Matematiikan ja systeemianalyysin laitos/ Department of Mathematics and Systems Analysis

Transkriptio

1 Matematiikan ja systeemianalyysin laitos/ Department of Mathematics and Systems Analysis MS-E1000 Crystal Flowers in Halls of Mirrors: Mathematics meets Art and Architecture (6-10 op) Vastuuopettaja: Kirsi Peltonen Kurssin asema: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Kurssin taso: master s level, doctoral level Opetusperiodi: III-IV (Spring 2017), every other year Työmäärä toteutustavoittain: Contact teaching: 6h/week x 12 weeks=72h, independent study: 90h (projects and reports) Osaamistavoitteet: Students will learn to find connections between mathematics and art and architecture. Real mathematics will be revealed through patterns, symmetries, structures, shapes and beauty in such a way that will enable the student to view our environment from a new perspective. By the end of the course, the students will be able to distinguish aspects from their own fields which can be presented, considered and developed using the language of modern mathematics. Sisältö: During the course we will consider methods offered by various fields of mathematics which meet the needs of art and architecture. Through concrete projects, we will find phenomena and interpretations of these phenomena from both classical and modern mathematics. Possible topics from mathematics which may appear: Symmetries, systems of proportions, projections and perspectives. Geometric inversion, conformality and more general mapping classes. Tilings in the plane, polyhedrons and duality. Hopf fibration and other structures. On different types of geometries: spherical, hyperbolic, geometry of surfaces and minimal surfaces. Fractal geometry and dynamics. Kleinian groups, knot theory and contact structures. We will also, depending on the interest of the students, introduce artists and architects from different cultures: M.C. Escher, V. Vasarely, István Orosz, Eero Saarinen, Islamic art, Celtic knots, Sangaku. Toteutus, työmuodot ja arvosteluperusteet: There are no prerequisites from mathematics or art. Students must participate in 80% of the contact teaching. The course consists of project work completed in groups of at most six people at Aalto Design Factory. All steps of the project: planning, implementation, written reports and presentations will have an impact on grades. Individual input is also taken into account by reflections, portfolios, exercises and essays. Optional forms of completing the course can be provided, if required. A complete evaluation and monitoring of learning is performed during the course. Oppimateriaali: To be determined at the beginning of the course. Korvaavuudet: Mat Crystal Flowers in Halls of Mirrors: Mathematics meets Art and Architecture (6-10 cr) Arvosteluasteikko: 1-5 Opetuskieli: English Lisätietoja: At most 36 participants can be accepted. Tentatively 12 art students, 12 architecture students and 12 others. No prerequisities from mathematics or art. The course can be included for example in methodological studies and it is suitable at every stage of studies and for students in every school. MS-E1010 Tieteen filosofia (5 op) Vastuuopettaja: Ilpo Halonen Kurssin taso: maisteriopinnot, jatko-opinnot Opetusperiodi: I-II (syksy 2015), joka toinen vuosi Työmäärä toteutustavoittain: 48 t (luennot), 56 t (luentojen kertaus), 26 t 1

2 (oheislukemisto) Osaamistavoitteet: Opintojakson jälkeen opiskelija pystyy ymmärtämään ja arvioimaan filosofista lähestymistapaa tieteeseen. Sisältö: Tieteen ja tieteellisen ajattelun tunnuspiirteiden systemaattinen tarkastelu sekä eri aikojen käsitykset tieteen tavoitteista, menetelmistä sekä tieteellisen tiedon luonteesta. Tieteen ja tekniikan suhde eri aikoina. Matemaattisten ja fysikaalisten tieteiden perusteet, tavoitteet ja menetelmät filosofisesta näkökulmasta. Myös historiallinen kehitys antiikin ajattelijoista tieteiden vallankumoukseen ja edelleen nykypäivän keskeisiin näkemyksiin asti. Logiikka ja argumentaatioteoria, tieteellinen päättely, sen luonne ja tavoitteet, tieteellinen selittäminen ja kausaliteetin ongelmat. Tieteen etiikka ja tieteellinen maailmankuva. Tiedon kasvuun ja tieteen kehitykseen liittyvät kysymykset. Toteutus, työmuodot ja arvosteluperusteet: Kirjallinen tentti. Oppimateriaali: Luentojen tiivistelmät ja tieto pakollisesta sekä suositeltavasta oheiskirjallisuudesta julkaistaan kurssin MyCourses-sivuilla kurssin kuluessa. Korvaavuudet: Mat , Mat , Mat Arvosteluasteikko: Hyväksytty/hylätty. Opetuskieli: Suomi pääosin. Pyydettäessä suoritettavissa englanniksi. Lisätietoja: MS-E1011 Tieteen historia (5 op) Vastuuopettaja: Ilpo Halonen Kurssin taso: maisteriopinnot, jatko-opinnot Opetusperiodi: I-II (syksy 2016), joka toinen vuosi Työmäärä toteutustavoittain: 48 t (luennot), 56 t (luentojen kertaus), 26 t (oheislukemisto) Osaamistavoitteet: Opintojakson jälkeen opiskelija pystyy ymmärtämään ja arvioimaan, miten tiede on eri aikoina vaikuttanut maailmankuvaan ja miten maailmankuva (mahdollisesti) on vaikuttanut tieteeseen. Sisältö: Valitut kohdat tiedehistoriasta antiikista uudelle ajalle asti. Asioiden ymmärtäminen tieteen metodisen kehityksen ja tieteellisestä metodista esitettyjen teorioiden kannalta. Järjestelmällisen tieteenharjoituksen synty Kreikassa, Aristoteleen rooli filosofian ja tieteiden isänä, hänen teostensa asema keskiajan eurooppalaisissa yliopistoissa. Luonnontieteen nousu 1600-luvulla, ns. tieteen suuri vallankumous. Tämän vallankumouksen yhtenä tärkeänä teemana oli tähtitieteen kehitys. Em. kauden keskeisten matemaattis-fysikaalisten keksintöjen takaa löytyvät hahmot: mm. Kopernikus, Brahe, Kepler, Bacon, Galilei, Descartes, Newton ja Leibniz. Valittuja kohtia uudemmasta tiedehistoriasta (mm. Einstein). Toteutus, työmuodot ja arvosteluperusteet: Kirjallinen tentti. Oppimateriaali: Luentojen tiivistelmät ja tieto pakollisesta sekä suositeltavasta oheiskirjallisuudesta julkaistaan kurssin Noppa-sivuilla kurssin kuluessa. Substitutes for courses: Kurssin voi suorittaa myös pyydettäessä englanniksi kirjallisuustentillä. Korvaavuudet: Mat , Mat , Mat Arvosteluasteikko: Hyväksytty/hylätty. Opetuskieli: Suomi pääosin. Pyydettäessä suoritettavissa englanniksi. Lisätietoja: ilpo.halonen@aalto.fi MS-E1050 Graph theory (5 cr) Responsible teacher: Alexander Engström Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: I (Autumn) 2

