In questo documento sono brevemente presentati i corsi di interesse per gli studenti di Matematica e Informatica attivati presso l'università norvegese.
Per i corsi di contenuto più prettamente informatico si veda

http://www.idi.ntnu.no/emner/nye_emnekoder.php

Di seguito sono evidenziati i corsi a contenuto numerico (Numerical Analysis Courses)(segnalo in particolare TMA4280 Introduction to Supercomputing).
L'elenco prosegue presentando i corsi raggruppati in matematici (Matematikk),  statistici (Statistikk) (segnalo in particolare ST1301 Computational biology ) e corsi avanzati (Doktorgradskurs)

• Numerical Analysis Courses:

TMA4205 Numerical linear algebra. The course emphasizes iterative techniques for solving large sparsesystems of equations which typically arise from discretizing PDEs.There will also be some discussion of eigenvalue problems and error analysis. Autumn

TMA4210 Numerisk solution of differential equations with the difference method. This course has recently been given as 50% Numerical PDEs and 50% term project in a general topic inMathematics. In the future it will be only numerical PDEs due tosome change decided this year. Difference methods for linear PDEs.Special attention to heat equation, Poissons equation, and wave equation. Discussion of different types of boundary conditions,and introduction to concepts like convergence, stability and consistency. Assignments with Matlab. Spring

TMA4215 Numerical mathematics. This course contains the basis numerial . Polynomial interpolation, Differentiation,Integration, least squares, splines, FFT and Numerical ODEs. Autumn.

TMA4220 Numerical solution of partial differential equations with the finite element method.

FEM in general. Special focus on convection-diffusion equations.Topics: Minimization principle, weak formulation, boundary conditions,quadrature, error analysis, stability, convergence, implementation,direct and indirect solution of resulting algebraic equations, and applications. Spring.

TMA4280 Introduction to Supercomputing.

The objective with this course is to give a brief introduction to modern computer architecture and to discuss numerical algorithms forvector and parallel processing. I recommend to look at

http://www.math.ntnu.no/~ronquist/kurs/super/2003h/course.html

where all info is in english. Autumn
 
 

T0218063526 v. 1 Numerical integration av time dependent differential equations

The course is given every second year if a sufficient number of students sign up. The course is given nexttime Spring 2006.The first part of the course is devoted to general techniques for solving ordinary differential equations,like Runge-Kutta and linear multistep methods. Then modern numerical methods for special applications arediscussed, for instance differential equations with conservation laws or other underlying geometric structures. The last part of the course will treat time integration of partial differential equations. Modern schemes based on splitting and exponentials will be presented and analyzed.

T0218064316 v. 1 Numerical Solution of Partial Differential Equations

The course is given every second year, depending on enough students signing up. Next time: Spring 2005.The course will treat selected topics within analysis and use of thefinite element method in computational mechanics with special focus on computational techniques for incompressible fluid flow. High order spectral element methods will be used in the spatial discretization. These methods will be discussed in conjunction with the solution of the Poisson problem, the stationary Stokes problem and problems which involve convection. Time discretization willinclude operator splitting methods. Treatment of general boundary conditions and deformed geometry will be discussed. Moreover, the efficient computation of derived quantities from the numerical solution will be discussed.

• Matematikk

MA0001 Mathematical methods A (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Two problem sets are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on 2MX from grammar school or equivalent. The course together with MA0002 will give a sufficient mathematical background for science courses only requiring basic mathematics. The course will introduce functions of one real variable, exponential-, logarithm and trigonometric functions, differentiation and integration with applications, implicit differentiation and linear approximation.

The course will not give additional credit to students who already have taken MNFMA001 Mathematical methods, MNFMA100 Basic calculus, MA1101 Basic calculus I, and MA0003 Mathematical methods for computer science.

MA0002 Mathematical methods B (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Two problem sets are approved .

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on MA0001 Mathematical methods A or equivalent. The course together with MA0002 will give a sufficient mathematical background for science courses only requiring basic mathematics. With some additional work this course should enable students to continue wity more advanced mathematics courses which normally require MA1101 Basic calculus I and MA1102 Basic calculus II. The course will introduce functions of several variables, partial differentiation, double integrals, linear approximation, linear equations, matrices, eigen-values, differential equations, numerical solution of differential equations (Eulerís method), and systems of differential equations.

The course will not give additional credit to students who already have taken MNFMA001 Mathematical methods. It will give 3,75 credit points to students who already have taken MA1101 Basic calculus I.

 MA0003 Mathematical methods for computer science (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Two problem sets are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on 2MX from grammar school or equivalent. The course will introduce functions of one real variable, exponential-, logarithm and trigonometric functions, differentiation and integration with applications, implicit differentiation and linear approximation.

The course includes similar topics to MA0001 Mathematical methods A and may be taken as an alternative to this, but the problem sessions are arranged for students in computer science.

