Search result: Catalogue data in Spring Semester 2020

Number	Title	Type	ECTS	Hours	Lecturers
Mathematics Bachelor
First Year
» 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.
Learning 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	T. H. Willwacher
Abstract	Eigenvalues and eigenvectors, Jordan normal form, bilinear forms, euclidean and unitary vector spaces, selected applications.
Learning 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.
Learning 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 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.
Learning 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
Learning 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.
Learning 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.
Learning 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: i) http://pi.math.cornell.edu/~hatcher/Top/TopNotes.pdf (for the part on General Topology) ii) http://pi.math.cornell.edu/~hatcher/AT/ATch1.pdf (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.
Learning 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/SS20/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	M. Schweizer
Abstract	- Discrete probability spaces - Continuous models - Limit theorems - Introduction to statistics
Learning 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	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.
Learning 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
Learning 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).
Learning objective	Learning Algebraic Geometry.
Literature	Primary reference: * Ulrich Görtz and Torsten Wedhorn: Algebraic Geometry I, Advanced Lectures in Mathematics, Springer. Secondary reference: * 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 Further readings: * 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.
Learning objective
Literature	1) 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 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.
Learning 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: https://www.merry.io/courses/dynamical-systems/
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: https://www.merry.io/courses/dynamical-systems/ 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 ¬
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
Learning 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.
Learning 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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