential equations of order n, the Cauchy problem, Picard's existence theorem, solution by power series and equations with singular points.  Fourier series: convergence point-wise, uniformly and in the mean-square, Parseval's equation. 

Partial differential equations: the heat equation and the wave equation solved on a finite interval by separation of variables and Fourier series and their solutions compared, the Dirichlet problem for the Laplace equation on the rectangle and the disc, the Poisson integral formula.

Galois theory (STÆ403M)

Selected topics in commutative algebra. 

Subject matter: Fields and field extensions.  Algeraic extensions, normal extensions and seperable extensions.  Galois theory.  Applications.  
Noetherian rings.  Hilbert's basis theorem and Hilbert's Nullstellensatz.

Linear algebra II (STÆ401M)

Modules and linear maps. Free modules and matrices. Quotient modules and short exact sequences. Dual modules. Finitely generated modules over a principal ideal domain. Linear operators on finite dimensional vector spaces.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Measure and Integration Theory (STÆ402M)

Measure spaces, measures, outer measures. The Lebesgue measure on Rn. Measurable functions, the monotone convergence theorem, Fatou’s Lemma. Integrable functions, Lebesgue’s  dominated convergence theorem and applications. Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces. Fourier series. Product of measure spaces, theorems of Tonelli and Fubini. Complex measures. Jordan decomposition and Lebesgue decomposition of measures, Radon-Níkodým theorem. Continuous linear functionals on Lp-spaces. Image measures, transformation formula for the Lebesgue measure on Rn.

General Geophysics (JEÐ201G)

An introduction to the physics of the Earth. Origin and age of the Earth. Dating with radioactive elements. Gravity, shape and rotation of the Earth, the geomagnetic field, magnetic anomalies, palaeomagnetism, electric conductivity. Earthquakes, seismograph and seismic waves. Layered structure of the Earth, heat transport and the internal heat of the Earth. Geophysical research in Iceland.

Practicals include solving problems set for each week and exercises in the use of geophysical instruments.  Students write one essay on a selected topic in geophysics.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Introduction to Quantum Mechanics (EÐL306G)

The course is devoted to theoretical foundations of quantum mechanics. 

Prelude to quantum physics. Wave functions and probability, Schrödinger's equation, momentum and the uncertainty principle, stationary states, one-dimensional quantum systems. Schrödinger's equation in spherical coordinates, the hydrogen atom, angular momentum and spin. Identical particles and the Pauli principle. Two-level systems, emission and absorbtion of radiation.

Classical Mechanics (EÐL302G)

Newtonian dynamics of a particle in various coordinate systems. Harmonic, damped and forced oscillations of a pendulum. Nonlinear oscillations and chaos. Gravitation and tidal forces. Calculus of variations. Lagrangian and Hamiltonian dynamics, generalized coordinates and constraints. Central force motion and planetary orbits. Dynamics of a system of particles, collisions in a center-of-mass coordinate system and in a lab system. Motion in a non-inertial reference frame, Coriolis and centrifugal forces. Motion relative to the Earth. Mechanics of rigid bodies, inertia tensors and principal axes of inertia. Eulerian angles, and Euler's equations for a rigid body. Precession, motion of a symmetric top and stability of rigid body rotations. Coupled oscillations, eigenfrequencies and normal modes.

Thermodynamics and Introduction to Statistical Mechanics (EFN307G)

Basic principles and mathematical methods in thermodynamics,laws of thermodynamics, state functions, Maxwell relations, equilibrium, phase transitions, quantum statistical mechanics, ideal and real gases, specific heat, rate theory, Bose and Fermi distributions.

Thermodynamics and chemical reactions (VÉL303G)

The objective of the course is to teach the student the basic concepts of thermodynamic systems. The students should also understand different forms of energy, energy transport and conversion from one state to another. The student should be able to calculate the rates of chemical reactions and energy balance.

Continuum Mechanics and Heat Transfer (JEÐ503M)

Objectives:   To introduce continuum mechanics, fluid dynamics and heat transfer and their application to problems in physics and geophysics. I. Stress and strain, stress fields, stress tensor, bending of plates, models of material behaviour: elastic, viscous, plastic materials. II. Fluids, viscous fluids, laminar and turbulent flow, equation of continuity, Navier-Stokes equation. III. Heat transfer: Heat conduction, convection, advection and geothermal resources. Examples and problems from various branches of physics will be studied, particularly from geophysics.

Teaching statement: To do well in this course, students should actively participate in the discussions, attend lectures, give student presentations and deliver the problem sets assigned in the course. Students will gain knowledge through the lectures, but it is necessary to do the exercises to understand and train the use of the concepts. The exercises are intergrated in the text of the book, it is recommended to do them while reading the text. Instructors will strive to make the concepts and terminology accessible, but it is expected that students study independently and ask questions if something is unclear. In order to improve the course and its content, it is appreciated that students participate in the course evaluation, both the mid-term and the end of term course evaluation.

Electromagnetism 1 (EÐL401G)

The equations of Laplace and Poisson. Magnetostatics. Induction.  Maxwell's equations. Energy of the electromagnetic field. Poynting's theorem. Electromagnetic waves. Plane waves in dielectric and conducting media, reflection and refraction.  Electromagnetic radiation and scattering. Damping.

General Relativity (EÐL610M)

This course provides a basic introduction to Einstein's relativity theory: Special relativity, four-vectors and tensors. General relativity, spacetime curvature, the equivalence principle, Einstein's equations, experimental tests within the solar system, gravitational waves, black holes, cosmology.

Teachers: Benjamin Knorr and Ziqi Yan, postdocs at Nordita

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Groundwater Hydrology (JEÐ502M)

A 7-week intensive course (first 7 weeks of fall term). 

Taught if sufficient number of students. May be taught as a reading course.

Occurrence of groundwater, the water content of soil, properties and types of aquifers (porosity, retention, yield, storage coefficients; unconfined, confined, leaky, homogeneous, isotropic aquifers). Principles of groundwater flow. Darcy's law, groundwater potential, potentiometric surface, hydraulic conductivity, transmissivity, permeability, determination of hydraulic conductivity in homogeneous and anisotropic aquifers, permeability, flow lines and flow nets, refraction of flow lines, steady and unsteady flow in confined, unconfined and leaky aquifers, general flow equations. Groundwater flow to wells, drawdown and recovery caused by pumping wells, determination of aquifer parameters from time-drawdown data, well loss, capacity and efficiency. Sea-water intrusion in coastal aquifers. Mass transport of solutes by groundwater flow. Quality and pollution of groundwater. Case histories from groundwater studies in Iceland. Numerical models of groundwater flow.   Students carry out an interdisciplinary project on groundwater hydrology and management.

Seismology (JEÐ505M)

Stress and strain tensors, wave-equations for P- and S-waves. Body waves and guided waves. Seismic waves: P-, S-, Rayleigh- and Love-waves. Free oscillations of the Earth. Seismographs, principles and properties. Sources of earthquakes: Focal mechanisms, seismic moment, magnitude scales, energy, frequency spectrum, intensity. Distribution of earthquakes and depths, geological framework. Seismic waves and the internal structure of the Earth.

