# Search result: Catalogue data in Autumn Semester 2019

Mathematics Bachelor | ||||||

First Year | ||||||

» First Year Compulsory Courses | ||||||

» Minor Courses | ||||||

» GESS Science in Perspective | ||||||

First Year Compulsory Courses | ||||||

First Year Examination Block 1 | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|---|

401-1151-00L | Linear Algebra I | O | 7 credits | 4V + 2U | T. H. Willwacher | |

Abstract | Introduction to the theory of vector spaces for students of mathematics or physics: Basics, vector spaces, linear transformations, solutions of systems of equations, matrices, determinants, endomorphisms, eigenvalues, eigenvectors. | |||||

Objective | - Mastering basic concepts of Linear Algebra - Introduction to mathematical methods | |||||

Content | - Basics - Vectorspaces and linear maps - Systems of linear equations and matrices - Determinants - Endomorphisms and eigenvalues | |||||

Literature | - R. Pink: Lineare Algebra I und II. Summary. Link: Link - G. Fischer: Lineare Algebra. Springer-Verlag 2014. Link: Link - K. Jänich: Lineare Algebra. Springer-Verlag 2004. Link: Link - H.-J. Kowalsky, G. O. Michler: Lineare Algebra. Walter de Gruyter 2003. Link: Link - S. H. Friedberg, A. J. Insel and L. E. Spence: Linear Algebra. Pearson 2003. Link - H. Schichl and R. Steinbauer: Einführung in das mathematische Arbeiten. Springer-Verlag 2012. Link: Link | |||||

402-1701-00L | Physics I | O | 7 credits | 4V + 2U | R. Grange | |

Abstract | This course gives a first introduction to Physics with an emphasis on classical mechanics. | |||||

Objective | Acquire knowledge of the basic principles regarding the physics of classical mechanics. Skills in solving physics problems. | |||||

252-0847-00L | Computer Science | O | 5 credits | 2V + 2U | M. Schwerhoff, F. Friedrich Wicker | |

Abstract | The course covers the fundamental concepts of computer programming with a focus on systematic algorithmic problem solving. Taught language is C++. No programming experience is required. | |||||

Objective | Primary educational objective is to learn programming with C++. After having successfully attended the course, students have a good command of the mechanisms to construct a program. They know the fundamental control and data structures and understand how an algorithmic problem is mapped to a computer program. They have an idea of what happens "behind the scenes" when a program is translated and executed. Secondary goals are an algorithmic computational thinking, understanding the possibilities and limits of programming and to impart the way of thinking like a computer scientist. | |||||

Content | The course covers fundamental data types, expressions and statements, (limits of) computer arithmetic, control statements, functions, arrays, structural types and pointers. The part on object orientation deals with classes, inheritance and polymorphism; simple dynamic data types are introduced as examples. In general, the concepts provided in the course are motivated and illustrated with algorithms and applications. | |||||

Lecture notes | English lecture notes will be provided during the semester. The lecture notes and the lecture slides will be made available for download on the course web page. Exercises are solved and submitted online. | |||||

Literature | Bjarne Stroustrup: Einführung in die Programmierung mit C++, Pearson Studium, 2010 Stephen Prata, C++ Primer Plus, Sixth Edition, Addison Wesley, 2012 Andrew Koenig and Barbara E. Moo: Accelerated C++, Addison-Wesley, 2000 | |||||

First Year Examination Block 2 | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |

401-1261-07L | Analysis I | O | 10 credits | 6V + 3U | P. S. Jossen | |

Abstract | Introduction to the differential and integral calculus in one real variable: fundaments of mathematical thinking, numbers, sequences, basic point set topology, continuity, differentiable functions, ordinary differential equations, Riemann integration. | |||||

Objective | The ability to work with the basics of calculus in a mathematically rigorous way. | |||||