The course will not give additional credit to students who already have taken MNFMA001 Mathematical methods, MNFMA100 Basic calculus, MA1101 Basic calculus I, and MA0001 Mathematical methods

MA0301 Elementary discrete mathematics (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam prerequisite: Two problem sets are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is primarily for computer science students, but should also be of interest to students who take courses in mathematics. There are no prerequesities except grammar school mathematics. The course gives an introduction to set theory, logic, induction and recursion, relations and functions, boolean algebra and graph theory.

The course will not give additional credit to students who already have taken MNFMA012 Elementary discrete mathematics.

MA1101 Basic calculus I (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Some compulsory are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on 3MX from grammar school or equivalent. Content of the course: Basic properties of real numbers and real functions of a real variable, limits, continuity, differentiation and integration. The fundamental theorem of calculus is central in the course, and so are its applications. In this context first order differential equations are introduced. Emphasize is made on rigour.

The course will not give additional credit to students who already have taken MNFMA100 Basic calculus, MA0001 Mathematical methods A, MA0002 Mathematical methods B and MA0003 Mathematical methods for computer science.

 MA1102 Basic calculus II (Grunnkurs i analyse II) (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam prerequisite: Problem sets which are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on MA1101 Basic calculus I. The course first introduces parametric curves, curvature and acceleration. Later on, Taylorís formula, L`Hôpitalís rule, improper integrals, infinite series, and uniform convergence are treated. Inn addition, the course contains certain aspects of numerical mathematics; the topics treated are Newtonís method, numerical integration and Simpsonís formula. A thorough treatment of first an second order differential equations is part of the course. Emphasize is made on rigour.

Students who already have taken MNFMA100 Basic calculus will get 1,5 credit points.

 MA1103 Vector calculus (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam prerequisite: Problem sets which are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

This course should be taken parallel to or after MA1102 Basic calculus II. Topics included are functions of several real variables, partial differentiation, directional derivative, gradient, extremal problem and Lagrangeís multiplier method. Other topics discussed are multiple integrals, line and surface integrals, with examples from applications. Furthermore, the course includes vector valued functions, divergence and curl of vector fields, the notion of flux, Greenís, Stokesí and Gaussí theorems, with examples of applications on.

The course will not give additional credit to students who already have taken MNFMA109 Vector calculus.

 MA1201 Linear algebra and geometry (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets which are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on 3MX from grammar school or equivalent. Topics included are: basic logic and method of proofs, systems of linear equations, Gaussian elimination, LU-decomposition, vectors in 2-space and 3-space (scalar and cross products), R^n, matrices, determinants (Cramer's rule, determinants as area and volume), linear transformations and their geometric properties in R^2, associated matrices for R^2 and R^3, some theory on eigenvalues for matrices, diagonal matrices, conic sections, numerial aspects (Gauss-Seidel),

complex numbers.

The course will not give additional credit to students who already have taken MNFMA108 Linear algebra.

 MA1202 Linear algebra with applications (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 3 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets which are approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on MA1201 Linear algebra and geometry. The course includes: general vector spaces (linear independence, basis), row spaces, column spaces, spaces with inner product, orthonormal basis, Gram-Schmidt, change of basis, orthogonal matrices, linear

transformations (kernel, image, dimension theorem, associated matrices), eigenvalues and eigenvectors for linear transformations and matrices, eigenspaces, complex vector spaces, complex inner products, unitary and Hermitian matrices, singular value decomposition. Several applications, among others; Markov chains, population growth (Leslie matrices), genetics.

The course will not give additional credit to students who already have taken MNFMA108 Linear algebra.

MA1301 Number theory (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 3 lessons a week

Problem sessions: 1 hour a week

Exam format: 4 hours written, or oral. Retake may be given as an oral examination.

There are no prerequesities except grammar school mathematics. This course gives an introduction to elementary number theory. Topics included are: greatest common divisor, Euclidean algorithm, linear diophantine equations, elementary prime number theory, linear congruences, Fermatís little theorem, Eulerís j -function, Eulerís theorem with application to cryptografi. Topics that may change from one year to another are number theoretical functions, Fermatís last theorem for n = 4, quadratic reciprocity and generation of random numbers.

The course will give 1,5 creditpoints to students who already have taken MNFMA104 Number theory.

 MA2001 Mathematical project (7,5 credit points)

Duration: 1 semester (spring)

This is a project to be done independently, with advice.

 MA2104 Differential equations and complex function theory(7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam format: 4 hours written. Retake may be given as an oral examination.

The course is based on MA1101 Basic calculus I, MA1102 Basic calculus II, MA1103 Vector calculus, MA1201 Linear algebra and geometry and MA1202 Linear algebra with applications.

The course includes applications of transformation methods to linear differential equations. In addition the course gives an introduction to complex analysis. Fourier series, Fourier transforms, solution of ordinary and partial differential equations, complex integration, Laurent series and Calculus of Residues are also included. Basic facts on Möbius transforms will also be covered.