The course is either tought in a traditional way (lectures, exercises, projects) or as a reading course where the students read textbooks and give a written or oral account of their studies.

Dynamic Meteorology (EÐL515M)

The primitive equations are derived and applied on atmospheric weather systems on various scales. Geostrophic wind, gradient wind, sea breeze, thermal wind, stability and wind profile of the atmospheric boundary layer. Vertical motion. Gravity waves and Rossby waves. Introduction to quasi-geostrophic theory, vorticity equation, potential vorticity, omega-equation and geopotential tendency equation. Quasi-geostrophic theory of mountain flows.

Tectonics (JAR315G)

Tectonic motions control the nature of the planet we inhabit and the location of continents, mountain ranges, volcanoes, where earthquakes occur and even are important for controlling the Earth's climate. Structural geology and crustal movements in the world, with special emphasis on movements in Iceland. This course introduces the techniques of structural geology through a survey of the mechanics of rock deformation, a survey of the features and geometries of faults and folds, and techniques of strain analysis. Regional structural geology and tectonics are introduced. The subject of the course is active tectonic movements and how this is manifested and recorded in the geological record with emphasis on processes currently active in Iceland. Lectures will be complimented with fieldwork and supportive examples will be given from a global perspective (e.g. compressional tectonics from the Andes and other extensional environments like the East Africa Rift). Methods to describe these processes will be taught and evaluated. Structural geology concepts including elastic, ductile, and brittle behavior of rocks in the crust and mantle will be discussed and discontinuities and brittle fracturing will be addressed. Plate tectonics, plate velocity models, both relative and absolute. Earthquakes. Plate boundary deformation including strike-slip, extensional, and compressive regimes with rifts and rifting structures and folds in addition to mountain building. (If time permits: microstructures, post-rifting and post-seismic movements, Isostasy, vertical crustal movements and sea level, and structural level. measuring crustal movements, GPS-geodesy, levelling, and analysis of seismic stratigraphy (i.e. active source seismic reflection and refraction profiles). Fieldwork will focus on discontinuity analysis and characterisation through a combination of exposure mapping with structural observations coupled with digital elevation (DEMs) model collection using drones and associated analysis to create a coherent assessment of active faults in Southwestern Iceland. Lectures are required as content in the lectures will be tested. Students visiting from abroad in Geology and Geophysics are encouraged to participate in this class as this will be held in English and provide excellent insight into the Iceland Tectonic and Plate Boundary system. 

Discrete mathematics (TÖL104G)

Propositions, predicates, inference rules. Set operations and Boolean algebra. Induction and recursion. Basic methods of analysis of algorithms and counting. Simple algorithms in number theory. Relations, their properties and representations. Trees and graphs and related algorithms. Strings, examples of languages, finite automata and grammars.

Innovation, IP protection, and application (VON101M)

Minimum number of students registered for the course to be taught: 10

This course will cover basics in intellectual property rights with emphasis on patents. What is intellectual property and when and how can they best be protected? How does intellectual property rights work (e.g. patents), and how can they be used and applied in innovation and industry. The system‘s organization will be covered, as well as the patent process, making a financial plan relating to patents, and database searches.

Operations Research 2 (IÐN508M)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover integer programming and modeling with stochastic programming. The student will become familiar with building mathematical models using Python.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbours, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course, the students work on a practical machine learning project.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Research Project (STÆ262L)

Research Project

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Introduction to Logic (STÆ528M)

Logical deductions and proofs. Propositional calculus, connectives, truth functions and tautologies. Formal languages, axioms, inference rules. Quantifiers. First-order logic. Interpretations. The compactness theorem. The Lövenheim-Skolem theorem. Computability, recursive functions. Gödel's theorem.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Combinatorics (STÆ533M)

This course is aimed at second and third year undergraduate mathematics students. The purpose is to introduce the student to several combinatorial structures, methods of their enumeration and useful properties. Particular emphasis will be placed on the systematic use of generating functions in enumeration.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Formal Languages and Computability (TÖL301G)

Finite state machines, regular languages and grammars, push-down automata, context-free languages and grammars, Turing machines, general languages and grammars, and their basic properties. Recursive and recursively enumerable languages, reduction between languages, connection to decision problems and proving unsolvability of such problems. The complexity classes P and NP, and NP-completeness. Examples of various models of computation.

Algorithms in Bioinformatics (TÖL504M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Atmospheric Physics (EÐL401M)

Taught every odd year.

Elementary atmospheric thermodynamics, radiation and motion. Atmospheric general circulation, atmosphere/ocean interaction, the role of polar areas in the atmospheric circulation, climate fluctuations. Introduction to recent research. Students deliver a written report on a selected topic.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Electromagnetism 1 (EÐL401G)

The equations of Laplace and Poisson. Magnetostatics. Induction.  Maxwell's equations. Energy of the electromagnetic field. Poynting's theorem. Electromagnetic waves. Plane waves in dielectric and conducting media, reflection and refraction.  Electromagnetic radiation and scattering. Damping.

General Oceanography 1 (JAR414M)

The aim is to introduce students to the disciplines of general oceanography, in particular marine geological, physical and chemical oceanography. To understand how the interactions of processes shape the characteristics of different ocean regions.
The course covers the distribution of land and water, the world oceans and their geomorphology. Instruments and techniques in oceanographic observations. Physical properties of sea water. Energy and water budgets. Distribution of properties in relation to turbulence and diffusion. Introductory dynamical oceanography. Chemical oceanography: Geochemical balance, major and minor elements, dissolved gases. Biogeochemical cycles. Biological processes in relation to the physical and chemical environment. Oceanography of the North Atlantic and Icelandic waters

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Algorithms in Bioinformatics (TÖL604M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Introduction to data science (REI202G)

The course provides an introduction to the methods at the heart of data science and introduces widely used software tools such as numpy, pandas, matplotlib and scikit-learn.

The course consists of 6 modules:

1. Introduction to the Python programming language.
2. Data wrangling and data preprocessing.
3. Exploratory data analysis and visualization.
4. Optimization.
5. Clustering and dimensionality reduction.
6. Regression and classification.

Each module concludes with a student project.

Note that there is an academic overlap with REI201G Mathematics and Scientific Computing and both courses cannot be valid for the same degree.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Statistical Consulting (MAS201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Applied data analysis (MAS202M)

The course focuses on statistical analysis using the R environment. It is assumed that students have basic knowledge of statistics and the statistical software R. Students will learn to apply a broad range of statistical methods in R (such as classification methods, resampling methods, linear model selection and tree-based methods). The course on 12 weeks and will be on "flipped" form. This means that no lectures will be given but students will read some material and watch videos before attending classes. Students will then work on assignments during the classes.