3 Workload: Lectures and tutored problem solving 36h (3x2h/week, 6 weeks), self-study about 100h. Learning Outcomes: The students will after the course understand the basic invariants of graphs and how they are related by regularity and structural graph theory. Content: Basic properties as connectivity, planarity and minor containment both in the deterministic and random setting. The Szemerédi regularity lemma, graph homomorphisms and graph limits; the graph minor theorem and the strong perfect graph theorem. Study Material: Graph Theory, Diestel, 4th edition; Large Networks and Graph Limits, Lovász. Substitutes for Courses: Mat Prerequisites: Mathematical maturity comparable to a bachelor in computer science, mathematics or operational research. Evaluation: 1-5 MS-E1051 Combinatorics (5 cr) Responsible teacher: Alexander Engström Status of the Course: Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: II (Autumn) Workload: Lectures and tutored problem solving 36h (3x2h/week, 6 weeks), self-study about 100h. Learning Outcomes: The students will learn how to analyse combinatorial problems using algebraic and analytic methods. Content: Enumeration and generating functions; posets and their algebraic properties. Study Material: Analytic Combinatorics, Flajolet and Sedgewick; Supplied lecture notes on algebraic combinatorics. Substitutes for Courses: MS-C1050 Prerequisites: Mathematical maturity comparable to a bachelor in computer science, mathematics or operational research. Preferably some basic algebra and complex analysis. Evaluation: 1-5. MS-E1059 Seminar on combinatorics (V) (V) (1-5 cr) Responsible teacher: Alexander Engström Status of the Course: Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional Teaching period: I-V (academic year) Learning Outcomes: An overview of contemporary research trends in algebraic and topological combinatorics and an understanding of the basic elements of a good math talk. Content: Seminar talks and discussions. Assessment Methods and Criteria: Active participation in seminars. Course Homepage: Evaluation: pass/fail Registration for Courses: Contact the teacher in charge. MS-E1089 Seminar on Algebra, Number Theory, and Applications to Communications and Computing V (V) (2 cr) 3

4 Responsible teacher: Camilla Hollanti Teaching period: I-V ( ) Learning Outcomes: To get familiar with the research topics of algebra, number theory and their applications. Content: Research topics of algebra, number theory and their applications Assessment Methods and Criteria: Attendance 6 times, including one presentation Evaluation: Pass/fail Further Information: MS-E1110 Number theory (5 cr) Responsible teacher: Camilla Hollanti Status of the Course: Major in Applied Mathematics & Major in Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor in Mathematics, optional Teaching period: II (Autumn). Workload: (4+2). Learning Outcomes: - The student understands the basic concepts of number theory and is able to solve simple Diophantine equations and perform modular arithmetics. - The student is familiar with some applications of number theory in cryptography. Content: integer factorization, primes, pseudo primes, Diophantine equations, modular arithmetics, squares and nonsquares in modular arithmetics, primititive roots, applications to cryptography. Assessment Methods and Criteria: Lectures, exercises, exam, essay. Study Material: K. H. Rosen: Elementary number theory and its applications Substitutes for Courses: Mat , MS-C1110. Prerequisites: High school mathematics. MS-A040X OR MS-C1080 is recommended. Evaluation: 1-5. MS-E1111 Galois theory (5 cr) Responsible teacher: Camilla Hollanti Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: IV (Spring 2016), every other year Workload: 6 hours x 6 weeks Content: To understand at an operative level the concepts of Galois extension and Galois correspondence. Ability to solve equations with algebraic methods. Assessment Methods and Criteria: Lectures, written exercises, possibility for an oral exam if needed. Study Material: Ian Stewart: Galois Theory, 3rd edition. Substitutes for Courses: Mat Prerequisites: MS-C1080 Algebran perusrakenteet or similar. Evaluation: 1-5. MS-E1280 Measure and integral (5 cr) Responsible teacher: Juha Kinnunen Status of the Course: Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional 4

