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.

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.

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.

This is a foundational course in single variable calculus. The prerequisites are high school courses on algebra, trigonometry. derivatives, and integrals. The course aims to create a foundation for understanding of subjects such as natural and physical sciences, engineering, economics, and computer science. Topics of the course include the following:

• Real numbers.
• Limits and continuous functions.
• Differentiable functions, rules for derivatives, derivatives of higher order, applications of differential calculus (extremal value problems, linear approximation).
• Transcendental functions.
• Mean value theorem, theorems of l'Hôpital and Taylor.
• Integration, the definite integral and rules/techniques of integration, primitives, improper integrals.
• Fundamental theorem of calculus.
• Applications of integral calculus: Arc length, area, volume, centroids.
• Ordinary differential equations: First-order separable and homogeneous differential equations, first-order linear equations, second-order linear equations with constant coefficients.
• Sequences and series, convergence tests.
• Power series, Taylor series.

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.

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.

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.

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.

The course will cover topics in algebra, school algebra and the history of algebra. The teaching of algebra at different school levels will be examined as well as the developement of algebraic thinking in different age groups.

 Part of the course are in-field studies where students prepare and teach algebra.

This course intertwines mathematics and mathematics education. Students get an introduction to probability and statistics, build mathematical models, and learn to approach the teaching of mathematics from a modelling perspective. Among topics covered are combinations, permutations, the counting of the number of outcomes, the binomial distribution, and probability distributions in general. Students will explore how conclusions are drawn about probability based on data and how computer simulations can be used for that purpose. Different types of mathematical models will be introduced, such as linear models, exponential growth models, inverse proportion models, power functions models, linear optimization models in two variables, and graph theoretical models.

The didactics of probability and statistics will be explored and analysed from a modelling perspective. Students will select and adapt exercises and problems for the teaching of probability and statistics, and plan teaching processes. An emphasis will be put on mathematics teaching that touches on important issues of the modern times, such as climate change and pandemics.

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.

Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobi matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flow. Cylindrical and spherical coordinates. Taylor polynomials. Extreme values and the classification of stationary points. Extreme value problems with constraints. Implicit functions and local inverses. Line integrals, primitive functions and exact differential equations. Double integrals. Improper integrals. Green's theorem. Simply connected domains. Change of variables in double integrals. Multiple integrals. Change of variables in multiple integrals. Surface integrals. Integration of vector fields. The theorems of Stokes and Gauss.

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.

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.

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.

The course aims to provide students with a comprehensive understanding of the developmental changes that occur from birth through adolescence.

Content:
The course will cover a broad range of developmental changes across different phases of childhood and adolescence, as well as theories that describe and explain them. Theories of cognitive, emotional, and social development, self-development, and moral development will be discussed. The approaches of behaviourism and ecological theory will also be highlighted. The origins and nature of individual differences, the continuity and discontinuity of development, and the plasticity of development will be discussed. The interaction between development and learning, motivation, parenting practices, culture, and different social environments will be addressed. A strong emphasis will be placed on understanding child development when working with children in applied settings.

Procedure:
The course will consist of lectures and recitations. Students will have a chance to discuss the course topics and deepen their understanding of the age group they plan to focus on during their studies and work.

Topics from Euclidian geometry. Foundations and systematic development of elementary geometry of the plane. Concepts, postulates, definitions, and theorems involving parallel lines polygons and circles. Attention is given to reasoning and proving theorems. A brief discussion of the geometry of solids. Calculation of area and volume.

Functions of a complex variable. Analytic functions. The exponential function, logarithms and roots. Cauchy's Integral Theorem and Cauchy's Integral Formula. Uniform convergence. Power series. Laurent series. Residue integration method. Application of complex function theory to fluid flows. Ordinary differential equations and systems of ordinary differential equations. Linear differential equations with constant coefficients. Systems of linear differential equations. The matrix exponential function. Various methods for obtaining a particular solution. Green's functions for initial value problems. Flows and the phase plane. Nonlinear systems of ordinary differential equations in the plane, equilibrium points, stability and linear approximations. Series solutions and the method of Frobenius. Use of Laplace transforms in solving differential equations.

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.

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.

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.

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.

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.