Research Project (STÆ262L)

Research Project

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Representation Theory of Finite groups (STÆ626M)

The definition of a linear representation of a group.  Schur's lemma. Group characters. The group algebra. Representation theory of the symmetric group. Applications of representation theory to group theory: Theorems of Frobenius and Burnside.

Analysis of Algorithms (TÖL403G)

Methodology for the design of algorithms and the analysis of their time conplexity. Analysis of algorithms for sorting, searching, graph theory and matrix computations. Intractable problems, heuristics, and randomized algorithms.

Financial Economics I (HAG106G)

The aim is to provide a theoretical as well as practical overview in financial economics. The efficient markets and the portfolio theory are covered as well as the Markowitz model. Risk, and risk assessment under uncertainty and using the utility function are introduced. Students will get practice in value assessment methods, CAPM, as well as fixed income analysis. Stock valuation and fundamentals of derivatives calculations such as the B&S model are covered. 
Projects are based on understanding of concepts introduced in the course and their usage.  In addition projects are based on Excel usage.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Introduction to Mathematics (STÆ110G)

The course covers the language of mathematics and the fundamentals of logic and set theory.
The treatment of logic and set theory is naive but sufficiently precise to serve as a foundation for the general use of logic and mathematics in further mathematical studies. Emphasis is placed on basic concepts such as quantifiers, implications, sets, mappings, injective and surjective functions. Training is provided in formulating simple proofs. The course is taught once a week, three class hours at a time. A written final exam will be held in teaching week 12. Students complete assignments during the semester that count for 30% of the final grade.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Linear Algebra (STÆ107G)

Basics of linear algebra over the reals.  

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullk space.  The dot product, length and angle measures.  Volumes in higher dimension and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Financial Economics II (HAG208G)

The aim of this course is threefold. First, to introduce the fundamentals of financial accounting in order for the students being able to read and understand corporate financial statements. Second, teach the students to analyse and calculate the main important multiples from financial statements as well as being able to interpret their meaning to potential users of this information. Third, the students should be able to conduct fair value estimates of the corporate entities using information from their financial accounts.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis IIA (STÆ207G)

Emphasis is laid on the theoretical aspects of the material. The aim is that the students acquire understanding of fundamental concepts and are able to use them, both in theoretical consideration and in calculations. Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobian matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flows. Cylindrical and spherical coordinates. Taylor polynomials. Extrema and classification of stationary points. Extrema with constraints. Implicit functions and local inverses. Line integrals and potential functions. Proper and improper multiple integrals. Change of variables in multiple integrals. Simply connected regions. Integration on surfaces. Theorems of Green, Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Theoretical Statistics (STÆ313M)

Likelihood, Sufficient Statistic, Sufficiency Principle, Nuisance Parameter, Conditioning Principle, Invariance Principle, Likelihood Theory. Hypothesis Testing, Simple and Composite Hypothesis, The Neyman-Pearson Lemma, Power, UMP-Test, Invariant Tests. Permutation Tests, Rank Tests. Interval Estimation, Confidence Interval, Confidence, Confidence Region. Point Estimation, Bias, Mean Square Error. Assignments constitute 30% of the final grade.

Complex Analysis I (STÆ301G)

Complex numbers and the topology of the complex plane. Sequences and series of complex numbers. Differentiable and holomorphic functions. Sequences and series of functions; power series. Path integration and primitives. The exponential function and related functions. Winding numbers. The Cauchy theorem, the integral formula of Cauchy and consequences. The identity theorem, the open mapping theorem and the maximum principle. Laurent series, isolated singularities and their classification. The theorem of residues and residue calculus. The argument principle and Rouché's theorem. Connections with real analysis: The Cauchy-Riemann equations, harmonic functions and the integral formulas of Poisson and Schwarz. Holomorphic functions defined by integrals (e. g. the Laplace transformation). Conformal mapping and the Riemann mapping theorem.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Mathematical Analysis IIIA (STÆ304G)

The course is an introduction to three important tools of applied mathematics, namely ordinary differential equations, Fourier-series and partial differential equations.  Some basic theoretical properties are proved and solution methods presented. 

Ordinary differential equations: linear differential equations of order n, the Cauchy problem, Picard's existence theorem, solution by power series and equations with singular points.  Fourier series: convergence point-wise, uniformly and in the mean-square, Parseval's equation. 

Partial differential equations: the heat equation and the wave equation solved on a finite interval by separation of variables and Fourier series and their solutions compared, the Dirichlet problem for the Laplace equation on the rectangle and the disc, the Poisson integral formula.

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Measure and Integration Theory (STÆ402M)

Measure spaces, measures, outer measures. The Lebesgue measure on Rn. Measurable functions, the monotone convergence theorem, Fatou’s Lemma. Integrable functions, Lebesgue’s  dominated convergence theorem and applications. Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces. Fourier series. Product of measure spaces, theorems of Tonelli and Fubini. Complex measures. Jordan decomposition and Lebesgue decomposition of measures, Radon-Níkodým theorem. Continuous linear functionals on Lp-spaces. Image measures, transformation formula for the Lebesgue measure on Rn.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Non-Life Insurance Mathematics (STÆ414G)

The course will give an overview of some important elements of non-life insurance and reinsurance. Models for claim numbers, the Poisson, mixed Poisson and renewal process. Stochastic models for non-life insurance risks, in particular the compound Poisson, compound mixed Poisson and renewal models. The Cramer-Lundberg and the renewal model as basic risk models. Methods for approximating the distribution of risk models. Small and large claim distributions and their properties. Premium calculation principles for the total claim amount of a portfolio. Experience rating: calculation of the premium in a policy. Reinsurance treaties and their properties. Bayesian methods in a non-life insurance context, in particular the Bayes and linear Bayes estimators for calculating the premium in a policy.

Life Insurance Mathematics (STÆ413G)

Payment flows; mortality theory; overview of the main forms of insurance; the principle of equivalence; prospective reserves and differential equations for these; cost; general Markov chains in life insurance with applications to disability insurance and multi-life insurance; profits and bonuses; market rate products.

Generalized Linear Models (STÆ421M)

Generalized linear regression models. Exponential dispersion models. Poisson processes and tests for overdispersion. Survival regression models. Nonlinear effects and basis expansions. Parametric, semiparametric and nonparametric likelihood methods. Partial likelihood methods. Generalized linear regression analysis in R.

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Financial Instruments (VIÐ503G)

This course starts with looking at interest rate markets and how the zero coupon curve is derived. Valuation of different kind of bonds is covered along with the characteristics and risk factors of the major listed bonds, with special emphasis on the Icelandic market. Next the valuation of derivatives is covered along with the main characteristics. Special emphasis is placed on futures/forwards, swaps and options.  The reasons behind derivatives trading are covered and what the main risk factors are.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Business Law B - Introduction to Financial Law (VIÐ601G)

The course reviews legislation and legal issues that concern the financial markets, corporate finance and operations. Legal environment of financial companies will be reviewed, securities law, liability for experts, a chapter in the penal code act regarding wealth deeds and legal issues related to acquisitions and sales of corporations, due diligence, etc. The course will also review contracts and documents in the financial market, including loan-, purchase- and shareholder agreements.