Information to students who already have passed exams in courses with former internal codes: The course is based on MNFMA100 Basic calculus, MNFMA108 Linear algebra and MNFMA109 Vector calculus.

Students who already have taken MNFMA211 Differential equations and Fourier analysis and MNFMA212 Function theory will get 3 credit points reduction.

 MA2201 Algebra (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 3 lessons a week

Problem sessions: 2 hours a week

Exam format: 5 hours written. Retake may be given as an oral examination.

The course is based on MA1201 Linear algebra and geometry and MA2101 Linear algebra with applications. It is also an advantage to be familiar with the material in MA1301 Number theory. Topics that will be treated are groups, subgroups, normal subgroups, factor groups, group homomorphisms, group action on sets, combinatorial counting results, Sylow theorems, rings and fields. Most of the course follows the lectures in SIF5021 Algebra and Number theory.

Information to students who already have passed exams in courses with former internal codes: The course is based on MNFMA108 Linear algebra. It is also an advantage to be familiar with the material in MNFMA104 Number theory. .

The course will not give additional credit to students who already have taken MNFMA205 Algebra.

 MA2301 Advanced discrete mathematics (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 3 lessons a week

Problem sessions: 2 hours a week

Exam format: 4 hours written, or oral.

The course is based on MA0301 Elementary discrete mathematics. This course provides parts of the theoretical background for computer science. The course includes formal languages, finite automata, Turing machines, computability, polynomical reduction and complexity-classes with examples.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA012 Elementary discrete mathematics.

The course will not give additional credit to students who already have taken MNFMA217 Advanced discrete mathematics.

MA2401 Geometry (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam format: 4 hours written, or oral.

The course is to some extent based on MA1101 Basic calculus I, MA1102 Basic calculus II, MA1201 Linear algebra and geometry and MA1202 Linear algebra with applications. Topics

are: Axiomatic foundation of Euclidean and hyperbolic geometries. The connection with high school geometry and also the historical development is covered to some extent. Other geometrical topics may also be included.

Information to students who have passed exams in courses with former internal codes: The course is to some extent based on MNFMA100 Basic calculus and MNFMA108 Linear algebra.

The course will not give additional credit to MNFMA220 Geometry.

 MA2501 Numerical methods (7,5 credit points)

Duration: 1 semester (spring). The course will be lectured for the first time in spring 2005.

Lectures: 3 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved

Exam format: 4 hours written (80 %) and a project (20 %). Retake may be given as an oral examination.

The students are supposed to know elementary part of mathematics like Taylor-expansions, integration and derivation. Among the topics of the course are interpolation and the least square methods. Numerical derivation and integration. None-linear equations and system of equations. Numerical solution of ordinary and partial differential equations.

 MA3001 Mathematical seminar for master degree students(7,5 credit points)

Duration: 1 semester

Seminar: Will be announced

Exam format: Oral.

The seminar provides an introduction to an essential mathematical topic which is not described in the course catalogue. Topics will vary.

 SIF5020 Linear methods (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam format: 5 hours written, or oral (80 %). In addition problem sets (20 %). The problem sets and the written exam must both be passed.

The course is based on MA1101 Basic calculus I, MA1102 Basic calculus II, MA1103 Vector calculus, MA1201 Linear algebra and geometry, MA1202 Linear algebra with applications and MA2104 Differential equations, or equivalent. The course includes further aspects of matrices and linear algebra, and also cover basic ideas and methods in linear analysis/functional analysis. Topics are: Linear algebra with and without coordinates, projections, Cayley-Hamiltons theorem, Spectral theorem and a brief introduction to Jordan decomposition. Further topics: Positive definite matrices, singular value decomposition and generalized inverse, problems on least squares, metric spaces, completeness and contraction principle, Banach spaces, Hilbert spaces, approximation, orthogonal systems and Fourier series. Finally, the course will include linear functionals, dual spaces and Rieszí representation theorem in Hilbert spaces.

 SIF5052 Foundation on analysis (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam format: 5 hours written, or oral.

The course is based on MA1101 Basic calculus I, MA1102 Basic calculus II, MA1102 Vector calculus, MA2104 Differential equations and complex function theory, and SIF5020 or equivalent. Topics are properties of real numbers (including supremum, infimum and completeness). Furthermore the course includes general measure theory including sigma algebras, measure spaces, measurable functions, outer measure. Lebesgue integration, product measures, the Fubini-Tonelli theorem, the classical convergence theorems and functions of bounded variation are also covered.

The course will not give additional credit to students who already have taken MNFMA320 Foundation on analysis.

 SIF5054 Function analysis (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam format: 5 hours written, or oral.

The course is based on SIF5052 Foundation of analysis. Topics are Hahn-Banachís threorem,

open mapping theorem, dual spaces, weak convergence, Banach-Alaoglusí theorem and the spectral theorem for bounded self-adjoint operators.