Literature | H. Amann, J. Escher: Analysis I Link J. Appell: Analysis in Beispielen und Gegenbeispielen Link R. Courant: Vorlesungen über Differential- und Integralrechnung Link O. Forster: Analysis 1 Link H. Heuser: Lehrbuch der Analysis Link K. Königsberger: Analysis 1 Link W. Walter: Analysis 1 Link V. Zorich: Mathematical Analysis I (englisch) Link A. Beutelspacher: "Das ist o.B.d.A. trivial" Link H. Schichl, R. Steinbauer: Einführung in das mathematische Arbeiten Link | |||||

Compulsory Courses | ||||||

Examination Block I In Examination Block I either the course unit 402-2883-00L Physics III or the course unit 402-2203-01L Classical Mechanics must be chosen and registered for an examination. (Students may also enrol for the other of the two course units; within the ETH Bachelor's programme in mathematics, this other course unit cannot be registered in myStudies for an examination nor can it be recognised for the Bachelor's degree.) | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |

401-2303-00L | Complex Analysis | O | 6 credits | 3V + 2U | P. Biran | |

Abstract | Complex functions of one variable, Cauchy-Riemann equations, Cauchy theorem and integral formula, singularities, residue theorem, index of closed curves, analytic continuation, special functions, conformal mappings, Riemann mapping theorem. | |||||

Objective | Working knowledge of functions of one complex variables; in particular applications of the residue theorem. | |||||

Literature | B. Palka: "An introduction to complex function theory." Undergraduate Texts in Mathematics. Springer-Verlag, 1991. E.M. Stein, R. Shakarchi: Complex Analysis. Princeton University Press, 2010 Th. Gamelin: Complex Analysis. Springer 2001 E. Titchmarsh: The Theory of Functions. Oxford University Press D. Salamon: "Funktionentheorie". Birkhauser, 2011. (In German) L. Ahlfors: "Complex analysis. An introduction to the theory of analytic functions of one complex variable." International Series in Pure and Applied Mathematics. McGraw-Hill Book Co. K.Jaenich: Funktionentheorie. Springer Verlag R.Remmert: Funktionentheorie I. Springer Verlag E.Hille: Analytic Function Theory. AMS Chelsea Publications | |||||

401-2333-00L | Methods of Mathematical Physics I | O | 6 credits | 3V + 2U | G. Felder | |

Abstract | Fourier series. Linear partial differential equations of mathematical physics. Fourier transform. Special functions and eigenfunction expansions. Distributions. Selected problems from quantum mechanics. | |||||

Objective | ||||||

402-2883-00L | Physics III | W | 7 credits | 4V + 2U | U. Keller | |

Abstract | Introductory course on quantum and atomic physics including optics and statistical physics. | |||||

Objective | A basic introduction to quantum and atomic physics, including basics of optics and equilibrium statistical physics. The course will focus on the relation of these topics to experimental methods and observations. | |||||

Content | Evidence for Quantum Mechanics: atoms, photons, photo-electric effect, Rutherford scattering, Compton scattering, de-Broglie waves. Quantum mechanics: wavefunctions, operators, Schrodinger's equation, infinite and finite square well potentials, harmonic oscillator, hydrogen atoms, spin. Atomic structure: Perturbation to basic structure, including Zeeman effect, spin-orbit coupling, many-electron atoms. X-ray spectra, optical selection rules, emission and absorption of radiation, including lasers. Optics: Fermat's principle, lenses, imaging systems, diffraction, interference, relation between geometrical and wave descriptions, interferometers, spectrometers. Statistical mechanics: probability distributions, micro and macrostates, Boltzmann distribution, ensembles, equipartition theorem, blackbody spectrum, including Planck distribution | |||||

Lecture notes | Lecture notes will be provided electronically during the course. | |||||

Literature | Quantum mechanics/Atomic physics/Molecules: "The Physics of Atoms and Quanta", H. Hakan and H. C. Wolf, ISBN 978-3-642-05871-4 Optics: "Optics", E. Hecht, ISBN 0-321-18878-0 Statistical mechanics: "Statistical Physics", F. Mandl 0-471-91532-7 | |||||

402-2203-01L | Classical Mechanics | W | 7 credits | 4V + 2U | M. Gaberdiel | |

Abstract | A conceptual introduction to theoretical physics: Newtonian mechanics, central force problem, oscillations, Lagrangian mechanics, symmetries and conservation laws, spinning top, relativistic space-time structure, particles in an electromagnetic field, Hamiltonian mechanics, canonical transformations, integrable systems, Hamilton-Jacobi equation. | |||||

