# Search result: Catalogue data in Spring Semester 2020

Number Title Type ECTS Hours Lecturers Mathematics Bachelor  Compulsory Courses  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 Measure4. 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.pdf6. 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 SocietyGalois 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: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. 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 basics1.1. What is a differential equation?1.2. Some methods of resolution1.3. Important examples of ODEsChapter 2. Existence, uniqueness, and regularity in the Lipschitz case2.1. Banach fixed point theorem2.2. Gronwall’s lemma2.3. Cauchy-Lipschitz theorem2.4. Stability2.5. RegularityChapter 3. Linear systems3.1. Exponential of a matrix3.2. Linear systems with constant coefficients3.3. Linear system with non-constant real coefficients3.4. Second order linear equations3.5. Linearization and stability for autonomous systems3.6 Periodic Linear SystemsChapter 4. Numerical solution of ordinary differential equations4.1. Introduction4.2. The general explicit one-step method4.3. Example of linear systems4.4. Runge-Kutta methods4.5. Multi-step methods4.6. Stiff equations and systems4.7. Perturbation theories for differential equationsChapter 5. Geometrical numerical integration methods for differential equation5.1. Introduction5.2. Structure preserving methods for Hamiltonian systems5.3. Runge-Kutta methods5.4. Long-time behaviour of numerical solutionsChapter 6. Finite difference methods6.1. Introduction6.2. Numerical algorithms for the heat equation6.3. Numerical algorithms for the wave equation6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimensionChapter 7. Stochastic differential equations7.1. Introduction7.2. Langevin equation7.3. Ornstein-Uhlenbeck equation7.4. Existence and uniqueness of solutions in dimension one7.5. 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 thecourse 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 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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