The course will not give additional credit to students who already have taken MNFMA325 Function analysis.

 SIF5029 Complex analysis (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam format: 5 hours written, or oral.

The course is based on MA2104 Differential equations and complex function theory. It is also an advantage to be familiar with the material in SIF5020 Linear methods. The course provides an introduction in fundamental theory for complex integration, conformal mappings and harmonic functions. The course also includes selected advanced topics as for example analytic continuation, eleptic functions, the theory of Hardy spacesø, Wiener-Hopf equations, harmonic functions, Bergman kernel function, interpolation and approximation, entire functions and applications on fluid dynamics.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA212 Function theory.

The course will not give additional credit to students who already have taken MNFMA326 Advanced complex analysis.

 MA3105 Advanced real analysis (7,5 credit points)

Duration: 1 semester (spring. The course will be lectured every other year, the first time in 2004)

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The course is based on SIF5052 Foundation on analysis. Topics are the classical L^p-rom, Radon-Nikodymís theorem, locally compact spaces, Stone Weierstrass theorem, Radon-measure, the foundation of probability theory, Fourier analysis, convolution. The course may be taken as a parallel to SIF5054 Functional analysis.

The course will not give additional credit to students who already have taken MNFMA325 Functional analysis.

 MA3201 Rings and modules (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The course is based on MA1201 Linear algebra and geometry, MA1202 Linear algebra with applications and MA2201 Algebra, or equivalent. The course includes rings, structure theorems for modules over principal ideal domains and for simple and semisimple rings.

Information to students who have passed exams in courses with former internal codes: The course is based on MFNMA108 Linear algebra and MNFMA205 Algebra or equivalent.

The course will not give additional credit to students who already have taken MNFMA318 Rings and modules and MNFMA321 Abstract algebra.

 MA3202 Commutativ algebra and Galois theory (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The course is based on MA1201 Linear algebra and geometry, MA1202 Linear algebra with applications and MA 2201 Algebra, or equivalent. The course includes commutative noetherian rings, Hilbertís basic theorem and Galois theory. The course will not give additional credit to students who already have taken MNFMA319 Commutative algebra and Galois theory or MNFMA321 Abstract algebra.

Information to students who have passed exams in courses with former internal codes: The course is based on MFNMA108 Linear algebra and MNFMA205 Algebra or equivalent.

MA3203 Ring theory (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The course is based on MA3201 Rings and modules or equivalent (the course may be taken parallel to MA3202 Commutative algebra and Galois theory). The content of the course may vary, but the core will consist of artinian, noetherian and local rings, projective and injective modules, the Jordan-Hölder Theorem, radical of modules and rings, socles, exact sequences, categories, functors, equivalence, duality, and adjoint functors.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA318 Rings and modules or equivalent.

The course will not give additional credit to students who already have taken MNFMA327 Ring theory.

MA3204 Homological algebra (7,5 credit points)

Duration: 1 semester (autumn. The course will be lectured every other year, the first time in 2003).

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The course is based on MA3201 Rings and modules and MA3202 Commutative algebra and Galois theory, or equivalent. The content of the course may vary, but the core will include categories of modules, the functors Hom and tensor product, free, projective, injective and flat resolutions, direct and inverse limits, projective, injective and flat dimension, homology and the functors Ext and Tor.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA318 Rings and modules and MNFMA319 Commutative algebra and Galois theory, or equivalent.

The course will not give additional credit to students who already have taken MNFMA330 Homological algebra.

 SIF5034 Manifolds (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 1 hour a week

Exam format: 5 hours written, or oral.

The course is based on MA1101 Basic calculus I, MA1102 Basic calculus II, MA1103 Vector calculus, MA1201 Linear algebra and geometry, MA1202 Linear algebra with applications and MA2104 Differential equations and complex function theory, or equivalent.

The goal with the course is to give the students basic insight in geometric concepts and methods in differential topology preparing the study of dynamical systems on manifolds. Contents: Some point set topology. Manifolds, differentiable structures and tangent spaces. Vector bundles. Riemannian manifolds. Partition of unity. Embeddings and immersions. Transversality. Integrability.

 MA3402 Analysis on manifolds (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The course is based on SIF5034 Manifolds. Some knowledge of basic analysis is an advantage. The subject is demanding and requires mathematical maturity. The goal with the course is to give basic insight in analysis on manifolds. Subjects to be treated are: Tensors, differential forms, exact and closed forms, Poincarés lemma, de Rham kohomology, integration on manifolds, Stokesí theorem and Mayer-Vietoris sequence.

The course will not give additional credit to students who already have taken MNFMA317 Analysis on manifolds.

 MA3403 Algebraic topology (7,5 credit points)

Duration: 1 semester (spring. The course will be lectured every other year, the first time in 2005).

