Search result: Catalogue data in Spring Semester 2021

Number	Title	Type	ECTS	Hours	Lecturers
Mathematics Bachelor
First Year
» 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.
Objective
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 https://link.springer.com/book/10.1007/3-7643-7402-0 J. Appell: Analysis in Beispielen und Gegenbeispielen https://link.springer.com/book/10.1007/978-3-540-88903-8 R. Courant: Vorlesungen über Differential- und Integralrechnung https://link.springer.com/book/10.1007/978-3-642-61973-1 O. Forster: Analysis 2 https://link.springer.com/book/10.1007/978-3-658-02357-7 H. Heuser: Lehrbuch der Analysis https://link.springer.com/book/10.1007/978-3-322-96826-5 K. Königsberger: Analysis 2 https://link.springer.com/book/10.1007/3-540-35077-2 W. Walter: Analysis 2 https://link.springer.com/book/10.1007/978-3-642-97614-8 V. Zorich: Mathematical Analysis II (englisch) https://link.springer.com/book/10.1007/978-3-662-48993-2
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, MATLAB implementation.
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: Completed course 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.

Compulsory Courses
Examination Block II
Number	Title	Type	ECTS	Hours	Lecturers
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.
Literature	Hauptreferenz: - Klaus Jänich: Topologie (Springer). https://link.springer.com/book/10.1007/978-3-662-10575-7 Weitere Referenzen: - Boto von Querenburg: Mengentheoretische Topologie (Springer). http://link.springer.com/book/10.1007/978-3-642-56860-2 - http://pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf (für den ersten Teil der Vorlesung über die allgemeine (/mengentheoretische) Topologie) - http://pi.math.cornell.edu/~hatcher/AT/ATch1.pdf (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 2.4. Stability 2.5. Regularity 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 4.1. Introduction 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 5.1. Introduction 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 6.1. Introduction 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 7.1. Introduction 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: https://www.sam.math.ethz.ch/~grsam/SS21/NAII/ 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 course subjects. 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
Core Courses: Pure Mathematics
Number	Title	Type	ECTS	Hours	Lecturers
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.
Objective
Lecture notes	I will produce full lecture notes, available on my website: https://www.merry.io/courses/differential-geometry/
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.
Objective
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: http://www.math.cornell.edu/~hatcher/AT/ATpage.html 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: https://www.uzh.ch/cmsssl/en/studies/application/deadlines.html	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.
Objective
» Core Courses: Pure Mathematics (Mathematics Master)

Core Courses: Applied Mathematics and Further Appl.-Oriented Fields ¬
Number	Title	Type	ECTS	Hours	Lecturers
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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