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Individual Taxation (VIÐ501G)

The course covers the principles of Icelandic tax law concerning tax liability and taxable income, including which items are tax deductible. A special emphasis will be placed on the filing of sources of income for individuals and the self-employed through solving problems and cases. The filing of tax returns for individuals, couples, and businesses will be introduced. The determination of benefits and tax credit will be discussed. The fundamental principles of tax law will be covered, along with re-assessment of taxes and the consequences of fraudulent filing. An overview will be given of the key principles of the laws on value added tax and the social insurance fee. Upon completion of the course a student shall be able to file tax returns for individuals and small businesses as well as appeal tax assessments that he/she deems incorrect.

Data Base Theory and Practice (TÖL303G)

Databases and database management systems. Physical data organization. Data modelling using the Entity-Relationship model and the Relational model. Relational algebra and calculus.  The SQL query language. Design theory for relational data bases, functional dependencies, decomposition of relational schemes, normal forms. Query optimization. Concurrency control techniques and crash recovery. Database security and authorization. Data warehousing.

Business Law A (VIÐ302G)

This course deals with law and regulation applicable to commercial transactions and business organizations. The purpose of the course is to prepare students for the legal challenges they can expect to encounter as entrepreneurs and managers of private businesses . Topics covered include contracts, torts, negotiable instruments, security and guarantees, and bankruptcy. Laws applicable to business organization will also be studied and the fundamentals of securities laws.

Engineering Economics (IÐN502G)

The objective of the course is that students get the skills to:

1.    Understand the main concepts in accounting, cost theory and investment theory.

2.    Be able to use methods of measuring the economic feasibility of technical projects.

3.    Be able to develop computer models to assess the profitability of investments, the value of companies and pricing of bonds

Among topics included are accounting, cost theory, cash flow analysis, investment theory, measures of profitability including net present value and internal rate of return, and the building of profitability models. The course ends with a group assignment where the students exercise the development of computer models for feasibility assessment of projects.

Microeconomics I (VIÐ105G)

The aim of the course is to teach students the basic principles of economic thinking and main theories and concepts in microeconomics. The topics covered include: Markets, specialisation and trade. Supply, demand, elasticity and government policies. Efficiency and welfare. The Icelandic tax system and the effects of taxation on market activity. Externalities, public goods and common resources. Firm behaviour and the organisation of industry. Consumer choice. Labour market, earnings and discrimination. Asymmetric information, political economy, behavioural economics.

Introduction to Financial Accounting (VIÐ103G)

This course is intended to do the student able to read corporate financial statements. Fundamentals of financial accounting and financial reporting are introduced. The double entry model explained through the accounting equation. Presentation of the conceptual framework for accounting: assumptions, principles and concepts.  The logical relationship between individual chapters in financial statements is in foreground. Whose things have influence on shareholders equity? Main methods of financial statement analysis are presented, especially ratio analysis. Extensive exercises are covered in separate group sessions.

Financial Markets (VIÐ505G)

Financial institutions are a pillar of civilized society, supporting people in their productive ventures and managing the economic risks they take on. The workings of these institutions are important to comprehend, if we are to predict their actions today and their evolution in the coming information age. The course strives to offer understanding of the theory of finance and its relation to the history, strengths and imperfections of such institutions as banking, insurance, securities, futures, and other derivatives markets, and the future of these institutions over the next century. The Icelandic Banking System collapse offers myriad of examples and cases that provide a fruitful ground for learning. A frequent reference will be made to those throughout the course.

Financial Statements A (VIÐ505M)

This course is designed for students on the F- and R-line (finance and accounting). The purpose with the course is that the students obtains knowledge and understanding on matters that management of companies needs to have to prepare financial statements in accordance with generally accepted accounting principles. In the course students, will learn about generally accepted accounting principles according to international accountings standards (IFRS) and icelandic GAAP. Among topics: Financial accounting and accounting standards, income statement, balance sheet and cash flow. Revenue recognition and cost accounting, inventories, accounts receivables, PPE, intangible assets, income tax, impairment test, accounting for financial instruments, liabilities and equity. Students will need to solve assignments during the course.

Innovation, IP protection, and application (VON101M)

Minimum number of students registered for the course to be taught: 10

This course will cover basics in intellectual property rights with emphasis on patents. What is intellectual property and when and how can they best be protected? How does intellectual property rights work (e.g. patents), and how can they be used and applied in innovation and industry. The system‘s organization will be covered, as well as the patent process, making a financial plan relating to patents, and database searches.

Operations Research 2 (IÐN508M)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover integer programming and modeling with stochastic programming. The student will become familiar with building mathematical models using Python.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbours, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course, the students work on a practical machine learning project.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Research Project (STÆ262L)

Research Project

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Introduction to Logic (STÆ528M)

Logical deductions and proofs. Propositional calculus, connectives, truth functions and tautologies. Formal languages, axioms, inference rules. Quantifiers. First-order logic. Interpretations. The compactness theorem. The Lövenheim-Skolem theorem. Computability, recursive functions. Gödel's theorem.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Combinatorics (STÆ533M)

This course is aimed at second and third year undergraduate mathematics students. The purpose is to introduce the student to several combinatorial structures, methods of their enumeration and useful properties. Particular emphasis will be placed on the systematic use of generating functions in enumeration.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Formal Languages and Computability (TÖL301G)

Finite state machines, regular languages and grammars, push-down automata, context-free languages and grammars, Turing machines, general languages and grammars, and their basic properties. Recursive and recursively enumerable languages, reduction between languages, connection to decision problems and proving unsolvability of such problems. The complexity classes P and NP, and NP-completeness. Examples of various models of computation.

Algorithms in Bioinformatics (TÖL504M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Measure and Integration Theory (STÆ402M)

Measure spaces, measures, outer measures. The Lebesgue measure on Rn. Measurable functions, the monotone convergence theorem, Fatou’s Lemma. Integrable functions, Lebesgue’s  dominated convergence theorem and applications. Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces. Fourier series. Product of measure spaces, theorems of Tonelli and Fubini. Complex measures. Jordan decomposition and Lebesgue decomposition of measures, Radon-Níkodým theorem. Continuous linear functionals on Lp-spaces. Image measures, transformation formula for the Lebesgue measure on Rn.

Mathematics for Physicists I (STÆ211G)

Order of magnitude estimates, scaling relations, and dimensional analysis. Plotting with matplotlib. Complex numbers, oscillations and Fourier-series. Mechanics and the time derivatives of vectors. Particle trajectories in polar coordinates. Derivatives, the chain rule and equations of state. Scalar and vector potentials and connection to electromagnetism. Stokes and divergence theorems and Maxwell's equations. We emphasize applications and problem solving.