5 Teaching period: II (Autumn) Workload: lectures 24h (2x2h/week, 6 weeks), exercises 12h (1x2h/week, 6 weeks), self-study ca 100h Learning Outcomes: After this course you will know basic methods in measure and integration theory. Content: Outer measure (properties of measurable sets, characterizations of measurable sets, Lebesgue outer measure), measurable functions (properties of measurable functions, approximation by simple functions, Egoroff and Lusin theorems), integration (construction and properties of integral, Lebesgue integral, convergence theorems), Fubini s theorem. Assessment Methods and Criteria: Homework assignments and attendance (50%), final exam (50%). Study Material: All material is available at the course homepage. Substitutes for Courses: MS-C1280. Prerequisites: MS-A000X, MS-A010X, MS-A020X, MS-A030X, MS-A050X, MS-C1540. Evaluation: 1-5 MS-E1281 Real analysis (5 cr) Responsible teacher: Juha Kinnunen Status of the Course: Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: IV (Spring 2016), every other year Workload: 24h (2x2h/week, 6 weeks), exercises 12h (1x2h/week, 6 weeks), self-study ca 100h Learning Outcomes: After this course you will know how to apply real analysis methods in research. Content: Lebesgue spaces (Hölder s and Minkowski s inequalities, Riesz-Fischer theorem, dual spaces and weak convergence), Hardy-Littlewood maximal function (Vitali covering theorem, Marcinkiewicz interpolation theorem, maximal function theorem, Lebesgue s differentiation theorem), convolution approximations, differentiation of Radon measures (Besicovitch covering theorem, Lebesgue points), Radon-Nikodym theorem, Riesz representation theorem, weak convergence and compactness for Radon measures, Sobolev spaces (Poincare and Sobolev inequalities). Assessment Methods and Criteria: Homework assignments and attendance (100%). Study Material: All material is available at the course homepage. Substitutes for Courses: Mat Prerequisites: MS-A000X, MS-A010X, MS-A020X, MS-A030X, MS-A050X, MS-C1280, MS-C1350, MS-C1540. Evaluation: 1-5 MS-E1289 Seminar on analysis and geometry (V) (V) (1-5 cr) Responsible teacher: Juha Kinnunen; Kirsi Peltonen Status of the Course: Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: I-V (academic year) Learning Outcomes: This is a research seminar. Content: We study the most recent methods and results in modern analysis and geometry. The talks are related, for example, to differential geometry, geometric analysis, harmonic analysis and partial differential equations. Substitutes for Courses: Mat

6 Evaluation: hyv Opintojaksot MS-E1460 Functional analysis (5 cr) Responsible teacher: Ville Turunen Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor in Mathematics, optional Teaching period: I (Autumn) Workload: Lectures 24h (2x2h/week, 6 weeks), exercises 12h (1x2h/week, 6 weeks), self-study ca 100h. Learning Outcomes: You will learn about norms and inner products in infinite-dimensional vector spaces. Related to these structures, you will understand basic properties of bounded linear operators and duality in Banach and Hilbert spaces, together with diagonalization of compact self-adjoint operators. Content: Bounded linear operators and functionals in Banach and Hilbert spaces, elementary spectral theory (Riesz Compactness Theorem, Uniform Boundedness Principle, Open Mapping and Closed Graph Theorems, Hahn-Banach Theorem, Riesz Hilbert Space Representation and Hilbert-Schmidt Spectral Theorems). Assessment Methods and Criteria: Weekly exercises (1/3) and an exam (2/3). Alternatively, just exam (100%). Study Material: Lecture notes (additional literature to be announced at the course homepage). Substitutes for Courses: Mat Principles of Functional Analysis. Prerequisites: MS-A000X, MS-A010X, MS-C1540. Evaluation: MS-E1531 Differential geometry (5 cr) Responsible teacher: Kirsi Peltonen Status of the Course: Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: III (Spring 2016), every other year Workload: (4 + 2) Learning Outcomes: This course is an introduction to the basic machinery behind the modern differential geometry: tensors, differential forms, smooth manifolds and vector bundles. The geometries lying above these structures are involved in several applications through mathematical analysis, physics, stochastics and statistical modells. The central goal is to become familiar with this particular language of abstract mathematics that opens the venue to apply geometric methods in different applications. A modern viewpoint to some of the classical Riemann, Finsler or Kähler model geometries is served in addition to the possibility to open the door to the beautiful worlds of contact and symplectic geometry that are present in the most recent progress of geometrization of applications. The course provides basic skills to recognize geometric phenomena in mathematical analysis and applications. Content: Topics related to differential geometry varying from classical Riemannian geometry to modern geometries. More specified topics will be announced later. Assessment Methods and Criteria: Active participation in lectures and weekly excercises. Individual research projects that are related to the topics of the course. Always discuss beforehand with the lecturer before starting such a project. A traditional exam is also possible. Study Material: All material related to the course can be found from MyCourses pages of the course. There is no special book the course is following but excellent treatments in 6

7 the spirit of the lectures are provided by: - John M. Lee: Introduction to Smooth Manifolds, Springer - John M. Lee: Riemannian Manifolds: An Introduction to Curvature, Springer. Substitutes for Courses: Mat Prerequisites: MS-A210, MS-A310, MS-C1530, MS-C1540 Evaluation: 1-5 Further Information: The content of the course is part of a good mathematical education, which should self-evidently belong to the curriculum of every math major student. A highly open mind is necessary to gain the capability to apply methods provided by differential geometry to other sciences. Suitable to everybody interested in geometrization, especially those with a focus on fields in natural sciences where the connection is most visible like in general relativity and electromagnetism. Other potential fields are all sciences that make use of statistical or stochastic methods. MS-E1600 Probability theory (5 cr) Responsible teacher: Kalle Kytölä; Lasse Leskelä Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: III (Spring) Workload: 2 x 2h lectures,1 x 2h exercises sessions Learning Outcomes: After completing the course, the participant - Can compute the expected value of a random number as an integral with respect to a probability measure - Can compute probabilities related to independent random variables by using a product measure - Recognizes different types of convergence of a random sequence - Can explain how and when a random sum can be approximated by a Gaussian distribution - Can represent conditional probabilities with respect to the information content of a sigma-algebra Content: - Random numbers, vectors, and sequences - Integration with respect to a probability measure - Stochastic independence and product measure - Law of large numbers and the central limit theorem - Conditional expectation with respect to a sigma-algebra Study Material: TBA Substitutes for Courses: Mat Prerequisites: Familiarity with continuous functions and open sets (e.g. MS-C1540 Euklidiset avaruudet) Evaluation: 1-5 MS-E1601 Brownian motion and stochastic analysis (5 cr) Responsible teacher: Lasse Leskelä Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: II (Autumn 2015), every other year Workload: - 2 x 2h lectures - 1 x 2h exercises sessions 7

