Search result: Catalogue data in Autumn Semester 2018

Mathematics Bachelor Information
First Year
» First Year Compulsory Courses
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First Year Compulsory Courses
First Year Examination Block 1
NumberTitleTypeECTSHoursLecturers
401-1151-00LLinear Algebra I Information O7 credits4V + 2UR. Pink
AbstractIntroduction 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
402-1701-00LPhysics IO7 credits4V + 2UA. Wallraff
AbstractThis course gives a first introduction to Physics with an emphasis on classical mechanics.
ObjectiveAcquire knowledge of the basic principles regarding the physics of classical mechanics. Skills in solving physics problems.
252-0847-00LComputer Science Information O5 credits2V + 2UM. Schwerhoff, F. Friedrich Wicker
AbstractThe 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.
ObjectivePrimary 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.
ContentThe 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 notesA script written in English will be provided during the semester. The script and the lecture slides will be made available for download on the course web page. Exercises are solved and submitted online.
LiteratureBjarne 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
NumberTitleTypeECTSHoursLecturers
401-1261-07LAnalysis I Information O10 credits6V + 3UP. S. Jossen
AbstractIntroduction 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.
ObjectiveThe ability to work with the basics of calculus in a mathematically rigorous way.
LiteratureH. 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.)
NumberTitleTypeECTSHoursLecturers
401-2303-00LComplex Analysis Information O6 credits3V + 2UM. Struwe
AbstractComplex 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.
ObjectiveWorking knowledge of functions of one complex variables; in particular applications of the residue theorem.
LiteratureE.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.

B. Palka: "An introduction to complex function theory."
Undergraduate Texts in Mathematics. Springer-Verlag, 1991.

R.Remmert: Theory of Complex Functions. Springer Verlag
401-2333-00LMethods of Mathematical Physics I Information O6 credits3V + 2UT. H. Willwacher
AbstractFourier series. Linear partial differential equations of mathematical physics. Fourier transform. Special functions and eigenfunction expansions. Distributions. Selected problems from quantum mechanics.
Objective
Prerequisites / NoticeDie Einschreibung in die Übungsgruppen erfolgt online. Melden Sie sich im Laufe der ersten Semesterwoche unter echo.ethz.ch mit Ihrem ETH Account an. Der Übungsbetrieb beginnt in der zweiten Semesterwoche.

Vorlesungshomepage: Link
402-2883-00LPhysics IIIW7 credits4V + 2US. Johnson
AbstractIntroductory course on quantum and atomic physics including optics and statistical physics.
ObjectiveA 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.
ContentEvidence 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 notesLecture notes will be provided electronically during the course.
LiteratureQuantum 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
402-2203-01LClassical Mechanics Information W7 credits4V + 2UC. Anastasiou
AbstractA 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
252-0851-00LAlgorithms and ComplexityO4 credits2V + 1UJ. Lengler, A. Steger
AbstractIntroduction: 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.
ObjectiveAfter 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.
ContentDie 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 notesJa. Wird zu Beginn des Semesters verteilt.
Examination Block II
NumberTitleTypeECTSHoursLecturers
401-2003-00LAlgebra I Information O7 credits4V + 2UR. Pandharipande
AbstractIntroduction and development of some basic algebraic structures - groups, rings, fields.
ObjectiveIntroduction to basic notions and results of group, ring and field
theory.
ContentGroup Theory: basic notions and examples of groups; Subgroups, Quotient groups and Homomorphisms, Sylow Theorems, Group actions and applications

Ring Theory: basic notions and examples of rings; Ring Homomorphisms, ideals and quotient rings, applications

Field Theory: basic notions and examples of fields; finite fields, applications
LiteratureJ. 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)
B.L. van der Waerden: Algebra I & II (Springer Verlag)
Core Courses
Core Courses: Pure Mathematics
NumberTitleTypeECTSHoursLecturers
401-3531-00LDifferential Geometry I Information
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.
W10 credits4V + 1UW. Merry
AbstractThis will be an introductory course in differential geometry.

Topics covered include:

- Smooth manifolds, submanifolds, vector fields,
- Lie groups, homogeneous spaces,
- Vector bundles, tensor fields, differential forms,
- Integration on manifolds and the de Rham theorem,
- Principal bundles.
Objective
Lecture notesI will produce full lecture notes, available on my website at

Link
LiteratureThere are many excellent textbooks on differential geometry. A friendly and readable book that covers everything in Differential Geometry I is:

John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag.