Objective | Fundamental understanding of the description of Mechanics in the Lagrangian and Hamiltonian formulation. Detailed understanding of important applications, in particular, the Kepler problem, the physics of rigid bodies (spinning top) and of oscillatory systems. | |||||

252-0851-00L | Algorithms and Complexity | O | 4 credits | 2V + 1U | J. Lengler, A. Steger | |

Abstract | Introduction: RAM machine, data structures; Algorithms: sorting, median, matrix multiplication, shortest paths, minimal spanning trees; Paradigms: divide & conquer, dynamic programming, greedy algorithms; Data Structures: search trees, dictionaries, priority queues; Complexity Theory: P and NP, NP-completeness, Cook's theorem, reductions. | |||||

Objective | After this course students know some basic algorithms as well as underlying paradigms. They will be familiar with basic notions of complexity theory and can use them to classify problems. | |||||

Content | Die Vorlesung behandelt den Entwurf und die Analyse von Algorithmen und Datenstrukturen. Die zentralen Themengebiete sind: Sortieralgorithmen, Effiziente Datenstrukturen, Algorithmen für Graphen und Netzwerke, Paradigmen des Algorithmenentwurfs, Klassen P und NP, NP-Vollständigkeit, Approximationsalgorithmen. | |||||

Lecture notes | Ja. Wird zu Beginn des Semesters verteilt. | |||||

Examination Block II | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |

401-2003-00L | Algebra I | O | 7 credits | 4V + 2U | R. Pink | |

Abstract | Introduction and development of some basic algebraic structures - groups, rings, fields. | |||||

Objective | Introduction to basic notions and results of group, ring and field theory. | |||||

Content | Group Theory: basic notions and examples of groups, subgroups, factor groups, homomorphisms, group actions, Sylow theorems, applications Ring Theory: basic notions and examples of rings, ring homomorphisms, ideals, factor rings, euclidean rings, principal ideal domains, factorial rings, applications Field Theory: basic notions and examples of fields, field extensions, algebraic extensions, applications | |||||

Literature | Karpfinger-Meyberg: Algebra, Spektrum Verlag S. Bosch: Algebra, Springer Verlag B.L. van der Waerden: Algebra I und II, Springer Verlag S. Lang, Algebra, Springer Verlag A. Knapp: Basic Algebra, Springer Verlag J. Rotman, "Advanced modern algebra, 3rd edition, part 1" Link J.F. Humphreys: A Course in Group Theory (Oxford University Press) G. Smith and O. Tabachnikova: Topics in Group Theory (Springer-Verlag) M. Artin: Algebra (Birkhaeuser Verlag) R. Lidl and H. Niederreiter: Introduction to Finite Fields and their Applications (Cambridge University Press) | |||||

Core Courses | ||||||

Core Courses: Pure Mathematics | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |

401-3531-00L | Differential Geometry I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | U. Lang | |

Abstract | Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in R^n, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. | |||||

Objective | ||||||

Lecture notes | Partial lecture notes are available from Link | |||||

Literature | Differential geometry in R^n: - Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces - Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten - Christian Bär: Elementare Differentialgeometrie Differential topology: - Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds - Victor Guillemin & Alan Pollack: Differential Topology - Morris W. Hirsch: Differential Topology | |||||

401-3461-00L | Functional Analysis I At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. | W | 10 credits | 4V + 1U | M. Struwe | |

Abstract | Baire category; Banach and Hilbert spaces, bounded linear operators; basic principles: Uniform boundedness, open mapping/closed graph theorem, Hahn-Banach; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; compact operators and Fredholm theory; closed range theorem; spectral theory of self-adjoint operators in Hilbert spaces. | |||||

Objective | Acquire a good degree of fluency with the fundamental concepts and tools belonging to the realm of linear Functional Analysis, with special emphasis on the geometric structure of Banach and Hilbert spaces, and on the basic properties of linear maps. | |||||