Course description in English:*  This course aims to prepare students to use programming and computational thinking strategies to create computer graphics and solve problems of various kinds, and furthermore to prepare students for teaching these same topics to pupils in compulsory school and upper secondary school. No previous experience or knowledge of programming is required before starting the course.

The courses mathematical topics are mainly coordinate geometry and the fundamentals of computational thinking: abstraction, decomposition, algorithmic thinking, debugging, automation and generalization. Students will learn about the use of variables, functions, loops, and logical operators in programming. Students will learn to use computational thinking strategies to build simple computer models, such as computer games or art, and to use programming for solving mathematical tasks.

Students will also learn to plan lessons in compulsory and upper secondary school, aiming for the development of students’ computational thinking and creative programming skills, along with the use of programming for investigating mathematical topics. The dynamic geometry software GeoGebra will be put into a computational thinking perspective and its possibilities for mathematics teaching will be explored. The position of programming and computational thinking in society and the educational system will be discussed, also in connection with other school subjects.

Course participation involves mostly the solving of tasks, reading, and participation in a critical discussion.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This is a foundational course in single variable calculus. The prerequisites are high school courses on algebra, trigonometry. derivatives, and integrals. The course aims to create a foundation for understanding of subjects such as natural and physical sciences, engineering, economics, and computer science. Topics of the course include the following:

• Real numbers.
• Limits and continuous functions.
• Differentiable functions, rules for derivatives, derivatives of higher order, applications of differential calculus (extremal value problems, linear approximation).
• Transcendental functions.
• Mean value theorem, theorems of l'Hôpital and Taylor.
• Integration, the definite integral and rules/techniques of integration, primitives, improper integrals.
• Fundamental theorem of calculus.
• Applications of integral calculus: Arc length, area, volume, centroids.
• Ordinary differential equations: First-order separable and homogeneous differential equations, first-order linear equations, second-order linear equations with constant coefficients.
• Sequences and series, convergence tests.
• Power series, Taylor series.

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.

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.

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.

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.

The course will cover topics in algebra, school algebra and the history of algebra. The teaching of algebra at different school levels will be examined as well as the developement of algebraic thinking in different age groups.

 Part of the course are in-field studies where students prepare and teach algebra.

This course intertwines mathematics and mathematics education. Students get an introduction to probability and statistics, build mathematical models, and learn to approach the teaching of mathematics from a modelling perspective. Among topics covered are combinations, permutations, the counting of the number of outcomes, the binomial distribution, and probability distributions in general. Students will explore how conclusions are drawn about probability based on data and how computer simulations can be used for that purpose. Different types of mathematical models will be introduced, such as linear models, exponential growth models, inverse proportion models, power functions models, linear optimization models in two variables, and graph theoretical models.

The didactics of probability and statistics will be explored and analysed from a modelling perspective. Students will select and adapt exercises and problems for the teaching of probability and statistics, and plan teaching processes. An emphasis will be put on mathematics teaching that touches on important issues of the modern times, such as climate change and pandemics.

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.

Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobi matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flow. Cylindrical and spherical coordinates. Taylor polynomials. Extreme values and the classification of stationary points. Extreme value problems with constraints. Implicit functions and local inverses. Line integrals, primitive functions and exact differential equations. Double integrals. Improper integrals. Green's theorem. Simply connected domains. Change of variables in double integrals. Multiple integrals. Change of variables in multiple integrals. Surface integrals. Integration of vector fields. The theorems of Stokes and Gauss.

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.

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.

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.

The course aims to provide students with a comprehensive understanding of the developmental changes that occur from birth through adolescence.

Content:
The course will cover a broad range of developmental changes across different phases of childhood and adolescence, as well as theories that describe and explain them. Theories of cognitive, emotional, and social development, self-development, and moral development will be discussed. The approaches of behaviourism and ecological theory will also be highlighted. The origins and nature of individual differences, the continuity and discontinuity of development, and the plasticity of development will be discussed. The interaction between development and learning, motivation, parenting practices, culture, and different social environments will be addressed. A strong emphasis will be placed on understanding child development when working with children in applied settings.

Procedure:
The course will consist of lectures and recitations. Students will have a chance to discuss the course topics and deepen their understanding of the age group they plan to focus on during their studies and work.

