Search result: Catalogue data in Spring Semester 2021
|» First Year Compulsory Courses|
|» Minor Courses|
|» GESS Science in Perspective|
|First Year Compulsory Courses|
| First Year Examination Block 1|
Offered in the Autumn Semester
|First Year Examination Block 2|
|401-1262-07L||Analysis II||O||10 credits||6V + 3U||G. Felder|
|Abstract||Introduction to differential and integral calculus in several real variables, vector calculus: differential, partial derivative, implicit functions, inverse function theorem, minima with constraints; Riemann integral, vector fields, differential forms, path integrals, surface integrals, divergence theorem, Stokes' theorem.|
|Content||Calculus in several variables; curves and surfaces in R^n; extrema with constraints; integration in n dimensions; vector calculus.|
|Literature||H. Amann, J. Escher: Analysis II|
J. Appell: Analysis in Beispielen und Gegenbeispielen
R. Courant: Vorlesungen über Differential- und Integralrechnung
O. Forster: Analysis 2
H. Heuser: Lehrbuch der Analysis
K. Königsberger: Analysis 2
W. Walter: Analysis 2
V. Zorich: Mathematical Analysis II (englisch)
|401-1152-02L||Linear Algebra II||O||7 credits||4V + 2U||M. Akka Ginosar|
|Abstract||Eigenvalues and eigenvectors, Jordan normal form, bilinear forms, euclidean and unitary vector spaces, selected applications.|
|Objective||Basic knowledge of the fundamentals of linear algebra.|
|Literature||Siehe Lineare Algebra I|
|Prerequisites / Notice||Linear Algebra I|
|401-1652-10L||Numerical Analysis I||O||6 credits||3V + 2U||C. Schwab|
|Abstract||This course will give an introduction to numerical methods, aimed at mathematics majors. It covers numerical linear algebra, quadrature, interpolation and approximation methods as well as their error analysis and implementation.|
|Objective||Knowledge of the fundamental numerical methods as well as |
`numerical literacy': application of numerical methods for the solution
of application problems, mathematical foundations of numerical
methods, and basic mathematical methods of the analysis of
stability, consistency and convergence of numerical methods,
|Content||Rounding errors, solution of linear systems of equations, nonlinear equations, |
interpolation (polynomial as well as trigonometric), least squares problems,
extrapolation, numerical quadrature, elementary optimization methods.
|Lecture notes||Lecture Notes and reading list will be available.|
|Literature||Lecture Notes (german or english) will be made available to students of ETH BSc MATH.|
Quarteroni, Sacco and Saleri, Numerische Mathematik 1 + 2, Springer Verlag 2002 (in German).
There is an English version of this text, containing both German volumes, from the same publisher. If you feel more comfortable with English, you can follow this text as well. Content and Indexing are identical in the German and the English text.
|Prerequisites / Notice||Admission Requirements:|
Linear Algebra I, Analysis I in ETH BSc MATH
Parallel enrolment in
Linear Algebra II, Analysis II in ETH BSc MATH
Weekly homework assignments involving MATLAB programming
are an integral part of the course.
Turn-in of solutions will be graded.
|402-1782-00L||Physics II||O||7 credits||4V + 2U||R. Wallny|
|Abstract||Introduction to theory of waves, electricity and magnetism. This is the continuation of Physics I which introduced the fundamentals of mechanics.|
|Objective||basic knowledge of mechanics and electricity and magnetism as well as the capability to solve physics problems related to these subjects.|
|Examination Block II|
|401-2284-00L||Measure and Integration||O||6 credits||3V + 2U||F. Da Lio|
|Abstract||Introduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces|
|Objective||Basic acquaintance with the abstract theory of measure and integration|
|Content||Introduction to abstract measure and integration theory, including the following topics: Caratheodory extension theorem, Lebesgue measure, convergence theorems, L^p-spaces, Radon-Nikodym theorem, product measures and Fubini's theorem, measures on topological spaces|
|Lecture notes||New lecture notes in English will be made available during the course.|
|Literature||1. L. Evans and R.F. Gariepy " Measure theory and fine properties of functions"|
2. Walter Rudin "Real and complex analysis"
3. R. Bartle The elements of Integration and Lebesgue Measure
4. The notes by Prof. Michael Struwe Springsemester 2013, https://people.math.ethz.ch/~struwe/Skripten/AnalysisIII-FS2013-12-9-13.pdf.