Financial Statements B (VIÐ604M)

This course is a continuation of Financial Statements A, which is taught in the fall semester. It is expected that students of this course are fully familiar with the content of the course Financial Statements A.
The course will cover the principles in accounting under both IFRS and Icelandic law. Topics: cash flow, income tax, earnings per share, financial instruments, finance leases, assets held for sale and discontinued operations, investment properties, provision, information in the financial statements and related parties.
Assignments are part of the course, and students will need submit them.

Reserved the righttochangethecoursedescription.

Finance II (VIÐ402G)

Good corporate governance and skilled financial management are the key ingredients for a successfully run corporation.  Finance II builds on the course Finance I, and has its main focus on the corporation and how it is being run from financial management point of view.  The course covers topics in corporate governance, how incentives are embedded in the operation of the firm and what economic and financial outcomes are to be expected from the incentive structure.  The main focus of the course is financial management; the firm’s capital structure, short and long term financing, capital budgeting, dividend policies, short term financial planning as well as financial distress.

Management Accounting (VIÐ204G)

Introduction to management accounting. Most important cost terms will be presented and cost-volume-profit analysis. Different accounting systems around manufacturing costs and allocation of indirect costs. The difference between absorption costing and variable costing. Budgeting, standard costing and variance analysis. Performance evaluation of different departments and products and cost allocation. After this course the students should understand well the importance of management accounting for decision making in business.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Portfolio Management (VIÐ604G)

The theory behind decisions of investors and corporations regarding building and managing asset and liability portfolios. Risk management of corporations will also be covered.

The course is taught in English

Financial Accounting (VIÐ401G)

This course is a continuation of the introductory course. The main emphasis here is on the preparation of financial statement, fx. allowance for doubtful accounts, depreciation af property, plant and equipment, goodwill and other intangible assets, inventories valuation, fair value of securities and equities, deferred taxes etc. Preparation of cash-flow statement. In this context the Icelandic legal regulation of accounting and International Financial Accounting Standards (IFRS/IAS) are being dealt with. Calculation of income tax will be presented. Extensive exercises are covered in separate group sessions. After this course students should be capable of preparing financial statement for a comparatively simple company.

Portfolio Management (VIÐ604G)

The theory behind decisions of investors and corporations regarding building and managing asset and liability portfolios. Risk management of corporations will also be covered.

The course is taught in English

Mathematics for Physicists II (EÐL408G)

Python tools related to general data and time series analysis. The method of least squares, linear and non-linear fitting. Fourier transforms, fast Fourier transforms (FFT), spectral analysis and convolution. Differential equations, including the Laplace equation and their use in the description of physical systems. Partial differential equations and boundary value problems. Special functions and their relation to important problems in physics. We will emphasize applications and problem solving.

Microeconomics II (HAG201G)

Intermediate microeconomic theory. Basic factors of price theory, uncertainty, including analysis of demand, costs of production and supply relationships, and price and output determination under various market structures, market failures and public choice.

Algorithms in Bioinformatics (TÖL604M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Introduction to data science (REI202G)

The course provides an introduction to the methods at the heart of data science and introduces widely used software tools such as numpy, pandas, matplotlib and scikit-learn.

The course consists of 6 modules:

1. Introduction to the Python programming language.
2. Data wrangling and data preprocessing.
3. Exploratory data analysis and visualization.
4. Optimization.
5. Clustering and dimensionality reduction.
6. Regression and classification.

Each module concludes with a student project.

Note that there is an academic overlap with REI201G Mathematics and Scientific Computing and both courses cannot be valid for the same degree.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Mathematical Physics (EÐL612M)

Continuum mechanics: Stress and strain, equations of motion. Seismic waves. Maxwell's equations and electromagnetic waves. Plane waves, reflection and refraction. Distributions and Fourier transforms. Fundamental solutions of linear partial differential equation. Waves in homogeneous media. Huygens' principle and Ásgeirsson's mean value theorem. Dispersion, phase and group velocities, Kramers-Kronig equations. The method of stationary phase. Surface waves on liquids.

Operations Research (IÐN401G)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover linear programming and the simplex algorithm, as well as related analytical topics. It will also introduce special types of mathematical models, including transportation, assignment, network, and integer programming models. The student will become familiar with a modeling language for linear programming.

Statistical Consulting (MAS201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.

Applied data analysis (MAS202M)

The course focuses on statistical analysis using the R environment. It is assumed that students have basic knowledge of statistics and the statistical software R. Students will learn to apply a broad range of statistical methods in R (such as classification methods, resampling methods, linear model selection and tree-based methods). The course on 12 weeks and will be on "flipped" form. This means that no lectures will be given but students will read some material and watch videos before attending classes. Students will then work on assignments during the classes.

Research Project (STÆ262L)

Research Project

Mathematical Analysis IV (STÆ401G)

Aim: To introduce the student to Fourier analysis and partial differential equations and their applications.
Subject matter: Fourier series and orthonormal systems of functions, boundary-value problems for ordinary differential equations, the eigenvalue problem for Sturm-Liouville operators, Fourier transform. The wave equation, diffusion equation and Laplace's equation solved on various domains in one, two and three dimensions by methods based on the first part of the course, separation of variables, fundamental solution, Green's functions and the method of images.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Representation Theory of Finite groups (STÆ626M)

The definition of a linear representation of a group.  Schur's lemma. Group characters. The group algebra. Representation theory of the symmetric group. Applications of representation theory to group theory: Theorems of Frobenius and Burnside.

Analysis of Algorithms (TÖL403G)

Methodology for the design of algorithms and the analysis of their time conplexity. Analysis of algorithms for sorting, searching, graph theory and matrix computations. Intractable problems, heuristics, and randomized algorithms.

Physics 1 R (EÐL107G)

Introduce students to methods and fundamental laws of mechanics, waves and thermodynamics, to the extent that they can apply their knowledge to solve problems. 

Concepts, units, scales and dimensions.  Vectors. Kinematics of particles. Particle dynamics, inertia, forces and Newton's laws. Friction. Work and energy, conservation of energy. Momentum, collisions. Systems of particles, center of mass. Rotation of a rigid body.  Angular momentum and moment of inertia. Statics. Gravity. Solids and fluids, Bernoulli's equation. Oscillations: Simple, damped and forced. Waves. Sound.  Temperature. Ideal gas. Heat and the first law of thermodynamics. Kinetic theory of gases. Entropy and the second law of thermodynamics.

Note that the textbook is accessible to students via Canvas free of charge.

Computer Science 1 (TÖL101G)

Note: Only one course of either TÖL101G Tölvunarfræði 1 or TÖL105G Tölvunarfræði 1a can count towards the BS degree.

The Java programming language is used to introduce basic concepts in computer programming: Expressions and statements, textual and numeric data types, conditions and loops, arrays, methods, classes and objects, input and output. Programming and debugging skills are practiced in quizzes and projects throughout the semester.