Topics from Euclidian geometry. Foundations and systematic development of elementary geometry of the plane. Concepts, postulates, definitions, and theorems involving parallel lines polygons and circles. Attention is given to reasoning and proving theorems. A brief discussion of the geometry of solids. Calculation of area and volume.

Functions of a complex variable. Analytic functions. The exponential function, logarithms and roots. Cauchy's Integral Theorem and Cauchy's Integral Formula. Uniform convergence. Power series. Laurent series. Residue integration method. Application of complex function theory to fluid flows. Ordinary differential equations and systems of ordinary differential equations. Linear differential equations with constant coefficients. Systems of linear differential equations. The matrix exponential function. Various methods for obtaining a particular solution. Green's functions for initial value problems. Flows and the phase plane. Nonlinear systems of ordinary differential equations in the plane, equilibrium points, stability and linear approximations. Series solutions and the method of Frobenius. Use of Laplace transforms in solving differential equations.

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.

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.

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.

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.

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.

Course description in English:*  This course aims to prepare students to use programming and computational thinking strategies to create computer graphics and solve problems of various kinds, and furthermore to prepare students for teaching these same topics to pupils in compulsory school and upper secondary school. No previous experience or knowledge of programming is required before starting the course.

The courses mathematical topics are mainly coordinate geometry and the fundamentals of computational thinking: abstraction, decomposition, algorithmic thinking, debugging, automation and generalization. Students will learn about the use of variables, functions, loops, and logical operators in programming. Students will learn to use computational thinking strategies to build simple computer models, such as computer games or art, and to use programming for solving mathematical tasks.

Students will also learn to plan lessons in compulsory and upper secondary school, aiming for the development of students’ computational thinking and creative programming skills, along with the use of programming for investigating mathematical topics. The dynamic geometry software GeoGebra will be put into a computational thinking perspective and its possibilities for mathematics teaching will be explored. The position of programming and computational thinking in society and the educational system will be discussed, also in connection with other school subjects.

Course participation involves mostly the solving of tasks, reading, and participation in a critical discussion.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This is a foundational course in single variable calculus. The prerequisites are high school courses on algebra, trigonometry. derivatives, and integrals. The course aims to create a foundation for understanding of subjects such as natural and physical sciences, engineering, economics, and computer science. Topics of the course include the following:

• Real numbers.
• Limits and continuous functions.
• Differentiable functions, rules for derivatives, derivatives of higher order, applications of differential calculus (extremal value problems, linear approximation).
• Transcendental functions.
• Mean value theorem, theorems of l'Hôpital and Taylor.
• Integration, the definite integral and rules/techniques of integration, primitives, improper integrals.
• Fundamental theorem of calculus.
• Applications of integral calculus: Arc length, area, volume, centroids.
• Ordinary differential equations: First-order separable and homogeneous differential equations, first-order linear equations, second-order linear equations with constant coefficients.
• Sequences and series, convergence tests.
• Power series, Taylor series.

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.

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.

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.

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.

The course will cover topics in algebra, school algebra and the history of algebra. The teaching of algebra at different school levels will be examined as well as the developement of algebraic thinking in different age groups.

 Part of the course are in-field studies where students prepare and teach algebra.

This course intertwines mathematics and mathematics education. Students get an introduction to probability and statistics, build mathematical models, and learn to approach the teaching of mathematics from a modelling perspective. Among topics covered are combinations, permutations, the counting of the number of outcomes, the binomial distribution, and probability distributions in general. Students will explore how conclusions are drawn about probability based on data and how computer simulations can be used for that purpose. Different types of mathematical models will be introduced, such as linear models, exponential growth models, inverse proportion models, power functions models, linear optimization models in two variables, and graph theoretical models.

The didactics of probability and statistics will be explored and analysed from a modelling perspective. Students will select and adapt exercises and problems for the teaching of probability and statistics, and plan teaching processes. An emphasis will be put on mathematics teaching that touches on important issues of the modern times, such as climate change and pandemics.

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.

Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobi matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flow. Cylindrical and spherical coordinates. Taylor polynomials. Extreme values and the classification of stationary points. Extreme value problems with constraints. Implicit functions and local inverses. Line integrals, primitive functions and exact differential equations. Double integrals. Improper integrals. Green's theorem. Simply connected domains. Change of variables in double integrals. Multiple integrals. Change of variables in multiple integrals. Surface integrals. Integration of vector fields. The theorems of Stokes and Gauss.