5. The notes by Prof. UrsLang Springsemester 2019. https://people.math.ethz.ch/~lang/mi.pdf
6. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis: http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf
|401-2004-00L||Algebra II||O||5 credits||2V + 2U||M. Burger|
|Abstract||The main topics are field extensions and Galois theory.|
|Objective||Introduction to fundamentals of field extensions, Galois theory, and related topics.|
|Content||The main topic is Galois Theory. Starting point is the problem of solvability of algebraic equations by radicals. Galois theory solves this problem by making a connection between field extensions and group theory. Galois theory will enable us to prove the theorem of Abel-Ruffini, that there are polynomials of degree 5 that are not solvable by radicals, as well as Galois' theorem characterizing those polynomials which are solvable by radicals.|
|Literature||Joseph J. Rotman, "Advanced Modern Algebra" third edition, part 1,|
Graduate Studies in Mathematics,Volume 165
American Mathematical Society
Galois Theory is the topic treated in Chapter A5.
|401-2554-00L||Topology||O||6 credits||3V + 2U||P. Feller|
|Abstract||Topics covered include: topological spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces.|
|Objective||An introduction to topology -- the domain of mathematics that studies how to define the notion of continuity on a mathematical structure, and how to use it to study and classify these structures.|
- Klaus Jänich: Topologie (Springer).
- Boto von Querenburg: Mengentheoretische Topologie (Springer).
(für den ersten Teil der Vorlesung über die allgemeine (/mengentheoretische) Topologie)
(für den zweiten Teil der Vorlesung über die Anfänge der algebraischen Topologie (Fundamentalgrupppe, Überlagerungen)).
- James Munkres: Topology (Pearson Modern Classics for Advanced Mathematics Series).
- Lynn Arthur Steen, J. Arthur Seebach Jr.: Counterexamples in Topology (Springer).
- Edwin Spanier: Algebraic Topology (Springer).
|401-2654-00L||Numerical Analysis II||O||6 credits||3V + 2U||H. Ammari|
|Abstract||The central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation.|
|Objective||The course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in Python and test them in numerical experiments.|
|Content||Chapter 1. Some basics|
1.1. What is a differential equation?
1.2. Some methods of resolution
1.3. Important examples of ODEs
Chapter 2. Existence, uniqueness, and regularity in the Lipschitz case
2.1. Banach fixed point theorem
2.2. Gronwall’s lemma
2.3. Cauchy-Lipschitz theorem
Chapter 3. Linear systems
3.1. Exponential of a matrix
3.2. Linear systems with constant coefficients
3.3. Linear system with non-constant real coefficients
3.4. Second order linear equations
3.5. Linearization and stability for autonomous systems
3.6 Periodic Linear Systems
Chapter 4. Numerical solution of ordinary differential equations
4.2. The general explicit one-step method
4.3. Example of linear systems
4.4. Runge-Kutta methods
4.5. Multi-step methods
4.6. Stiff equations and systems
4.7. Perturbation theories for differential equations
Chapter 5. Geometrical numerical integration methods for differential equation
5.2. Structure preserving methods for Hamiltonian systems
5.3. Runge-Kutta methods
5.4. Long-time behaviour of numerical solutions
Chapter 6. Finite difference methods
6.2. Numerical algorithms for the heat equation
6.3. Numerical algorithms for the wave equation
6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimension
Chapter 7. Stochastic differential equations
7.2. Langevin equation
7.3. Ornstein-Uhlenbeck equation
7.4. Existence and uniqueness of solutions in dimension one
7.5. Numerical solution of stochastic differential equations
|Lecture notes||Lecture notes including supplements will be provided electronically.|
Please find the lecture homepage here:
All assignments and some previous exam problems will be available for download on lecture homepage.
|Literature||Note: Extra reading is not considered important for understanding the|
Deuflhard and Bornemann: Numerische Mathematik II - Integration gewöhnlicher Differentialgleichungen, Walter de Gruyter & Co., 1994.