8 Learning Outcomes: After completing the course the participant: - Can compute probabilities and expectations related to Brownian motion - Can define the stochastic integral - Recognizes random processes which are integrable with respect to Brownian motion - Can apply Itō s formula to various functionals of Brownian motion Content: - Brownian motion - Stochastic integral - Itō s formula and applications Study Material: TBA Substitutes for Courses: Mat Prerequisites: MS-E1600 Evaluation: hyv Opintojaksot MS-E1602 Large random systems (5 cr) Responsible teacher: Kalle Kytölä Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: IV (Spring 2016), every other year Workload: - 2 x 2h lectures - 1 x 2h exercises sessions Learning Outcomes: After completing the course the participant is able to: - Formulate mathematical models of various systems with a large number of interacting random components - Incorporate spacial structure and dynamics into probabilistic models - Estimate the asymptotics of probabilities and expected values in models with a size parameter - Formulate qualitative phase transitions in stochastic models and recognize them - Verify if a sequence of probability distributions on a metric space converges Content: Stochastic models with spatial and temporal structure laws - Large deviation estimates of rare events - Phase transitions in stochastic models - Convergence and tightness of probability measures Study Material: TBA Substitutes for Courses: Prerequisites: MS-E1600 Evaluation: hyv Opintojaksot MS-E1609 Seminar on stochastics and statistics (V) (V) (1-5 cr) Responsible teacher: Sirkku Ilmonen; Kalle Kytölä; Lasse Leskelä Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional Teaching period: I-V (academic year) Evaluation: hyv Opintojaksot 8

9 MS-E1651 Numerical matrix computations (5 cr) Responsible teacher: Antti Hannukainen Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: I (Autumn) Workload: 24h (2x2h/week, 6 weeks), exercises 12h (1x2h/week, 6 weeks), self-study ca 100h Learning Outcomes: Students learn to analyze and solve problems in linear algebra that occur often in scientific computing, data fitting and optimization. The main focus is on solution of linear systems, least squares problems and eigenvalue problems. After the course, the students can choose the best solution method for each problem and have a good understanding on issues related to numerical stability of the applied algorithms. Content: Matrix decompositions and their numerical computation, eigenvalue iterations, sparse matrices, iterative solution of linear systems. Assessment Methods and Criteria: weekly exercises (33.3%), an exam (66.6%) Study Material: All essential material is included in the lecture notes that are available at the course s homepage. Substitutes for Courses: Mat Prerequisites: MS-A000X, MS-A010X, MS-A020X, MS-A030X, MS-A050X. The courses MS- C1540 may also be useful. Evaluation: 1-5 MS-E1652 Computational methods for differential equations (5 cr) Responsible teacher: Nuutti Hyvönen Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: II (Autumn) Workload: lectures 24h (2x2h/week, 6 weeks), exercises 12h (1x2h/week, 6 weeks), self-study ca 100h Learning Outcomes: You will familiarize yourself with the basic properties of initial value problems for systems of ordinary differential equations. You will learn the fundamental theory about linear multistep methods (definition, consistency, zero-stability, convergence) and Runge-Kutta methods (definition, order conditions, convergence). You will learn to identify a stiff system and to understand the difference between explicit and implicit numerical schemes. You will understand the signifigance of absolute stability and A-stability, and know how to examine the region of absolute stability for a given numerical method. You will get to know the basic principles of the discrete Fourier transform. You will familiarize yourself with simple parabolic and hyperpolic initial/boundary value problems and learn how to discretize them with the help of difference schemes. You will practice implementing the introduced methods numerically. Content: Basic existence and uniqueness results for systems of ordinary differential equations. Linear multistep methods and Runge-Kutta methods: stability, convergence and numerical implementation. Discrete Fourier transform. Discretization of simple initial/boundary value problems for parabolic and hyperbolic partial differential equations. Assessment Methods and Criteria: Weekly exercises (33.3%) and an exam (66.7%). Study Material: All essential material is included in the lecture notes that are available at the course s homepage. Substitutes for Courses: Mat

10 Prerequisites: S-A000X, MS-A010X, MS-A020X, MS-A030X, MS-A050X. The courses MS-C1340, MS-C1350, MS-C1650, MS-E1651 may also be useful. Evaluation: 1-5 MS-E1653 Finite element method (5 cr) Responsible teacher: Antti Hannukainen Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: III-IV (Spring) Workload: 48h (2x2h/week, 12 weeks), self-study ca 100h, project ca 20h Learning Outcomes: Students learn to derive and analyze weak form of an elliptic partial differential equation and to implement finite element solver in 2D. They will develop understanding on the principles of the error analysis of the finite element method and different factors affecting the accuracy of the solution. Content: The topic of the course is solution of elliptic partial differential equation using finite element method. Both algorithmic and theoretical aspects of the method are covered. Assessment Methods and Criteria: weekly exercises (50%), an exam (50%), project (pass/fail) Study Material: All essential material is included in the lecture notes that are available at the course s homepage. Substitutes for Courses: MS-C1741 Prerequisites: MS-A000X, MS-A010X, MS-A020X, MS-A030X, MS-A050X. The courses MS-C1350, MS- C1540 may also be useful. Evaluation: 1-5 MS-E1654 Computational inverse problems (5 cr) Responsible teacher: Nuutti Hyvönen Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; Master s level Minor of Mathematics, optional Teaching period: IV (Spring) Workload: lectures 24h (2x2h/week, 6 weeks), exercises 12h (1x2h/week, 6 weeks), self-study ca 100h Learning Outcomes: You will learn to identify an ill-posed inverse problem and to understand the restrictions its nature imposes on the solution process. You will familiarize yourself with several classical regularization methods for finding approximate solutions to linear ill-posed problems. You will learn to formulate an inverse problem as a Bayesian problem of statistical inference and to interpret the information contained in the resulting posterior probability distribution. You will learn to numerically implement the introduced solution techniques. Content: The course s topic is computational methods for solving inverse problems arising from practical applications. The course consists of two parts: the first three weeks focus on classic regularization techniques, the latter three weeks discuss statistical methods. Assessment Methods and Criteria: weekly exercises (25%), a home exam (75%). Study Material: All essential material is included in the lecture notes that are available at the course s homepage. Substitutes for Courses: Mat Prerequisites: MS-A000X, MS-A010X, MS-A020X, MS-A030X, MS-A050X. The courses 10