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.

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.

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.

The course aims to provide students with a comprehensive understanding of the developmental changes that occur from birth through adolescence.

Content:
The course will cover a broad range of developmental changes across different phases of childhood and adolescence, as well as theories that describe and explain them. Theories of cognitive, emotional, and social development, self-development, and moral development will be discussed. The approaches of behaviourism and ecological theory will also be highlighted. The origins and nature of individual differences, the continuity and discontinuity of development, and the plasticity of development will be discussed. The interaction between development and learning, motivation, parenting practices, culture, and different social environments will be addressed. A strong emphasis will be placed on understanding child development when working with children in applied settings.

Procedure:
The course will consist of lectures and recitations. Students will have a chance to discuss the course topics and deepen their understanding of the age group they plan to focus on during their studies and work.

Topics from Euclidian geometry. Foundations and systematic development of elementary geometry of the plane. Concepts, postulates, definitions, and theorems involving parallel lines polygons and circles. Attention is given to reasoning and proving theorems. A brief discussion of the geometry of solids. Calculation of area and volume.

Functions of a complex variable. Analytic functions. The exponential function, logarithms and roots. Cauchy's Integral Theorem and Cauchy's Integral Formula. Uniform convergence. Power series. Laurent series. Residue integration method. Application of complex function theory to fluid flows. Ordinary differential equations and systems of ordinary differential equations. Linear differential equations with constant coefficients. Systems of linear differential equations. The matrix exponential function. Various methods for obtaining a particular solution. Green's functions for initial value problems. Flows and the phase plane. Nonlinear systems of ordinary differential equations in the plane, equilibrium points, stability and linear approximations. Series solutions and the method of Frobenius. Use of Laplace transforms in solving differential equations.

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.

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.

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.

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.

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.

Course description in English:*  This course aims to prepare students to use programming and computational thinking strategies to create computer graphics and solve problems of various kinds, and furthermore to prepare students for teaching these same topics to pupils in compulsory school and upper secondary school. No previous experience or knowledge of programming is required before starting the course.

The courses mathematical topics are mainly coordinate geometry and the fundamentals of computational thinking: abstraction, decomposition, algorithmic thinking, debugging, automation and generalization. Students will learn about the use of variables, functions, loops, and logical operators in programming. Students will learn to use computational thinking strategies to build simple computer models, such as computer games or art, and to use programming for solving mathematical tasks.

Students will also learn to plan lessons in compulsory and upper secondary school, aiming for the development of students’ computational thinking and creative programming skills, along with the use of programming for investigating mathematical topics. The dynamic geometry software GeoGebra will be put into a computational thinking perspective and its possibilities for mathematics teaching will be explored. The position of programming and computational thinking in society and the educational system will be discussed, also in connection with other school subjects.

Course participation involves mostly the solving of tasks, reading, and participation in a critical discussion.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

This is a foundational course in single variable calculus. The prerequisites are high school courses on algebra, trigonometry. derivatives, and integrals. The course aims to create a foundation for understanding of subjects such as natural and physical sciences, engineering, economics, and computer science. Topics of the course include the following:

• Real numbers.
• Limits and continuous functions.
• Differentiable functions, rules for derivatives, derivatives of higher order, applications of differential calculus (extremal value problems, linear approximation).
• Transcendental functions.
• Mean value theorem, theorems of l'Hôpital and Taylor.
• Integration, the definite integral and rules/techniques of integration, primitives, improper integrals.
• Fundamental theorem of calculus.
• Applications of integral calculus: Arc length, area, volume, centroids.
• Ordinary differential equations: First-order separable and homogeneous differential equations, first-order linear equations, second-order linear equations with constant coefficients.
• Sequences and series, convergence tests.
• Power series, Taylor series.

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.

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.

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.

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.

The course will cover topics in algebra, school algebra and the history of algebra. The teaching of algebra at different school levels will be examined as well as the developement of algebraic thinking in different age groups.

 Part of the course are in-field studies where students prepare and teach algebra.