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The goal with the subject is to give the students basic insight in concepts and methods in algebraic topology. The course is a foundation for studies in topology, geometry, algebra or theoretical physics. An introduction is given to homology theory, cell-complexes, homotopy theory, category theory, homology and cohomology, covering spaces, duality and concrete homological calculations. Some knowledge of manifolds and algebra is an advantage.

The course will not give additional credit to students who already have taken MNFMA333 Algebraic topology.

 MA3404 Non-linear dynamical systems (7,5 credit points)

Duration: 1 semester (autumn).

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The goal with the subject is to give a basic introduction to the modern theory of dynamical systems and give a basic for further study in this area. It will also be a valuable support for other courses using dynamical systems. A series of modern techniques will be discussed, both for continuous and discrete systems (iterated maps). The will be emphasis on the interplay between differentiable and symbolic dynamics. Central themes will be chaos, attractors and bifurcation theory. The course is based on SIF5025 Differential equations and dynamical systems.

The course will not give additional credit to students who already have taken MNFMA350 Non-linear dynamical systems.

MA3405 Cohomology theory (7,5 credit points)

Duration: 1 semester (spring. The course will be lectured every other year, the first time in 2005.).

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The goal with the subject is to give a solid background for studies in topology, and support studies in related subjects as algebra, geometry and theoretical physics. Subjects to be treated are: Cohomology, vector bundles, K-theory, characteristic classes, cobordism theory, spectra and combinatorial methods. The course is based on MA3403 Algebraic topology.

The course will not give additional credit to students who already have taken MNFMA351 Cohomology theory.

• Statistikk

ST0101 Probability with applications (7,5 credit points)

Duration: 1 semester (autumn)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course provides a general introduction to probability calculation, with applications from science and medicine, and includes the following topics; events and sample space, the uniform probability model, the probability axioms, probability laws, conditional probability, independence, combinatorics, the urn model, random variables, mean value, variance, standard deviation, discrete and continuous probability distributions, bivariate distributions, covariance and correlation, independent random variables, computing the mean by conditioning, binomial distribution, hyper-geometric distribution, geometric distribution, Poisson distribution, exponential distribution, normal distribution, the central limit

theorem, chi-squared distribution, Students t-distribution, Fishers F-distribution, multinomial distribution and binormal distribution.

The course will not give additional credit to students who already have taken ST1101 Probability. Students are thus recommended to take only one of these courses. Furthermore, the course will not give additional credit to students who already have taken MNFST001 Statistics with applications and MNFST101 Probability and statistics I. Credit points will be reduced with 3,5 for students who already have taken ST0202 Statistics for social sciences.
 

ST0201 Statistics with applications (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on ST0101 Probability with applications. The course provides an introduction to the basic definitions and concepts for statistical inference and gives an introduction to some basic statistical methods. The topics covered include; point estimation, properties of point estimators, some simple examples of stratification, confidence interval for the mean in the normal distribution with known and unknown variance, confidence interval for the variance in a normal distribution, confidence interval based on normal approximation, testing a statistical hypothesis, test statistic, critical region, significance level, power of a test, power function, testing in normal- and binomial models, tests based on the normal approximation, correlation, simple linear regression, T-tests for comparison of two samples, one-way analysis of variance, non-parametric methods, analysis of categorical variables.

The course will not give additional credit to students who already have taken ST1201 Statistical methods. Students are thus recommended to take only one of these courses. Furthermore, the course will not give additional credit to students who already have taken MNFST001 Statistics with applications and MNFST101 Probability and statistics I. Credit points will be reduced with 4 to students who already have taken ST0202 / MNFSIB1 Statistics for social sciences.
 

ST0202 Statistics for sociologists (7,5 credit points)

Duration: 1 semester (autumn/spring)

Lectures: 36 hours

Problem sessions: 2 hours a week through the semester

Exam prerequisite: Problem sets approved.

Exam format: 5 hours written, or oral.

The course provides a short introduction to probability calculation and basic statistics. The number of practical assignments and deadlines for submission will be given at the start of the semester.

Credit points will be reduced with 3,5 to students who already have taken ST0101 Probability with applications and ST1101 Probability, and with 4 to those who already have taken ST0201 Statistics with applications and ST1201 Statistical methods. The course will not give additional credit to students who already have taken MNFSIB1 Statistics for social sciences, MNFST001 Statistics with applicatons and MNFST101 Probability and statistics I.
 

ST1101 Probability (7,5 credit points)

Duration: 1 semester (spring)

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on MA1101 Basic calculus I. The course provides an introduction to probability calculations and includes the following topics; the probability axioms, probability laws, conditional probability, independence, combinatorics, the urn model, random variables, mean value, variance, standard deviation, discrete and continuous univariate probability distributions, transformation of random variables, discrete and continuous bivariate distributions, covariance and correlation, independent random variables, computing the mean by conditioning, moment generating functions, cumulant generating functions, order statistics,

binomial distribution, hyper-geometric distribution, geometric distribution, Poisson distribution, exponential distribution, normal distribution, the central limit theorem, chi-squared distribution, Students t-distribution, Fishers F-distribution, multinomial distribution and binormal distribution.