11 MS-C1340, MS-C1650, MS-E1460, MS-E1651, MS-E1652, MS-E2112 may also be useful. Evaluation: 1-5 MS-E1659 Seminar on applied mathematics (V) (V) (1-5 cr) Responsible teacher: Nuutti Hyvönen; Antti Hannukainen Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional Teaching period: I-V (academic year) Learning Outcomes: An overview of research on applied mathematics and mechanics at Aalto University and collaborating units. Content: Seminar talks and discussion on current research topics in applied mathematics and mechanics. The seminar usually convenes once a week during the academic year. Assessment Methods and Criteria: A seminar talk and active participation. Substitutes for Courses: Mat , Mat Prerequisites: MS-A000X, MS-A010X, MS-A020X, MS-A030X, MS-A050X. The courses MS-C1340, MS-C1350, MS-C1650, MS-E1460, MS-E1651, MS-E1652, MS-E1653, MS-E1654, MS-1740, MS-1741, MS-1742, MS-1743 may also be useful. Evaluation: pass/fail Registration for Courses: Contact the teachers in charge. MS-E1740 Continuum mechanics 1 (5 cr) Responsible teacher: Rolf Stenberg Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: I (Autumn) Workload: 24 h (2x2h/week, 6weeks), exercises 12 h (1x2h/week, 6 weeks) Learning Outcomes: You know the mathematical tools for modeling a continuum, i.e. a solid or a fluid. Content: Tensor calculus. The concepts of continuum mass and force. Kinematics. Balance laws. Assessment Methods and Criteria: Weekly exercises (1/3) and an exam (2/3). Study Material: O. Gonzalez, A. Stuart. A first course in continuum mechanics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, Substitutes for Courses: Mat Prerequisites: MS-A000X, MS-A010X,MS-A020X,MS-A030X Evaluation: 1-5. MS-E1741 Continuum mechanics 2 (5 cr) Responsible teacher: Rolf Stenberg Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: II (Autumn) Workload: Lecture 24 h (2x2h/week, 6weeks), exercises 12 h (1x2h/week, 6 weeks) Learning Outcomes: You are able to derive and analyze the main mathematical models 11

12 for fluids and solids. Content: Constitutive laws. Inviscid fluids, Navier-Stokes equations. Linear and nonlinear elasticity. Assessment Methods and Criteria: Weekly exercises (1/3) and an exam (2/3). Study Material: O. Gonzalez, A. Stuart. A first course in continuum mechanics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, Substitutes for Courses: Mat Prerequisites: MS-E1740 Continuum mechanics 1. Evaluation: 1-5. MS-E1742 Computational mechanics 1 (5 cr) Responsible teacher: Rolf Stenberg Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: IV (Spring) Workload: 24 h (2x2h/week, 6weeks), exercises 12 h (1x2h/week, 6 weeks) Learning Outcomes: You will learn how the finite element method is applied for problems which are constrained minimization problems with a Lagrange multiplier. Content: General variational problems. The finite element theory for approximating saddle-point problems. Applications to Stokes equations. Assessment Methods and Criteria: Weekly exercises (1/3) and an exam (2/3). Study Material: Larson, Mats G.; Bengzon, Fredrik. The finite element method: theory, implementation, and applications. Texts in Computational Science and Engineering, 10. Springer, Heidelberg, Substitutes for Courses: Mat Prerequisites: MS-E1653 Finite element method (5 cr). MS-E1743 Computational mechanics 2 (5 cr) Responsible teacher: Rolf Stenberg Status of the Course: Major of Applied Mathematics, Master s Programme in Mathematics and Operations Research, optional; master s level Minor of Mathematics, optional Teaching period: V (Spring) Workload: 24 h (2x2h/week, 6weeks), exercises 12 h (1x2h/week, 6 weeks) Learning Outcomes: You will be able to apply the theory from MS-E1742 Computational mechanics 1 to a variety of problems in continuum mechanics. Content: Stabilized finite element methods; applications to the convection diffusion and Stokes equations. Finite element methods in solid mechanics; the Timoshenko beam and the Reissner-Mindlin plate model Assessment Methods and Criteria: Weekly exercises (1/3) and an exam (2/3). Study Material: Lecture notes. Substitutes for Courses: Mat Prerequisites: MS-E1742 Evaluation: 1-5. MS-E1980 Special assignment in mathematics (V) (V) (5-10 cr) 12

13 Responsible teacher: Nuutti Hyvönen Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional Level of the Course: master s level Teaching period: I-V (academic year) Workload: self-study ca 135h Learning Outcomes: Understands the mathematics related to the assignment. Is able to write a scientific report on the project. Content: An individual research assignment or a literature survey. Substitutes for Courses: Mat , Mat Evaluation: 1-5 /Finnish/Swedish (to be agreed with the teacher) Further Information: Before starting the special assignment, the topic must be agreed with a member of the faculty at the Department of Mathematics and Systems Analysis. MS-E1981 Individual studies in mathematics (V) (V) (1-10 cr) Responsible teacher: Nuutti Hyvönen Status of the Course: Major of Applied Mathematics & Major of Mathematics, Master s Programme in Mathematics and Operations Research, optional Teaching period: I-V (academic year) Content: Guest lectures, web-based teaching or other individual studies. The content and scope must be settled with the teacher in charge. Substitutes for Courses: Mat , Mat , Mat Evaluation: 1-5 or pass/fail /Finnish/Swedish (to be agreed with the teacher) MS-E1990 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1991 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1992 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1993 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1994 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1995 Course with Varying Content V (V) (1-10 cr) 13