Discrete mathematics (TÖL104G)

Propositions, predicates, inference rules. Set operations and Boolean algebra. Induction and recursion. Basic methods of analysis of algorithms and counting. Simple algorithms in number theory. Relations, their properties and representations. Trees and graphs and related algorithms. Strings, examples of languages, finite automata and grammars.

Computer Science 1a (TÖL105G)

Programming in Python (for computations in engineering and science): Main commands and statements (computations, control statements, in- and output), definition and execution of functions, datatypes (numbers, matrices, strings, logical values, records), operations and built-in functions, array and matrix computation, file processing, statistics, graphics. Object-oriented programming: classes, objects, constructors and methods. Concepts associated with design and construction of program systems: Programming environment and practices, design and documentation of function and subroutine libraries, debugging and testing of programmes.

Introduction to Mathematics (STÆ110G)

The course covers the language of mathematics and the fundamentals of logic and set theory.
The treatment of logic and set theory is naive but sufficiently precise to serve as a foundation for the general use of logic and mathematics in further mathematical studies. Emphasis is placed on basic concepts such as quantifiers, implications, sets, mappings, injective and surjective functions. Training is provided in formulating simple proofs. The course is taught once a week, three class hours at a time. A written final exam will be held in teaching week 12. Students complete assignments during the semester that count for 30% of the final grade.

Mathematical Analysis IA (STÆ101G)

Main emphasis is on the differential and integral calculus of functions of a single variable. The systems of real and complex numbers. Least upper bound and greatest lower bound. Natural numbers and induction. Mappings and functions. Sequences and limits. Series and convergence tests. Conditionally convergent series. Limits and continuous functions. Trigonometric functions. Differentiation. Extreme values. The mean value theorem and polynomial approximation. Integration. The fundamental theorem of calculus. Logarithmic and exponential functions, hyperbolic and inverse trigonometric functions. Methods for finding antiderivatives. Real power series. First-order differential equations. Complex valued functions and second-order differential equations.

Linear Algebra (STÆ107G)

Basics of linear algebra over the reals.  

Subject matter: Systems of linear equations, matrices, Gauss-Jordan reduction.  Vector spaces and their subspaces.  Linearly independent sets, bases and dimension.  Linear maps, range space and nullk space.  The dot product, length and angle measures.  Volumes in higher dimension and the cross product in threedimensional space.  Flats, parametric descriptions and descriptions by equations.  Orthogonal projections and orthonormal bases.  Gram-Schmidt orthogonalization.  Determinants and inverses of matrices.  Eigenvalues, eigenvectors and diagonalization.

Physics 2 R (EÐL206G)

Introduction to electrodynamics in material; from insulators to superconductors.  Charge and electric field. Gauss' law. Electric potential. Capacitors and dielectrics. Electric currents and resistance. Circuits. Magnetic fields. The laws of Ampère and Faraday. Induction. Electric oscillation and alternating currents. Maxwell's equations. Electromagnetic waves. Reflection and refraction. Lenses and mirrors. Wave optics.

Computer Science 2 (TÖL203G)

The programming language Java will be used in the course. Various data structures, algorithms and abstract data types will be covered. Among the data types and structures covered are lists, stacks, queues, priority queues, trees, binary trees, binary search trees and heaps along with related algorithms. Various search and sort algorithms will be covered. Algorithms will be analysed for their space and time complexity. There will be programming assignments in Java using the given data structures and algorithms. There will be many small assignments.

Sets and Metric Spaces (STÆ202G)

Elements of set theory: Sets. Mappings. Relations, equivalence relations, orderings. Finite, infinite, countable and uncountable sets. Equipotent sets. Construction of the number systems. Metric spaces: Open sets and closed sets, convergent sequences and Cauchy sequences, cluster points of sets and limit points of sequences. Continuous mappings, convergence, uniform continuity. Complete metric spaces. Uniform convergence and interchange of limits. The Banach fixed point theorem; existence theorem about solutions of first-order differential equations. Completion of metric spaces. Compact metric spaces. Connected sets. Infinite series, in particular function series.

Probability and Statistics (STÆ203G)

Basic concepts in probability and statistics based on univariate calculus. 

Topics: 
Sample space, events, probability, equal probability, independent events, conditional probability, Bayes rule, random variables, distribution, density, joint distribution, independent random variables, condistional distribution, mean, variance, covariance, correlation, law of large numbers, Bernoulli, binomial, Poisson, uniform, exponential and normal random variables. Central limit theorem. Poisson process. Random sample, statistics, the distribution of the sample mean and the sample variance. Point estimate, maximum likelihood estimator, mean square error, bias. Interval estimates and hypotheses testing form normal, binomial and exponential samples. Simple linear regression. Goodness of fit tests, test of independence.

Mathematical Analysis IIA (STÆ207G)

Emphasis is laid on the theoretical aspects of the material. The aim is that the students acquire understanding of fundamental concepts and are able to use them, both in theoretical consideration and in calculations. Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobian matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flows. Cylindrical and spherical coordinates. Taylor polynomials. Extrema and classification of stationary points. Extrema with constraints. Implicit functions and local inverses. Line integrals and potential functions. Proper and improper multiple integrals. Change of variables in multiple integrals. Simply connected regions. Integration on surfaces. Theorems of Green, Stokes and Gauss.

Introduction to Probability Theory (STÆ210G)

This is an extension of the course "Probability and Statistics" STÆ203G. The basic concepts of probability are considered in more detail with emphasis on definitions and proofs. The course is a preparation for the two M-courses in probability and the two M-courses in statistics that are taught alternately every other year.

Topics beyond those discussed in the probability part of STÆ203G:

Kolmogorov's definition. Proofs of propositions on compound events and conditional probability. Proofs for discrete and continuous variables of propositions on expectation, variance, covariance, correlation, and conditional expectation and variance. Proofs of propositions for Bernoulli, binomial, Poisson, geometric, uniform, exponential, and gamma variables. Proof of the tail-summing proposition for expectation and the application to the geometric variable. Proof of the proposition on memoryless and exponential variables. Derivation of the distribution of sums of independent variables such as binomial, Poisson, normal, and gamma variables. Probability and moment generating functions.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Combinatorics (STÆ533M)

This course is aimed at second and third year undergraduate mathematics students. The purpose is to introduce the student to several combinatorial structures, methods of their enumeration and useful properties. Particular emphasis will be placed on the systematic use of generating functions in enumeration.

Complex Analysis I (STÆ301G)

Complex numbers and the topology of the complex plane. Sequences and series of complex numbers. Differentiable and holomorphic functions. Sequences and series of functions; power series. Path integration and primitives. The exponential function and related functions. Winding numbers. The Cauchy theorem, the integral formula of Cauchy and consequences. The identity theorem, the open mapping theorem and the maximum principle. Laurent series, isolated singularities and their classification. The theorem of residues and residue calculus. The argument principle and Rouché's theorem. Connections with real analysis: The Cauchy-Riemann equations, harmonic functions and the integral formulas of Poisson and Schwarz. Holomorphic functions defined by integrals (e. g. the Laplace transformation). Conformal mapping and the Riemann mapping theorem.