The course will not give additional credit to students who already have taken ST0101 Probability with applications. Students are thus recommended to take only one of these courses. Furthermore, credit points will be reduced with 3,5 to students who already have taken ST0202/MNFSIB1 Statistics for social sciences. The course will not give additional credit to students who already have taken MNFST001 Statistics with applications and MNFST101 Probability and statistics I.
 

ST1201 Statistical methods (7,5 credit points)

Duration: 1 semester (autumn). The course will be lectured for the first time in autumn 2004.

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved.

Exam format: 4 hours written (80 %) and midterm test (20 %). Retake may be given as an oral examination.

The course is based on ST1101 Probability and MA1101 Basic calculus I. The course provides an introduction to statistical methods and includes point estimation, the method of moments, least squares estimation, maximum likelihood estimation, general introduction to interval estimation and testing of statistical hypothesis, methods for normal distributed random variables based on T-distribution, chi-squared distribution and F-distribution,

testing in binomial models, group comparison, methods for normal approximation, simple regression and analysis of variance for normally distributed data, analysis of correlation, goodness of fit tests, non-parametric tests.

The course will not give additional credit to students who already have taken ST0201 Statistics with applications. Furthermore, credit points will be reduced with 4 to students who already have taken ST0202/MNFSIB1 Statistics for social sciences. Finally, the course will not give additional credit to students who already have taken MNFST001 Statistics with applications and MNFST101 Probability and statistics I.
 

ST1301 Computational biology (7,5 credit points)

Duration: 1 semester (spring).

Lectures: 2 lessons a week

Problem sessions: 4 hours a week

Exam prerequisite: Problem sets approved.

Exam format: Approved problem sets (20 %), 4 hours written (80 %). Retake may be given as an oral examination.

Basic programming and use of computer packages. Data types, control flow, functions, graphical techniques. Methods for simulation of random variables, solution of non-linear equations, numerical maximum likelihood computation, methods for non-parametric and parametric bootstrapping. Simulation of deterministic, stochastic and chaotic dynamic model. Age structure, genetic drift, environmental and demographic variance, time to extinction, catch-recatch models, invasion models.

ST2101 Stochastic modelling and simulation (15 credit points)

Duration: 1 semester (spring). The course will be lectured for the first time in spring 2005.

Lectures: 6 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved.

Exam format: Problem sets approved (20 %), 4 hours written (80 %). Retake may be given as an oral examination.

The course is based on knowledge in mathematics similar to MA0001/0002 Mathematical methods A/B or MA1101/1102 Basic calculus I/II and knowledge of statistics similar to ST0101/0202 Probability with applications/Statistics for social sciences or ST1101/1201 Probability/Statistical methods. The course includes methods for simulation of random variables (including the inverse transformation method, rejection sampling and transformation methods), generating functions, Markov processes with discrete and continuous time parameter and discrete and continuous sample space, Poisson processes, birth and death processes, queueing processes, renewal processes, statistical inference for stochastic processes, simulation of stochastic processes, an introduction to diffusion processes.
 

ST2201 Mathematical statistics (15 credit points)

Duration: 1 semester (spring). The course will be lectured for the first time in spring 2005.

Lectures: 6 lessons a week

Problem sessions: 2 hours a week

Exam format: 4 hours written (80 %), and midterm test (20 %). Retake may be given as an oral examination.

The course is based on knowledge in mathematics similar to MA0001/0002 Mathematical methods A/B or MA1101/1102 Basic calculus I/II and knowledge of statistics similar to

ST0101/0202 Probability with applications/Statistics for social sciences or ST1101/1201 Probability/Statistical methods. The course includes transformation of random variables (including transformation of vector variables), theory of point estimation, interval estimation and testing of statistical hypothesis, inference for normal populations, asymptotic properties for statistical methods.

The course will not give additional credit to students who already have taken MNFST102 Probability and statistics II.
 

ST2202 Applied statistics (15 credit points)

Duration: 1 semester (autumn). The course will be lectured for the first time in autumn 2005.

Lectures: 6 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved.

Exam format: Problem sets approved (20 %), 4 hours written (80 %). Retake may be given as an oral examination.

The course is based on knowledge in mathematics similar to MA0001/0002 Mathematical methods A/B or MA1101/1102 Basic calculus I/II and knowledge of statistics similar to ST0101/0202 Probability with applications/Statistics for social sciences or ST1101/1201 Probability/Statistical methods. The course includes chi-squared tests (test for probabilities in multinomial models, model test with known and unknown parameters, test for homogeneity, test for independence) regression analysis, design of experiments (including analysis of variance and 2^k experiments), generalised linear models, transformations for normal approximations, variance stabilising transformations, non-parametric tests, approximations for mean and variance (including for estimators implicitly defined by an equation), combination of estimators, combination of tests, multiple testing (including Bonferroni correction), simulation tests and bootstrapping.
 