14 Responsible teacher: Pekka Alestalo MS-E1996 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1997 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1998 Course with Varying Content V (V) (1-10 cr) Responsible teacher: Pekka Alestalo MS-E1999 Matemaattiset ohjelmistot V (V) (1-5 op) Vastuuopettaja: Heikki Apiola Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: suomi Lisätietoja: Vaihtuvasisältöinen. MS-E2108 Systeemianalyysin erikoistyöt (V) (V) (5-8 op) Vastuuopettaja: Enrico Bartolini; Harri Ehtamo; Raimo Hämäläinen; Ahti Salo; Kai Virtanen Kurssin taso: Maisteritaso Opetusperiodi: I, II, III, IV, V Työmäärä toteutustavoittain: Itsenäinen työskentely 130h Osaamistavoitteet: Kirjallisen ja tieteellisen raportointitaidon kehittäminen. Sisältö: Yksilöllinen itsenäinen tutkimustehtävä; aihe teollisuudesta, laboratoriosta tai muualta korkeakoulusta. Tarkemmat ohjeet kurssin www-sivuilta. Toteutus, työmuodot ja arvosteluperusteet: Työselostus Korvaavuudet: Mat Sovelletun matematiikan erikoistyöt Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: sopimuksen mukaan MS-E2112 Multivariate statistical analysis (5 cr) Responsible teacher: Sirkku Ilmonen Status of the Course: Major of Applied Mathematics & Major of Systems and Operations Research, Master s Programme in Mathematics and Operations Research, optional; Minor of Systems and Operations Research, optional Teaching period: III-IV (Spring) Workload: Lectures 24h (2), Exercises 24h (2), Project work 40h, Other autonomous studies 40h. Learning Outcomes: This course is an introduction to multivariate statistical analysis. The goal is to learn basics of common multivariate data analyzing techniques and to use the methods in practice. Software R is used in the exercises of this course. Content: Multivariate Location and Scatter, Principal Component Analysis (PCA), Bivariate Correspondence Analysis, Multivariate Correspondence Analysis (MCA), 14

15 Canonical Correlation Analysis, Discriminant Analysis, Classification, Clustering. Assessment Methods and Criteria: Exam and compulsory project work. Study Material: K. V. Mardia, J. T. Kent, J. M. Bibby: Multivariate Analysis, Academic Press, London, 2003 (reprint of 1979) and lecture slides. Substitutes for Courses: Mat Statistical Multivariate Methods P Prerequisites: At least one statistics/probability course and one matrix algebra course. Evaluation: 1-5 MS-E2113 Jonoteoria (3-6 op) Vastuuopettaja: Harri Ehtamo Kurssin taso: Maisteritaso Opetusperiodi: Ei luennoida tänä lukuvuonna Osaamistavoitteet: Johdatteleva kurssi jonoteoriaan. Sisältö: Jonoilmiöiden tarkastelu stokastisena prosessina, ääretön tai äärellinen käyttäjäjoukko, yksi tai useampi palveluyksikkö, jonokurit, prioriteetit, sisäkkäiset jonot, jonojen käsittely Markov-prosesseina. Sovellutuksia palvelujärjestelmistä ja tietoliikennetekniikan piiristä. Korvaavuudet: Korvaa kurssin Mat Jonoteoria L Korvaava kurssi ELEC-E7450 Performance analysis Esitiedot: 1. ja 2. vuoden matematiikka, MS-C2111 Stokastiset prosessit Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: suomi MS-E2114 Investment science (5 cr) Responsible teacher: Eeva Vilkkumaa Status of the Course: Optional course of the Systems and Operations Research major. Optional course of the Systems and Operations Research minor. Teaching period: IV (Spring) Workload: Lectures 24h (4) Exercises 24h (4) Assignment 25h Autonomous studies 55h Content: Instruments of investment science and finance, risk analysis, term structure of interest rates, pricing of derivatives, optimization of investment portfolio return. Assessment Methods and Criteria: Exam and assignments Study Material: D.G. Luenberger: Investment Science, Oxford University Press, Substitutes for Courses: Mat Investment science Prerequisites: 1st and 2nd years math, applied probability MS-E2117 Riskianalyysi (5 op) Vastuuopettaja: Ahti Salo Kurssin asema: Systems and Operations Research -pääaineen valinnainen kurssi. Systems and Operations Research ja Multi-Disciplinary Energy Studies -sivuaineiden valinnainen kurssi. Kurssin taso: Maisteritaso Opetusperiodi: III - IV Työmäärä toteutustavoittain: Luento-opetus 24h (2) Laskuharjoitukset 24h (2) Harjoitustyöt 20h Itsenäinen työskentely 60h 15