Literature | We will be using the Lecture Notes on "Funktionalanalysis I" by Michael Struwe. Other useful, and recommended references include the following books: Haim Brezis: "Functional analysis, Sobolev spaces and partial differential equations". Springer, 2011. Manfred Einsiedler and Thomas Ward: "Functional Analysis, Spectral Theory, and Applications", Graduate Text in Mathematics 276. Springer, 2017. Peter D. Lax: "Functional analysis". Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. Elias M. Stein and Rami Shakarchi: "Functional analysis" (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011. Walter Rudin: "Functional analysis". International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991. Dirk Werner, "Funktionalanalysis". Springer-Lehrbuch, 8. Auflage. Springer, 2018 | |||||

Prerequisites / Notice | Solid background on the content of all Mathematics courses of the first two years of the undergraduate curriculum at ETH (most remarkably: fluency with measure theory, Lebesgue integration and L^p spaces). | |||||

401-3371-00L | Dynamical Systems I | W | 10 credits | 4V + 1U | W. Merry | |

Abstract | This course is a broad introduction to dynamical systems. Topic covered include topological dynamics, ergodic theory and low-dimensional dynamics. | |||||

Objective | Mastery of the basic methods and principal themes of some aspects of dynamical systems. | |||||

Content | Topics covered include: 1. Topological dynamics (transitivity, attractors, chaos, structural stability) 2. Ergodic theory (Poincare recurrence theorem, Birkhoff ergodic theorem, existence of invariant measures) 3. Low-dimensional dynamics (Poincare rotation number, dynamical systems on [0,1]) | |||||

Literature | The most relevant textbook for this course is Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002. I will also produce full lecture notes, available from my website Link | |||||

Prerequisites / Notice | The material of the basic courses of the first two years of the program at ETH is assumed. In particular, you should be familiar with metric spaces and elementary measure theory. | |||||

401-3001-61L | Algebraic Topology I | W | 8 credits | 4G | A. Sisto | |

Abstract | This is an introductory course in algebraic topology, which is the study of algebraic invariants of topological spaces. Topics covered include: singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms. | |||||

Objective | ||||||

Literature | 1) A. Hatcher, "Algebraic topology", Cambridge University Press, Cambridge, 2002. Book can be downloaded for free at: Link See also: Link 2) G. Bredon, "Topology and geometry", Graduate Texts in Mathematics, 139. Springer-Verlag, 1997. 3) E. Spanier, "Algebraic topology", Springer-Verlag | |||||

Prerequisites / Notice | You should know the basics of point-set topology. Useful to have (though not absolutely necessary) basic knowledge of the fundamental group and covering spaces (at the level covered in the course "topology"). Some knowledge of differential geometry and differential topology is useful but not strictly necessary. Some (elementary) group theory and algebra will also be needed. | |||||

401-3114-69L | Introduction to Algebraic Number Theory | W | 8 credits | 3V + 1U | Ö. Imamoglu | |

Abstract | This is an introductory course in algebraic number theory covering algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory. | |||||

Objective | ||||||

Content | This is an introductory course in algebraic number theory covering algebraic integers, discriminant, ideal class group, Minkowski's theorem on the finiteness of the ideal class group, Dirichlet's unit theorem, ramification theory. | |||||

401-3132-00L | Commutative Algebra | W | 10 credits | 4V + 1U | E. Kowalski | |

Abstract | This course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry. | |||||

Objective | We shall cover approximately the material from --- most of the textbook by Atiyah-MacDonald, or --- the first half of the textbook by Bosch. Topics include: * Basics about rings, ideals and modules * Localization * Primary decomposition * Integral dependence and valuations * Noetherian rings * Completions * Basic dimension theory | |||||

Literature | Primary Reference: 1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969) Secondary Reference: 2. "Algebraic Geometry and Commutative Algebra" by S. Bosch (Springer 2013) Tertiary References: 3. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995) 4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989) 5. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer) | |||||

Prerequisites / Notice | Prerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory). | |||||

» Core Courses: Pure Mathematics (Mathematics Master) |

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