Hairer and Wanner: Solving ordinary differential equations II - Stiff and differential-algebraic problems, Springer-Verlag, 1996.
Hairer, Lubich and Wanner: Geometric numerical integration - Structure-preserving algorithms for ordinary differential equations}, Springer-Verlag, Berlin, 2002.
L. Gruene, O. Junge "Gewoehnliche Differentialgleichungen", Vieweg+Teubner, 2009.
Hairer, Norsett and Wanner: Solving ordinary differential equations I - Nonstiff problems, Springer-Verlag, Berlin, 1993.
Walter: Gewöhnliche Differentialgleichungen - Eine Einführung, Springer-Verlag, Berlin, 1972.
Walter: Ordinary differential equations, Springer-Verlag, New York, 1998.
|Prerequisites / Notice||Homework problems involve Python implementation of numerical algorithms.|
|401-2604-00L||Probability and Statistics||O||7 credits||4V + 2U||J. Teichmann|
|Abstract||- Discrete probability spaces|
- Continuous models
- Limit theorems
- Introduction to statistics
|Objective||The goal of this course is to provide an introduction to the basic ideas and concepts from probability theory and mathematical statistics. This includes a mathematically rigorous treatment as well as intuition and getting acquainted with the ideas behind the definitions. The course does not use measure theory systematically, but does point out where this is required and what the connections are.|
|Content||- Discrete probability spaces: Basic concepts, Laplace models, random walks, conditional probabilities, independence|
- Continuous models: general probability spaces, random variables and their distributions, expectation, multivariate random variables
- Limit theorems: weak and strong law of large numbers, central limit theorem
- Introduction to statistics: What is statistics?, point estimators, statistical tests, confidence intervals
|Lecture notes||There will be lecture notes (in German) that are continuously updated during the semester.|
|Literature||A. DasGupta, Fundamentals of Probability: A First Course, Springer (2010)|
J. A. Rice, Mathematical Statistics and Data Analysis, Duxbury Press, second edition (1995)
|Core Courses: Pure Mathematics|
|401-3532-08L||Differential Geometry II||W||10 credits||4V + 1U||W. Merry|
|Abstract||This is a continuation course of Differential Geometry I.|
Topics covered include:
- Connections and curvature,
- Riemannian geometry,
- Gauge theory and Chern-Weil theory.
|Lecture notes||I will produce full lecture notes, available on my website: |
|Literature||There are many excellent textbooks on differential geometry. |
A friendly and readable book that contains everything covered in Differential Geometry I is:
John M. Lee "Introduction to Smooth Manifolds" 2nd ed. (2012) Springer-Verlag.
For Differential Geometry II, the textbooks:
- S. Kobayashi, K. Nomizu "Foundations of Differential Geometry" Volume I (1963) Wiley,
- I. Chavel, "Riemannian Geometry: A Modern Introduction" 2nd ed. (2006), CUP,
are both excellent. The monograph
- A. L. Besse "Einstein Manifolds", (1987), Springer,
gives a comprehensive overview of the entire field, although it is extremely advanced. (By the end of the course you should be able to read this book.)
|Prerequisites / Notice||Familiarity with all the material from Differential Geometry I will be assumed (smooth manifolds, Lie groups, vector bundles, differential forms, integration on manifolds, principal bundles and so on). Lecture notes for Differential Geometry I can be found on my website.|
|401-3462-00L||Functional Analysis II||W||10 credits||4V + 1U||A. Carlotto|
|Abstract||Sobolev spaces, weak solutions of elliptic boundary value problems, basic results in elliptic regularity theory (including Schauder estimates), maximum principles.|
|Objective||Acquire fluency with Sobolev spaces and weak derivatives on the one hand, and basic elliptic regularity on the other. Apply these methods for studying elliptic boundary value problems.|
|Literature||Michael Struwe. Funktionalanalysis I und II. Lecture notes, ETH Zürich, 2013/14.|
Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.
Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi. Lectures on elliptic partial differential equations. Springer - Edizioni della Normale, Pisa, 2018.
David Gilbarg, Neil Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001.
Qing Han, Fanghua Lin. Elliptic partial differential equations. Second edition. Courant Lecture Notes in Mathematics, 1. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.