This course intertwines mathematics and mathematics education. Students get an introduction to probability and statistics, build mathematical models, and learn to approach the teaching of mathematics from a modelling perspective. Among topics covered are combinations, permutations, the counting of the number of outcomes, the binomial distribution, and probability distributions in general. Students will explore how conclusions are drawn about probability based on data and how computer simulations can be used for that purpose. Different types of mathematical models will be introduced, such as linear models, exponential growth models, inverse proportion models, power functions models, linear optimization models in two variables, and graph theoretical models.

The didactics of probability and statistics will be explored and analysed from a modelling perspective. Students will select and adapt exercises and problems for the teaching of probability and statistics, and plan teaching processes. An emphasis will be put on mathematics teaching that touches on important issues of the modern times, such as climate change and pandemics.

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.

Open and closed sets. Mappings, limits and continuity. Differentiable mappings, partial derivatives and the chain rule. Jacobi matrices. Gradients and directional derivatives. Mixed partial derivatives. Curves. Vector fields and flow. Cylindrical and spherical coordinates. Taylor polynomials. Extreme values and the classification of stationary points. Extreme value problems with constraints. Implicit functions and local inverses. Line integrals, primitive functions and exact differential equations. Double integrals. Improper integrals. Green's theorem. Simply connected domains. Change of variables in double integrals. Multiple integrals. Change of variables in multiple integrals. Surface integrals. Integration of vector fields. The theorems of Stokes and Gauss.

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.

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.

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.

The course aims to provide students with a comprehensive understanding of the developmental changes that occur from birth through adolescence.

Content:
The course will cover a broad range of developmental changes across different phases of childhood and adolescence, as well as theories that describe and explain them. Theories of cognitive, emotional, and social development, self-development, and moral development will be discussed. The approaches of behaviourism and ecological theory will also be highlighted. The origins and nature of individual differences, the continuity and discontinuity of development, and the plasticity of development will be discussed. The interaction between development and learning, motivation, parenting practices, culture, and different social environments will be addressed. A strong emphasis will be placed on understanding child development when working with children in applied settings.

Procedure:
The course will consist of lectures and recitations. Students will have a chance to discuss the course topics and deepen their understanding of the age group they plan to focus on during their studies and work.

Topics from Euclidian geometry. Foundations and systematic development of elementary geometry of the plane. Concepts, postulates, definitions, and theorems involving parallel lines polygons and circles. Attention is given to reasoning and proving theorems. A brief discussion of the geometry of solids. Calculation of area and volume.

Functions of a complex variable. Analytic functions. The exponential function, logarithms and roots. Cauchy's Integral Theorem and Cauchy's Integral Formula. Uniform convergence. Power series. Laurent series. Residue integration method. Application of complex function theory to fluid flows. Ordinary differential equations and systems of ordinary differential equations. Linear differential equations with constant coefficients. Systems of linear differential equations. The matrix exponential function. Various methods for obtaining a particular solution. Green's functions for initial value problems. Flows and the phase plane. Nonlinear systems of ordinary differential equations in the plane, equilibrium points, stability and linear approximations. Series solutions and the method of Frobenius. Use of Laplace transforms in solving differential equations.

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.

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.

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.

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.

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.

Course description in English:*  This course aims to prepare students to use programming and computational thinking strategies to create computer graphics and solve problems of various kinds, and furthermore to prepare students for teaching these same topics to pupils in compulsory school and upper secondary school. No previous experience or knowledge of programming is required before starting the course.

The courses mathematical topics are mainly coordinate geometry and the fundamentals of computational thinking: abstraction, decomposition, algorithmic thinking, debugging, automation and generalization. Students will learn about the use of variables, functions, loops, and logical operators in programming. Students will learn to use computational thinking strategies to build simple computer models, such as computer games or art, and to use programming for solving mathematical tasks.

Students will also learn to plan lessons in compulsory and upper secondary school, aiming for the development of students’ computational thinking and creative programming skills, along with the use of programming for investigating mathematical topics. The dynamic geometry software GeoGebra will be put into a computational thinking perspective and its possibilities for mathematics teaching will be explored. The position of programming and computational thinking in society and the educational system will be discussed, also in connection with other school subjects.

Course participation involves mostly the solving of tasks, reading, and participation in a critical discussion.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.