ST2301 Mathematical evolutionary genetics (7,5 credit points)

Duration: 1 semester (autumn). The course will be lectured for the first time in autumn 2004.

Lectures: 4 lessons a week

Problem sessions: 2 hours a week

Exam prerequisite: Problem sets approved.

Exam format: Problem sets approved (20 %), 4 hours written (80 %). Retake may be given as an oral examination.

Topics include allele and genotype frequencies, linkage and linkage disequilibrium, identity by descent and inbreeding, dominance, epistacy, selection, X-linkage, Fisher's fundamental theorem, quantitative traits, equilibria between evolutionary forces, genetic load, genetic drift, effective population size, and frequency distributions in stochastic models.
 

ST2302 Stochastic population models (7,5 credit points)

Duration: 1 semester (autumn).

Lectures: 4 lessons a week

Exam format: Written or oral (80 %), and midterm test (20 %).

The course deals with models and probability distributions in statistical ecology, birth and death processes and life history models.  Stochastic processes and their diffusion. Approximations play a central role in the course and applications of such models in ecology and population genetics are given.  Concepts such as environmental and demographic variance are defined.  Topics important in conservation biology are emphasised.
 

ST3201 Generalised linear methods (7,5 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources).

Lectures: 4 lessons a week

Exam format: 5 hours written, or oral.

The course deals with testing of models, residual analysis, model formulae for linear presictors, analysis of linear models including retrospective collection of data and over dispersion, multinomial models, log-linear models for Poisson distributed observations, conditional likelihood, quasi-likelihood, control of model fitting and analysis of survival data.
 

ST3202 Topics in statistics (up to 15 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources).

Lectures: 4 lessons a week

Exam format: 4-6 hours written, or oral.

The course will include topics which are not presented in the course catalogue. Further information about credit points, literature and examination will be given at the beginning of the semester.

• Doktorgradskurs

MA80001 Mathematical seminar for Ph-students (7,5 credit points)

Duration: 1 semester.

Seminar: Will be announced

Exam format: Oral.

The course provides an introduction to an essential mathematical topic that is not described in the course catalogue.
 

MA8101 Chaos and fractal geometry (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written and problem sets in a computer laboratory, or oral.

The course is based on parts of SIF5052 Foundation of analysis or equivalent. The course deals with dynamical systems, periodic and chaotic systems, specific discrete models (e.g. logistic behaviour),fractal sets and dynamics of fractal sets,Mandelbrot and Julia sets, iterated systems of functions, image compression, Hausdorff dimension and fractal dimension. Computer lab is part of the course.

Information to students who already have passed exams in courses with former internal codes: The course is based on parts of MNFMA320 Foundation of analysis or equivalent.
 

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on SIF5052 Foundation on analysis. The course includes studier av transformasjoner av topologiske rom, eventuelt målrom, og asymptotiske egenskaper til slike transformasjoner. Opprinnelsen til ergodeteorien var den såkalte ergodehypotesen, som lå til grunn for klassisk statistisk mekanikk slik den ble grunnlagt av Bolztmann og Gibbs. Stikkord er målbevarende systemer, Birkhoffs punktvise ergodeteorem, rekurrens, systemer med diskret spektrum, entropi, og minimale dynamiske systemer.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA320 Foundation on analysis.
 

MA8102 Univalent functions (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

Fundamental (classical) theory, mainly based on flatesatsen. Løwnerkjeder, konveksitetsteori samt variasjonsmetoder behandles.
 

MA8103 Analytic theory for continued fractions (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

Emnet omfatter grunnleggende teori for kjedebrøker, med spesiell vekt på konvergensteori og beregningsalgoritmer. Videre behandles konvergens og korrespondanse av kjedebrøkutviklinger av funksjoner, Padè-approksimasjoner og momentteori. Det gis også eksempler på anvendelse i tallteori, digitalfilter og differensiallikninger.

MA8104 Functions of several complex variables (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

Emnet omfatter holomorfe funksjoner, Cauchys formel og noen konsekvenser av denne, Weierstrass' og Montels teoremer, analytisk fortsettelse og Reinhardt-områder, subharmoniske funksjoner og Hartogs teorem, samt holomorfitets-områder. Andre tema som kan være aktuelle er Stein-mangfoldigheter og forbindelsen med Banach-algebraer.
 