Algebra (STÆ303G)

Groups, examples and basic concepts. Symmetry groups. Homomorphisms and normal subgroups. Rings, examples and basic concepts. Integral domains. Ring homomorphisms and ideals. Polynomial rings and factorization of polynomials. Special topics.

Mathematical Analysis IIIA (STÆ304G)

The course is an introduction to three important tools of applied mathematics, namely ordinary differential equations, Fourier-series and partial differential equations.  Some basic theoretical properties are proved and solution methods presented. 

Ordinary differential equations: linear differential equations of order n, the Cauchy problem, Picard's existence theorem, solution by power series and equations with singular points.  Fourier series: convergence point-wise, uniformly and in the mean-square, Parseval's equation. 

Partial differential equations: the heat equation and the wave equation solved on a finite interval by separation of variables and Fourier series and their solutions compared, the Dirichlet problem for the Laplace equation on the rectangle and the disc, the Poisson integral formula.

Galois theory (STÆ403M)

Selected topics in commutative algebra. 

Subject matter: Fields and field extensions.  Algeraic extensions, normal extensions and seperable extensions.  Galois theory.  Applications.  
Noetherian rings.  Hilbert's basis theorem and Hilbert's Nullstellensatz.

Linear algebra II (STÆ401M)

Modules and linear maps. Free modules and matrices. Quotient modules and short exact sequences. Dual modules. Finitely generated modules over a principal ideal domain. Linear operators on finite dimensional vector spaces.

Theoretical Numerical Analysis (STÆ412G)

This is an extension of the course "Numerical Analysis" STÆ405G. The material of Numerical Analysis (STÆ405G) is studied in more detail and more theoretically with emphasis on proofs.

Numerical Analysis (STÆ405G)

Fundamental concepts on approximation and error estimates. Solutions of systems of linear and non-linear equations. PLU decomposition. Interpolating polynomials, spline interpolation and regression. Numerical differentiation and integration. Extrapolation. Numerical solutions of initial value problems of systems of ordinary differential equations. Multistep methods. Numerical solutions to boundary value problems for ordinary differential equations.

Grades are given for programning projects and in total they amount to 30% of the final grade. The student has to receive the minimum grade of 5 for both the projects and the final exam.

Measure and Integration Theory (STÆ402M)

Measure spaces, measures, outer measures. The Lebesgue measure on Rn. Measurable functions, the monotone convergence theorem, Fatou’s Lemma. Integrable functions, Lebesgue’s  dominated convergence theorem and applications. Inequalities of Hölder and Minkowski, Lp-spaces, simple facts about Banach and Hilbert spaces. Fourier series. Product of measure spaces, theorems of Tonelli and Fubini. Complex measures. Jordan decomposition and Lebesgue decomposition of measures, Radon-Níkodým theorem. Continuous linear functionals on Lp-spaces. Image measures, transformation formula for the Lebesgue measure on Rn.

Topology (STÆ419M)

General topology: Topological spaces and continuous maps. Subspaces, product spaces and quotient spaces. Connected spaces and compact spaces. Separation axioms, the lemma of Urysohn and a metrization theorem. Completely regular spaces and compactifications.

Functional Analysis (STÆ507M)

Banach spaces and Hilbert spaces and their main properties. Dual of Banach spaces.
Principles of functional analysis: Hahn--Banach theorem, Uniform boundedness Principle and Banach--Steinhaus, open mapping and closed graph theorems.

Weak topologies. Banach--Alaoglu theorem.
Riesz's lemma. Orthogonality in Banach spaces.

Operators and their adjoints. Compact operators.
Spectrum and resolvent. Lax--Milgram. The spectral theorem for compact, self-adjoint operators.

Sobolev spaces. Sobolev spaces on the real line and immersions. Approximation and dense subspaces. Poincaré inequality. Sobolev spaces in higher dimension. Extension theorem. Sobolev embeddings. Rellich--Kondrachov. Poincaré inequality in higher dimension. Poincaré--Wirtinger.

Differential Geometry (STÆ519M)

Curves, surfaces and manifolds in Euclidean space. Differentiable manifolds, vector fields and tensor fields. Hypersurfaces in Euclidean space, first and second fundamental forms, curvature, convex hypersurfaces. Inner geometry of surfaces. Riemannian manifolds, geodesics. The Gauss-Bonnet theorem for surfaces.

Mathematical Seminar (STÆ402G)

This course is intended for students who have completed at least 120 ECTS credits. Students who have not completed 120 ECTS credits and are interested in taking the course must obtain the approval of the supervisor prior to signing up for the course.

Each student prepares and studies a selected well-defined topic of mathematics or statistics and will be assigned a mentor related to that topic. Topics vary from year to year. A list of possible topics is released at the start of or prior to the course and students can also suggest topics (provided that a mentor can be found). Students write a thesis on their selected topic and prepare and give a lecture on the topic at a student conference. During the course, students provide each other with constructive critique both regarding the thesis writing and the preparation of the lecture. In addition to presenting their own projects at the student conference, students take active part, listen to their fellow course members and ask questions.

Innovation, IP protection, and application (VON101M)

Minimum number of students registered for the course to be taught: 10

This course will cover basics in intellectual property rights with emphasis on patents. What is intellectual property and when and how can they best be protected? How does intellectual property rights work (e.g. patents), and how can they be used and applied in innovation and industry. The system‘s organization will be covered, as well as the patent process, making a financial plan relating to patents, and database searches.

Operations Research 2 (IÐN508M)

This course will introduce the student to decision and optimization models in operations research. On completing the course the student will be able to formulate, analyze, and solve mathematical models, which represent real-world problems, and critically interpret their results. The course will cover integer programming and modeling with stochastic programming. The student will become familiar with building mathematical models using Python.

R Programming (MAS102M)

Students will perform traditional statistical analysis on real data sets. Special focus will be on regression methods, including multiple regression analysis. Students will apply sophisticated methods of graphical representation and automatic reporting. Students will hand in a projects where they apply the above mentioned methods on real datasets in order to answer research questions

Machine Learning (REI505M)

An overview of some of the main concepts, techniques and algorithms in machine learning. Supervised learning and unsupervised learning. Data preprocessing and data visualization. Model evaluation and model selection. Linear regression, nearest neighbours, support vector machines, decision trees and ensemble methods. Deep learning. Cluster analysis and the k-means algorithm. The students implement simple algorithms in Python and learn how to use specialized software packages. At the end of the course, the students work on a practical machine learning project.