16 Sisältö: Kurssi perehdyttää riskinalyysin keskeisimpiin menetelmiin ja antaa valmiudet soveltaa niitä erilaisten teknis-taloudellisten järjestelmien riskitarkasteluissa (ml. riskien tunnistaminen ja arviointi, riskienhallintatoimenpiteiden vertailu ja priorisointi, riskiviestintä). Käsiteltäviä menetelmiä ovat muun muassa vikapuut, syy-seuraus-kaaviot sekä todennäköisyyspohjainen turvallisuusanalyysi. Kurssilla perehdytään myös todennäköisyyksien estimointiin sekä asiantuntija-arvioiden että tilastollisten menetelmien pohjalta. Luennot ja laskuharjoitukset sisältävät riskianalyysien laatimista ja käyttöä havainnollistavia esimerkkejä. Toteutus, työmuodot ja arvosteluperusteet: Tentti ja harjoitustyöt Oppimateriaali: M. Modarres: Risk Analysis in Engineering: Techniques, Tools and Trends; Bilal M. Ayyub: Risk Analysis in Engineering and Economics Korvaavuudet: Mat Riskianalyysi Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: suomi MS-E2129 Systeemien identifiointi (5 op) Vastuuopettaja: Kai Virtanen Kurssin asema: Systems and Operations Research -pääaineen valinnainen kurssi. Systems and Operations Research -sivuaineen valinnainen kurssi. Kurssin taso: Maisteritaso Opetusperiodi: I - II Työmäärä toteutustavoittain: Luento-opetus 24h (2) Laskuharjoitukset 24h (2) Harjoitustyöt 20h Itsenäinen työskentely 60h Osaamistavoitteet: Kurssi antaa perusvalmiudet dynaamisten systeemien matemaattiseen mallintamiseen ja identifiointiin. Sisältö: Dynaamisten järjestelmien siirtofunktio- ja tilaesitysmallit ja mallintaminen; systeemiteoriaa. Dynaamisten järjestelmien identifiointi: epäparametriset menetelmät, herätteet ja koesuunnittelu, mallirakenteet, ennustevirhemenetelmät, parametrien estimointi, mallin rakenteen valinta, mallin validointi. Toteutus, työmuodot ja arvosteluperusteet: Tentti, harjoitustyöt ja laskuharjoitukset. Kaksi tenttitehtävää voi korvata harjoitustöillä ja kaksi tenttitehtävää voi korvata aktiivisella osallistumisella laskuharjoituksiin. Oppimateriaali: L. Ljung, T. Glad: Modeling of Dynamic Systems, Prentice Hall, Saatavissa myös ruotsinkielisenä, kustantaja Studentlitteratur Korvaavuudet: Mat Systeemien identifiointi Esitiedot: MS-C2128 Ennustaminen ja aikasarja-analyysi Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: suomi MS-E2130 Matemaattinen malliajattelu (3-6 op) Vastuuopettaja: Kai Virtanen Kurssin asema: Systems and Operations Research pääaineen ja sivuaineen valinnainen kurssi. Kurssin taso: Maisteritaso Opetusperiodi: I - II Työmäärä toteutustavoittain: Itsenäinen työskentely 100h (viikottaiset verkkoluennot, harjoitukset ja kommentoinnit) Harjoitustyö 25h Sisältö: Johdatus systeemiajatteluun ja matemaattisten mallien käyttöön eri alojen sovellutuksissa. Mallin muodostaminen, differentiaali- ja differenssiyhtälömallit, stokastiset mallit, optimointiin perustuva mallintaminen, dynaamisten järjestelmien simulointi. Toteutus, työmuodot ja arvosteluperusteet: Viikottaiset verkkoluennot, 16

17 harjoitustehtävät ja kommentoinnit sekä vapaaehtoinen harjoitustyö. Oppimateriaali: S. Pohjolainen (toim.): Matemaattinen mallinnus, WSOYpro, Lisälukemistona: F.R. Giordano, M.D. Weir, W.P. Fox: A First Course in Mathematical Modeling, Brooks/Cole, 1997 Korvaavuudet: Mat Matemaattinen malliajattelu L Esitiedot: 1. ja 2. vuoden matematiikka Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: suomi Lisätietoja: Laajuus 4 op ja vapaaehtoisen harjoitustyön kanssa 5 op. Kurssi on verkkopohjainen. Syksyllä järjestetään mallinnuksen peruskurssi ja sekä syksyllä että keväällä vaihtuva-aiheisia jatkokursseja. Jatkokurssien suoritukset (3 op) kurssikoodilla MS-E2195. MS-E2133 Systems analysis laboratory II (5 cr) Responsible teacher: Kai Virtanen Status of the Course: Compulsory course of the Systems and Operations Research major. Optional course of the Systems and Operations Research minor. Teaching period: I - II (Autumn) Workload: Lectures 4h Exercises 48h (4) Autonomous studies 75h Learning Outcomes: Get familiar with problem solving in operations research. Content: Two fairly large assignments on the implementation and analysis of mathematical models. The assignments deal with optimization and stabilization and control of a large scale system. Assessment Methods and Criteria: Assignments and written reports Substitutes for Courses: Mat Systems analysis laboratory II Prerequisites: MS-E2139 Nonlinear Programming, MS-E2148 Dynamic Optimization. Taking the course simultaneously with MS-E2129 System Identification is recommended MS-E2134 Decision making and problem solving (5 cr) Responsible teacher: Eeva Vilkkumaa Status of the Course: Compulsory course of the Systems and Operations Research major. Teaching period: I (Autumn) Workload: Lectures 24h (4) Exercises 24h (4) Assignment 40h Autonomous studies 40h Learning Outcomes: After completing this course the student 1. knows the central concepts in decision analysis, 2. can structure and model problems with multiple attributes, decision dynamics and uncertainty for aiding decision-making, 3. understands the assumptions underlying decision analytic models and why behavior of real decision makers may differ from the behavior that these models would predict, and 4. knows how to use optimization methods in conjunction with decision analysis. Content: Models for decision making: subjective values, multi-objective decision making and optimization, group decision making, decision under uncertainty, dynamics of decision making. Assessment Methods and Criteria: Exam and assignments Study Material: Lecture slides and exercises are the primary course material. Additional 17

18 literature includes F. Eisenführ, M. Weber, T. Langer: Rational Decision-Making, Springer, 2010, Clemen, R.T. (1996): Making Hard Decisions: An Introduction to Decision Analysis, 2nd edition and French, S. (1988): Decision Theory: An Introduction to the Mathematics of Rationality. Substitutes for Courses: Mat Decision Making and Problem Solving P Prerequisites: MS-C2105 Introduction to Optimization, applied probability MS-E2136 Special topics in decision making (V) (V) (3-6 cr) Responsible teacher: Raimo Hämäläinen; Ahti Salo Status of the Course: Optional course of the Systems and Operations Research major. Optional course of the minors Systems and Operations Research and Multi-Disciplinary Energy Studies. Teaching period: will be announced later Content: Annually varying topics on decision making. Substitutes for Courses: Mat Special Topics in Decision Making Evaluation: 0-5 or pass/fail MS-E2139 Nonlinear programming (5 cr) Responsible teacher: Kimmo Berg Status of the Course: Alternative course of the Systems and Operations Research major. Alternative course of the Systems and Operations Research minor (MSc). Optional course of the Mathematics minor (MSc). Teaching period: II (Autumn) Workload: Lectures 24h (4) Exercises 24h (4) Voluntary home exercises 15h Autonomous studies 70h Learning Outcomes: Present different convexity properties and concepts of convex analysis. Interpret and explain the optimality conditions and use them to calculate optimal solutions. Analyze different optimization algorithms and use them to solve optimization problems. Content: The first part of the course teaches the optimization theory: convexity, necessary and sufficient optimality condition and their derivation, the interpretation of Lagrange multipliers, and duality. The second part teaches numerical optimization: unconstrained, convex, and constrained optimization. Applications from natural sciences, engineering and economics. Assessment Methods and Criteria: Exam. Extra points can be earned by doing homework. Study Material: M.S. Bazaraa, H.D. Sherali, C.M. Shetty: Nonlinear Programming, Theory and Algorithms, Wiley and Sons 1993/ nd (blue or red) or 3rd (green) edition is ok. Substitutes for Courses: Mat Nonlinear Programming P Prerequisites: 1st and 2nd year math MS-E2140 Linear programming (5 cr) Responsible teacher: Enrico Bartolini 18