MA8105 Harmonic analysis (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on MA2104 Differential equations and complex function theory, and SIF5052 Foundation of analysis. Den klassiske Fourieranalysen skjer på enhetssirkelen, de hele tall og den reelle tallinjen. Den rette rammen for Fourieranalyse er klassen av alle lokalkompakte abelske grupper. Fra dette abstrakte utgangspunktet utledes nøkkelbegrepene i harmonisk analyse: Haarmålet, konvolusjon, den duale gruppen og Fouriertransformen, positiv-definite funksjoner, inversjonsteoremet, Plancherels teorem, Pontryagins dualitetsteorem, og Bohr-kompaktifiseringen.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA211 Differential equations and Fourier analysis and MNFMA320 Foundation of analysis.

MA8106 Theory for H^p-spaces (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

Emnet omfatter harmoniske og subharmoniske funksjoner, randegenskaper og Poisson-integralet, maksimalfunksjoner, kanonisk faktorisering, Nevanlinna-klassen, F. & M. Riesz`s teorem, Beurlings teorem samt dualitet. Blant tema som kan inngå kan nevnes: H^p-rom over generelle områder, Feffermans resultat om BMO som det duale til H¹, koronateoremet samt forbindelsen med operator-teori.
 

MA8107 Operatoralgebras (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on SIF5054 Functional analysis or equivalent. The course provides an introduction to den grunnleggende teorien for C*-algebraer og von Neumann algebraer. Teorien vil bli illustrert ved konkrete eksempler: Approksimative endelig-dimensjonale (AF-) algebraer, type I, II og III faktorer, samt den hyperendelige II1-faktoren.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA325 Functional analysis.
 

MA8108 Kvasikonforme avbildninger (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on SIF5029 Complex analysis or equivalent. The course includes de klassiske problemstillinger, ekstremal lengde, geometrisk og analytisk definisjon av kvasikonforme avbildninger, Beltramis differensial likning, kvasidisker, Schwarzisk derivert og det universielle Teichmüller-rom. Blant tema som kan inngå nevnes forbindelsen med Riemannske flater og kvasikonforme avbildninger i høyere dimensjoner.
 

MA8202 Commutative algebra (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on MA3201 Rings and modules and MA3202 Commutative algebra and Galois theory. The content of the course may vary, but the core will include ideals, modules, chain conditions, the spectrum of a ring, Hilbert Nullstellensatz, associated primes and primary decomposition, valuation rings, graded rings, dimension theory, regular sequences, Koszul complex, and regular, Cohen-Macaulay and Gorenstein rings.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA318 Rings and modules and MNFMA319 Commutative algebra and Galois theory.
 

MA8203 Algebraic geometry (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on MA3201 Rings and modules and MA3202 Commutative algebra and Galois theory. The course includes affine and projective varieties, projective plane curves, rational morphisms, resolution of singularities and Riemann-Roch Theorem.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA318 Rings and modules and MNFMA319 Commutative algebra and Galois theory.
 

MA8204 Representation theory for finite groups (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on MA3201 Rings and modules, MA3202 Commutative algebra and Galois theory and MA3203 Ring theory. The course includes character theory, the theory of vertices and sources, and the Brauer correspondence.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA318 Rings and modules and MNFMA319 Commutative algebra and Galois theory.
 

MA8205 Representation theory for algebras (12 credit points)

Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on MA3203 Ring theory and MA3204 Homological algebra. The content of the course may vary, but the core will include algebras given by a quiver and relations, representations of quivers, almost split sequences, Brauer-Thrall I, classification of hereditary

algebras of finite representation type, categories of functors and tilting theory for artin algebras.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA327 Ring theory and MNFMA330 Homological algebra.
 

MA8401 Differential geometry (12 credit points) Duration: 1 semester (the course will be lectured depending upon student registration and resources)

Lectures: 4 lessons a week

Exam format: 6 hours written, or oral.

The course is based on MA1101 Basic calculus I, MA1102 Basic calculus II, MA1201 Linear algebra and geometry, MA1202 Linear algebra with applications, MA1103 Vector calculus and partial MA3402 Analysis on manifolds, or equivalent. The course provides an introduction to Riemannsk geometri, men innholdet kan variere. Sentrale tema er kurve- og flateteori, fundamentalformer, krumning, kovariant derivasjon, geodetiske kurver, differensialformer, Stokes' teorem og Gauss-Bonnets teorem.

Information to students who have passed exams in courses with former internal codes: The course is based on MNFMA100 Basic calculus, MNFMA108 Linear algebra, MNFMA109 Vector analysis and partial MNFMA317 Analysis on manifolds.

MA8402 Lie-groups and Lie-algebras

Duration: 1 semester (spring. The course will be lectured every other year, the first time in 2005.).

Lectures: 4 lessons a week

Exam format: 4 hours written, or oral.

The course gives a basic introduction to classical Lie-groups with emphasis on matrix-groups and special examples as SU(2), SO(3), Lorentz- and Poincaré-groups, their structure, Lie-algebras and representatives. Furhtermore, aøølication of Lie-theory will be given by examples from geometry, differential equations, classical physics or quantum mechanics.