Various approaches to teaching mathematics in upper secondary schools (SNU503M)

In this course, students learn to plan mathematics teaching in upper secondary school using various approaches to provide access for all. An emphasis will be put on exploring different teaching environments and teaching methods that build on research on the teaching and learning of mathematics. In the course, the aims of learning mathematics both in Iceland and its neighboring countries will be discussed based on curricular and governmental documents. Students will read about and get a chance to try out various ways to assess and analyze students’ mathematical achievements. The course format includes lectures, project work, presentations, topic studies connected to practice, and critical topic discussion. An emphasis will be put on students’ discussion about challenges and their search for solutions to problems related to the teaching and learning of mathematics.

Research Project (STÆ262L)

Research Project

Theory of linear models (STÆ310M)

Simple and multiple linear regression, analysis of variance and covariance, inference, variances and covariances of estimators, influence and diagnostic analyses using residual and influence measures, simultaneous inference. General linear models as projections with ANOVA as special case, simultaneous inference of estimable functions. R is used in assignments. Solutions to assignments are returned in LaTeX and PDF format.

In addition selected topics will be visited, e.g. generalized linear models (GLMs), nonlinear regression and/or random/mixed effects models and/or bootstrap methods etc.

Students will present solutions to individually assigned
projects/exercises, each of which is handed in earlier through a web-page.

This course is taught in semesters of even-numbered years.

Applied Linear Statistical Models (STÆ312M)

The course focuses on simple and multiple linear regression as well as analysis of variance (ANOVA), analysis of covariance (ANCOVA) and binomial regression. The course is a natural continuation of a typical introductory course in statistics taught in various departments of the university.

We will discuss methods for estimating parameters in linear models, how to construct confidence intervals and test hypotheses for the parameters, which assumptions need to hold for applying the models and what to do when they are not met.

Students will work on projects using the statistical software R.

Numerical Linear Algebra (STÆ511M)

Iterative methods for linear systems of equations.  Decompositions of matrices: QR, Cholesky, Jordan, Schur, spectral and singular value decomposition (SVD) and their applications.  Discrete Fourier transform (DFT) and the fast Fourier transform (FFT).  Discrete cosine transform (DCT) in two-dimensions and its application for the compression of images (JPEG) and audio (MP3, AAC).  Sparse matrices and their representation.

Special emphasis will be on the application and implementation of the methods studied.

Graph Theory (STÆ520M)

Graphs, homomorphisms and isomorphisms of graphs. Subgraphs, spanning subgraphs. Paths, connected graphs. Directed graphs. Bipartite graphs. Euler graphs and Hamilton graphs; the theorems of Chvátal, Pósa, Ore and Dirac. Tournaments. Trees, spanning trees, the matrix-tree theorem, Cayley's theorem. Weighted graphs, the algorithms of Kruskal and Dijkstra. Networks, the max-flow-min-cut theorem, the algorithm of Ford and Fulkerson, Menger's theorem. Matchings, Berge's theorem, Hall's marriage theorem, the König-Egerváry theorem, the Kuhn-Munkres algorithm. Inseparable and two-connected graphs. Planar graphs, Euler's formula, Kuratowski's theorem, dual graphs. Embeddings of graphs in surfaces, the Ringel-Youngs-Mayer theorem. Colourings, Heawood's coloring theorem, Brooks's theorem, chromatic polynomial; edge colourings, Vizing's theorem.

Introduction to Logic (STÆ528M)

Logical deductions and proofs. Propositional calculus, connectives, truth functions and tautologies. Formal languages, axioms, inference rules. Quantifiers. First-order logic. Interpretations. The compactness theorem. The Lövenheim-Skolem theorem. Computability, recursive functions. Gödel's theorem.

Bayesian Data Analysis (STÆ529M)

Goal: To train students in applying methods of Bayesian statistics for analysis of data. Topics: Theory of Bayesian inference, prior distributions, data distributions and posterior distributions. Bayesian inference  for parameters of univariate and multivariate distributions: binomial; normal; Poisson; exponential; multivariate normal; multinomial. Model checking and model comparison: Bayesian p-values; deviance information criterion (DIC). Bayesian computation: Markov chain Monte Carlo (MCMC) methods; the Gibbs sampler; the Metropolis-Hastings algorithm; convergence diagnostistics. Linear models: normal linear models; hierarchical linear models; generalized linear models. Emphasis on data analysis using software, e.g. Matlab and R.

Combinatorics (STÆ533M)

This course is aimed at second and third year undergraduate mathematics students. The purpose is to introduce the student to several combinatorial structures, methods of their enumeration and useful properties. Particular emphasis will be placed on the systematic use of generating functions in enumeration.

Numerical Methods for Partial Differential Equations (STÆ537M)

The aim of the course is to study numerical methods to solve partial differential equations and their implementation.

Formal Languages and Computability (TÖL301G)

Finite state machines, regular languages and grammars, push-down automata, context-free languages and grammars, Turing machines, general languages and grammars, and their basic properties. Recursive and recursively enumerable languages, reduction between languages, connection to decision problems and proving unsolvability of such problems. The complexity classes P and NP, and NP-completeness. Examples of various models of computation.

Algorithms in Bioinformatics (TÖL504M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Stochastic Processes (STÆ415M)

Introduction to stochastic processes with main emphasis on Markov chains.

Subject matter: Hitting time, classification of states, irreducibility, period, recurrence (positive and null), transience, regeneration, coupling, stationarity, time-reversibility, coupling from the past, branching processes, queues, martingales, Brownian motion.

Introduction to Measure-Theoretic Probability (STÆ418M)

Probability based on measure-theory.

Subject matter: Probability, extension theorems, independence, expectation. The Borel-Cantelli theorem and the Kolmogorov 0-1 law. Inequalities and the weak and strong laws of large numbers. Convergence pointwise, in probability, with probability one, in distribution, and in total variation. Coupling methods. The central limit theorem. Conditional probability and expectation.

Algorithms in Bioinformatics (TÖL604M)

This course will cover the algorithmic aspects of bioinformatic. We  will start with an introduction to genomics and algorithms for students new to the field. The course is divided into modules, each module covers a single problem in bioinformatics that will be motivated based on current research. The module will then cover algorithms that solve the problem and variations on the problem. The problems covered are motif finding, edit distance in strings, sequence alignment, clustering, sequencing and assembly and finally computational methods for high throughput sequencing (HTS).

Introduction to data science (REI202G)

The course provides an introduction to the methods at the heart of data science and introduces widely used software tools such as numpy, pandas, matplotlib and scikit-learn.

The course consists of 6 modules:

1. Introduction to the Python programming language.
2. Data wrangling and data preprocessing.
3. Exploratory data analysis and visualization.
4. Optimization.
5. Clustering and dimensionality reduction.
6. Regression and classification.

Each module concludes with a student project.

Note that there is an academic overlap with REI201G Mathematics and Scientific Computing and both courses cannot be valid for the same degree.

Statistical Consulting (LÝÐ201M)

Participants in the course will obtain training in practical statistics as used when providing general statistical counselling. The participants will be introduced to actual statistical projects by assisting students in various departments within the university. The participants will report on the projects in class, discuss options for solving the projects and subsequently assist the students with analyses using R and interpretation of results.