# Search result: Catalogue data in Spring Semester 2021

Mathematics Bachelor | ||||||

Compulsory Courses | ||||||

Examination Block II | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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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) |

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