A more advanced (and far less friendly) series of books that covers everything in both Differential Geometry I and II is:

S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volumes I and II (1963, 1969) Wiley.
401-3461-00LFunctional Analysis I Information
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.
W10 credits4V + 1UM. Einsiedler
AbstractBaire 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; Fourier transform and applications.
ObjectiveAcquire 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.
LiteratureWe will be using the book
Functional Analysis, Spectral Theory, and Applications
by Manfred Einsiedler and Thomas Ward
and available by SpringerLink.

Other useful, and recommended references include the following:

Lecture Notes on "Funktionalanalysis I" by Michael Struwe

Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

Elias M. Stein and Rami Shakarchi. Functional analysis (volume 4 of Princeton Lectures in Analysis). Princeton University Press, Princeton, NJ, 2011.

Peter D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.

Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
Prerequisites / NoticeSolid 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).
401-3001-61LAlgebraic Topology I Information W8 credits4GP. Biran
AbstractThis is an introductory course in algebraic topology. Topics covered include:
singular homology, cell complexes and cellular homology, the Eilenberg-Steenrod axioms, cohomology. Along the way we will introduce the basics of homological algebra and category theory.
Objective
Literature1) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.

2) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.

Book can be downloaded for free at:
Link

See also:
Link

3) E. Spanier, "Algebraic topology", Springer-Verlag
Prerequisites / NoticeYou 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 usually covered in the course "topology").

Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.

Some (elementary) group theory and algebra will also be needed.
401-3132-00LCommutative Algebra Information W10 credits4V + 1UP. D. Nelson
AbstractThis course provides an introduction to commutative algebra as a foundation for and first steps towards algebraic geometry.
ObjectiveWe 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
LiteraturePrimary 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 / NoticePrerequisites: Algebra I (or a similar introduction to the basic concepts of ring theory).
» Core Courses: Pure Mathematics (Mathematics Master)
Core Courses: Applied Mathematics and Further Appl.-Oriented Fields
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NumberTitleTypeECTSHoursLecturers
401-3651-00LNumerical Methods for Elliptic and Parabolic Partial Differential Equations (University of Zurich)
Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students.
Other ETH-students are advised to attend the course "Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester.

No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT802

Mind the enrolment deadlines at UZH:
Link
W9 credits4V + 2US. Sauter
AbstractThis course gives a comprehensive introduction into the numerical treatment of linear and non-linear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods.
ObjectiveParticipants of the course should become familiar with
* concepts underlying the discretization of elliptic and parabolic boundary value problems
* analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems
* methods for the efficient solution of discrete boundary value problems
* implementational aspects of the finite element method
ContentA selection of the following topics will be covered:

* Elliptic boundary value problems
* Galerkin discretization of linear variational problems
* The primal finite element method
* Mixed finite element methods
* Discontinuous Galerkin Methods
* Boundary element methods
* Spectral methods
* Adaptive finite element schemes
* Singularly perturbed problems
* Sparse grids
* Galerkin discretization of elliptic eigenproblems
* Non-linear elliptic boundary value problems
* Discretization of parabolic initial boundary value problems
Lecture notesCourse slides will be made available to the audience.
LiteratureS. C. Brenner and L. Ridgway Scott: The mathematical theory of Finite Element Methods. New York, Berlin [etc]: Springer-Verl, cop.1994.

A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods,
Springer Applied Mathematical Sciences Vol. 159, Springer,
1st Ed. 2004, 2nd Ed. 2015.

R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013

Additional Literature:
D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007).
(Also available in German.)

D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications,
Springer, 2012 [DOI: 10.1007/978-3-642-22980-0]

V. Thomee: Galerkin Finite Element Methods for Parabolic Problems,
SECOND Ed., Springer Verlag (2006).
Prerequisites / NoticePractical exercises based on MATLAB
401-3601-00LProbability Theory Information
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.
W10 credits4V + 1UA.‑S. Sznitman
AbstractBasics of probability theory and the theory of stochastic processes in discrete time
ObjectiveThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
ContentThis course presents the basics of probability theory and the theory of stochastic processes in discrete time. The following topics are planned:
Basics in measure theory, random series, law of large numbers, weak convergence, characteristic functions, central limit theorem, conditional expectation, martingales, convergence theorems for martingales, Galton Watson chain, transition probability, Theorem of Ionescu Tulcea, Markov chains.
Lecture notesavailable, will be sold in the course
LiteratureR. Durrett, Probability: Theory and examples, Duxbury Press 1996
H. Bauer, Probability Theory, de Gruyter 1996
J. Jacod and P. Protter, Probability essentials, Springer 2004
A. Klenke, Wahrscheinlichkeitstheorie, Springer 2006
D. Williams, Probability with martingales, Cambridge University Press 1991
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