Search result: Catalogue data in Spring Semester 2020
|» First Year Compulsory Courses|
|» GESS Science in Perspective|
|» Minor Courses|
|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||P. S. Jossen|
|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||T. H. Willwacher|
|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.
Accompanying the lecture course "Physics II", among GESS Science in Perspective is offered: 851-0147-01L Philosophical Reflections on 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||R. Pink|
|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||A. Carlotto|
|Abstract||Topics covered include: Topological and metric spaces, continuity, connectedness, compactness, product spaces, separation axioms, quotient spaces, homotopy, fundamental group, covering spaces.|
|Objective||An introduction to topology i.e. 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.|
|Literature||We will follow these, freely available, standard references by Allen Hatcher:|
(for the part on General Topology)
(for the part on basic Algebraic Topology).
Additional references include:
"Topology" by James Munkres (Pearson Modern Classics for Advanced Mathematics Series)
"Counterexamples in Topology" by Lynn Arthur Steen, J. Arthur Seebach Jr. (Springer)
"Algebraic Topology" by Edwin Spanier (Springer).
|Prerequisites / Notice||The content of the first-year courses in the Bachelor program in Mathematics. In particular, each student is expected to be familiar with notion of continuity for functions from/to Euclidean spaces, and with the content of the corresponding basic theorems (Bolzano, Weierstrass etc..). In addition, some degree of scientific maturity in writing rigorous proofs (and following them when presented in class) is absolutely essential.|
|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||M. Schweizer|
|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||U. Lang|
|Abstract||Introduction to Riemannian geometry in combination with some elements of modern metric geometry. Contents: Riemannian manifolds, Levi-Civita connection, geodesics, Hopf-Rinow Theorem, curvature, second fundamental form, Riemannian submersions and coverings, Hadamard-Cartan Theorem, triangle and volume comparison, relations between curvature and topology, spaces of Riemannian manifolds.|
|Objective||Learn the basics of Riemannian geometry and some elements of modern metric geometry.|
|Literature||- M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992|
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer 2004
- B. O'Neill, Semi-Riemannian Geometry, With Applications to Relativity, Academic Press 1983
|Prerequisites / Notice||Prerequisite is a working knowledge of elementary differential geometry (curves and surfaces in Euclidean space), differentiable manifolds, and differential forms.|
|401-3462-00L||Functional Analysis II||W||10 credits||4V + 1U||M. Struwe|
|Abstract||Sobolev spaces, weak solutions of elliptic boundary value problems, elliptic regularity|
|Objective||Acquiring the methods for solving elliptic boundary value problems, Sobolev spaces, Schauder estimates|
|Lecture notes||Funktionalanalysis II, Michael Struwe|
|Literature||Funktionalanalysis II, Michael Struwe |
Functional Analysis, Spectral Theory and Applications.
Manfred Einsiedler and Thomas Ward, GTM Springer 2017
|Prerequisites / Notice||Functional Analysis I and a solid background in measure theory, Lebesgue integration and L^p spaces.|
|401-3146-12L||Algebraic Geometry||W||10 credits||4V + 1U||D. Johnson|
|Abstract||This course is an Introduction to Algebraic Geometry (algebraic varieties and schemes).|
|Objective||Learning Algebraic Geometry.|
* Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer.
* Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Science Publications.
* Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics, Springer.
* Siegfried Bosch: Algebraic Geometry and Commutative Algebra (Springer 2013).
Other good textbooks and online texts are:
* David Eisenbud, Joe Harris: The Geometry of Schemes, Graduate Texts in Mathematics, Springer.
* Ravi Vakil, Foundations of Algebraic Geometry, http://math.stanford.edu/~vakil/216blog/
* Jean Gallier and Stephen S. Shatz, Algebraic Geometry http://www.cis.upenn.edu/~jean/algeom/steve01.html
"Classical" Algebraic Geometry over an algebraically closed field:
* Joe Harris, Algebraic Geometry, A First Course, Graduate Texts in Mathematics, Springer.
* J.S. Milne, Algebraic Geometry, http://www.jmilne.org/math/CourseNotes/AG.pdf
* Günter Harder: Algebraic Geometry 1 & 2
* I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
* Alexandre Grothendieck et al.: Elements de Geometrie Algebrique EGA
* Saunders MacLane: Categories for the Working Mathematician, Springer-Verlag.
|Prerequisites / Notice||Requirement: Some knowledge of Commutative Algebra.|
|401-3002-12L||Algebraic Topology II||W||8 credits||4G||A. Sisto|
|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) A. Hatcher, "Algebraic topology",|
Cambridge University Press, Cambridge, 2002.
The book can be downloaded for free at:
2) G. Bredon, "Topology and geometry",
Graduate Texts in Mathematics, 139. Springer-Verlag, 1997.
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.
|401-3372-00L||Dynamical Systems II||W||10 credits||4V + 1U||W. Merry|
|Abstract||This course is a continuation of Dynamical Systems I. This time the emphasis is on hyperbolic and complex dynamics.|
|Objective||Mastery of the basic methods and principal themes of some aspects of hyperbolic and complex dynamical systems.|
|Content||Topics covered include:|
- Hyperbolic linear dynamical systems, hyperbolic fixed points, the Hartman-Grobman Theorem.
- Hyperbolic sets, Anosov diffeomorphisms.
- The (Un)stable Manifold Theorem.
- Shadowing Lemmas and stability.
- The Lambda Lemma.
- Transverse homoclinic points, horseshoes, and chaos.
- Complex dynamics of rational maps on the Riemann sphere
- Julia sets and Fatou sets.
- Fractals and the Mandelbrot set.
|Lecture notes||I will provide full lecture notes, available here:|
|Literature||The most useful textbook is|
- Introduction to Dynamical Systems, Brin and Stuck, CUP, 2002.
|Prerequisites / Notice||It will be assumed you are familiar with the material from Dynamical Systems I. Full lecture notes for this course are available here:|
However we will only really use material covered in the first 10 lectures of Dynamical Systems I, so if you did not attend Dynamical Systems I, it is sufficient to read through the notes from the first 10 lectures.
In addition, it would be useful to have some familiarity with basic differential geometry and complex analysis.
|» 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).
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