Michael Taylor. Partial differential equations I. Basic theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York, 2011.
Lars Hörmander. The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer, Berlin, 2003.
|Prerequisites / Notice||Functional Analysis I plus a solid background in measure theory, Lebesgue integration and L^p spaces.|
|401-3002-12L||Algebraic Topology II||W||8 credits||4G||P. Biran|
|Abstract||This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including:|
cohomology of spaces, operations in homology and cohomology, duality.
|Literature||1) G. Bredon, "Topology and geometry",|
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.
2) A. Hatcher, "Algebraic topology",
Cambridge University Press, Cambridge, 2002.
The book can be downloaded for free at:
3) E. Spanier, "Algebraic topology", Springer-Verlag
|Prerequisites / Notice||General topology, linear algebra, singular homology of topological spaces (e.g. as taught in "Algebraic topology I").|
Some knowledge of differential geometry and differential topology
is useful but not absolutely necessary.
|401-8142-21L||Algebraic Geometry II (University of Zurich)|
No enrolment to this course at ETH Zurich. Book the corresponding module directly at UZH.
UZH Module Code: MAT517
Mind the enrolment deadlines at UZH:
|W||9 credits||4V + 1U||University lecturers|
|Abstract||We continue the development of scheme theory. Among the topics that will be discussed are: properties of schemes and their morphisms (flatness, smoothness), coherent modules, cohomology, etc.|
|» Core Courses: Pure Mathematics (Mathematics Master)|
| Core Courses: Applied Mathematics and Further Appl.-Oriented Fields|
|401-3052-10L||Graph Theory||W||10 credits||4V + 1U||B. Sudakov|
|Abstract||Basics, trees, Caley's formula, matrix tree theorem, connectivity, theorems of Mader and Menger, Eulerian graphs, Hamilton cycles, theorems of Dirac, Ore, Erdös-Chvatal, matchings, theorems of Hall, König, Tutte, planar graphs, Euler's formula, Kuratowski's theorem, graph colorings, Brooks' theorem, 5-colorings of planar graphs, list colorings, Vizing's theorem, Ramsey theory, Turán's theorem|
|Objective||The students will get an overview over the most fundamental questions concerning graph theory. We expect them to understand the proof techniques and to use them autonomously on related problems.|
|Lecture notes||Lecture will be only at the blackboard.|
|Literature||West, D.: "Introduction to Graph Theory"|
Diestel, R.: "Graph Theory"
Further literature links will be provided in the lecture.
|Prerequisites / Notice||Students are expected to have a mathematical background and should be able to write rigorous proofs.|
|401-3642-00L||Brownian Motion and Stochastic Calculus||W||10 credits||4V + 1U||W. Werner|
|Abstract||This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.|
|Objective||This course covers some basic objects of stochastic analysis. In particular, the following topics are discussed: construction and properties of Brownian motion, stochastic integration, Ito's formula and applications, stochastic differential equations and connection with partial differential equations.|
|Lecture notes||Lecture notes will be distributed in class.|
|Literature||- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer (2016).|
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer (1991).
- D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (2005).
- L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales, vol. 1 and 2, Cambridge University Press (2000).
- D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer (2006).
|Prerequisites / Notice||Familiarity with measure-theoretic probability as in the standard D-MATH course "Probability Theory" will be assumed. Textbook accounts can be found for example in |
- J. Jacod, P. Protter, Probability Essentials, Springer (2004).
- R. Durrett, Probability: Theory and Examples, Cambridge University Press (2010).
|401-3632-00L||Computational Statistics||W||8 credits||3V + 1U||M. Mächler|
|Abstract||We discuss modern statistical methods for data analysis, including methods for data exploration, prediction and inference. We pay attention to algorithmic aspects, theoretical properties and practical considerations. The class is hands-on and methods are applied using the statistical programming language R.|
|Objective||The student obtains an overview of modern statistical methods for data analysis, including their algorithmic aspects and theoretical properties. The methods are applied using the statistical programming language R.|
|Content||See the class website|
|Prerequisites / Notice||At least one semester of (basic) probability and statistics.|
Programming experience is helpful but not required.
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