19 Status of the Course: Alternative course of the Systems and Operations Research major. Optional course of the Systems and Operations Research minor. Teaching period: I (Autumn) Workload: Lectures 24h (4) Exercises 24h (4) Assignments 20h Autonomous studies 60h Learning Outcomes: After completing this course the student 1. can formulate a wide variety of optimization problems, which solutions can be used for making better decisions (e.g. allocating resources, selecting routes and assigning tasks), as (mixed integer) linear programming problems, 2. understands the theoretical foundation of the Simplex algorithm and duality, and knows the special characteristics of network and integer programming problems, and 3. can solve (mixed integer) linear programming problems using optimization software. Content: The simplex method, dual of the linear program, interior point algorithms, integer programming. Applications to transportation problems, network problems and production planning. Assessment Methods and Criteria: Exam and assignments. Bonus points from home work and exercise sessions Study Material: D. Bertsimas, J.N. Tsitsiklis: Introduction to Linear Optimization, Athena Scientific 1997 Substitutes for Courses: Mat Linear Programming P Prerequisites: MS-C2105 Introduction to Optimization MS-E2142 Optimointiopin seminaari (V) (V) (5 op) Vastuuopettaja: Enrico Bartolini; Raimo Hämäläinen; Ahti Salo; Kai Virtanen Kurssin asema: Systems and Operations Research pää- ja sivuaineen valinnainen kurssi. Kurssin taso: Maisteritaso. Opetusperiodi: mahdollisesta luennoinnista ilmoitetaan myöhemmin Työmäärä toteutustavoittain: Seminaari 36h (3) Itsenäinen työskentely 90h Osaamistavoitteet: Syventää systeemi- ja operaatiotutkimuksen opintoja ja kehittää valmiuksia hyvien seminaariesitysten pitämiseen. Sisältö: Vaihtuva-alainen seminaari, joka voidaan suorittaa toistuvasti. Aihe ja järjestäminen ilmoitetaan myöhemmin. Toteutus, työmuodot ja arvosteluperusteet: toteutus: seminaari työmuodot ja arvostelu: läsnäolo, esitelmät ja kotitehtävät Korvaavuudet: Mat Optimointiopin seminaari L Esitiedot: MS-C2105 Optimoinnin perusteet Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: suomi MS-E2143 Network optimization (5 cr) Responsible teacher: Enrico Bartolini Teaching period: Not lectured this academic year. Exam according to agreement Workload: Autonomous studies Learning Outcomes: An advanced course in optimization. Content: Graphs and network flow formulations. Maximal flow, transportation, assignment, and shortest path problems. Linear programming simplex algorithm, the 19

20 dual-, and primal-dual algorithm for network problem. Assessment Methods and Criteria: Homework exercises from the course book. Study Material: D. Bertsekas: Network Optimization, Athena Scientific, Substitutes for Courses: Mat Network Optimization P Prerequisites: MS-C2105 Introduction to Optimization or MS-E2140 Linear Programming Evaluation: pass/fail MS-E2144 Optimoinnin matemaattinen teoria (V) (V) (3-6 op) Vastuuopettaja: Kimmo Berg Kurssin taso: Maisteritaso Opetusperiodi: Ei luennoida tänä lukuvuonna Sisältö: Vuosittain vaihtuva-aiheinen optimoinnin teorian jotain osa-aluetta, kuten konveksi optimointi tai funktionaalien optimointi, käsittelevä opintojakso. Toteutus, työmuodot ja arvosteluperusteet: Tentti Korvaavuudet: Mat Optimoinnin matemaattinen teoria L Esitiedot: MS-E2139 Nonlinear programming tai MS-E2140 Linear programming tai MS-E2148 Dynamic optimization Arvosteluasteikko: 1-5 Opintojaksot Opetuskieli: suomi MS-E2146 Integer programming (5 cr) Responsible teacher: Enrico Bartolini Status of the Course: Optional course of the Systems and Operations Research major and minor. Teaching period: IV (Spring) Workload: Lectures 24h (4) Exercises 24h (4) Home exercises 20h Assignment 20h Autonomous studies 40h Learning Outcomes: Be able to formulate a wide variety of optimization problems in integer and binary variables. Explain, interpret, and compare the most important families of algorithms in the general case and in special cases. Discuss the concept of complexity theory and relate it with the topics of the course. Content: Optimization problems including integer variables together with most common algorithms: branch and bound, cutting plane, dynamic programming, approximation and heuristics. Complexity analysis. Dual problem and its relation to the network problem. Lagrangian relaxation. Column generation algorithms. Assessment Methods and Criteria: Exam, assignment and home exercises Study Material: Laurence A. Wolsey: Integer Programming, Wiley-Interscience Publication, Substitutes for Courses: Mat Integer Programming P Prerequisites: MS-E2140 Linear Programming MS-E2148 Dynamic optimization (5 cr) Responsible teacher: Kimmo Berg Status of the Course: Compulsory course of the Systems and Operations Research major. Optional course of the Systems and Operations Research minor. Teaching period: III (Spring) 20