## Francesca Da Lio: Catalogue data in Autumn Semester 2021 |

Name | Prof. Dr. Francesca Da Lio |

Address | Dep. Mathematik ETH Zürich, HG G 37.2 Rämistrasse 101 8092 Zürich SWITZERLAND |

Telephone | +41 44 632 86 96 |

Fax | +41 44 632 10 85 |

francesca.dalio@math.ethz.ch | |

URL | http://www.math.ethz.ch/~fdalio |

Department | Mathematics |

Relationship | Adjunct Professor |

Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|

401-0251-00L | Mathematics I | 6 credits | 4V + 2U | F. Da Lio | |

Abstract | This course covers mathematical concepts and techniques necessary to model, solve and discuss scientific problems - notably through ordinary differential equations. | ||||

Objective | Mathematics is of ever increasing importance to the Natural Sciences and Engineering. The key is the so-called mathematical modelling cycle, i.e. the translation of problems from outside of mathematics into mathematics, the study of the mathematical problems (often with the help of high level mathematical software packages) and the interpretation of the results in the original environment. The goal of Mathematics I and II is to provide the mathematical foundations relevant for this paradigm. Differential equations are by far the most important tool for modelling and are therefore a main focus of both of these courses. | ||||

Content | 1. Single-Variable Calculus: review of differentiation, linearisation, Taylor polynomials, maxima and minima, antiderivative, fundamental theorem of calculus, integration methods, improper integrals. 2. Linear Algebra and Complex Numbers: systems of linear equations, Gauss-Jordan elimination, matrices, determinants, eigenvalues and eigenvectors, cartesian and polar forms for complex numbers, complex powers, complex roots, fundamental theorem of algebra. 3. Ordinary Differential Equations: separable ordinary differential equations (ODEs), integration by substitution, 1st and 2nd order linear ODEs, homogeneous systems of linear ODEs with constant coefficients, introduction to 2-dimensional dynamical systems. | ||||

Literature | - Thomas, G. B.: Thomas' Calculus, Part 1 (Pearson Addison-Wesley). - Bretscher, O.: Linear Algebra with Applications (Pearson Prentice Hall). | ||||

Prerequisites / Notice | Prerequisites: familiarity with the basic notions from Calculus, in particular those of function and derivative. | ||||

401-5350-00L | Analysis Seminar | 0 credits | 1K | A. Carlotto, F. Da Lio, A. Figalli, N. Hungerbühler, M. Iacobelli, T. Ilmanen, L. Kobel-Keller, T. Rivière, J. Serra, University lecturers | |

Abstract | Research colloquium | ||||

Objective | |||||

406-0251-AAL | Mathematics I Enrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | 6 credits | 13R | F. Da Lio | |

Abstract | This course covers mathematical concepts and techniques necessary to model, solve and discuss scientific problems - notably through ordinary differential equations. | ||||

Objective | Mathematics is of ever increasing importance to the Natural Sciences and Engineering. The key is the so-called mathematical modelling cycle, i.e. the translation of problems from outside of mathematics into mathematics, the study of the mathematical problems (often with the help of high level mathematical software packages) and the interpretation of the results in the original environment. The goal of Mathematics I and II is to provide the mathematical foundations relevant for this paradigm. Differential equations are by far the most important tool for modelling and are therefore a main focus of both of these courses. | ||||

Content | 1. Linear Algebra and Complex Numbers: systems of linear equations, Gauss-Jordan elimination, matrices, determinants, eigenvalues and eigenvectors, cartesian and polar forms for complex numbers, complex powers, complex roots, fundamental theorem of algebra. 2. Single-Variable Calculus: review of differentiation, linearisation, Taylor polynomials, maxima and minima, antiderivative, fundamental theorem of calculus, integration methods, improper integrals. 3. Ordinary Differential Equations: separable ordinary differential equations (ODEs), integration by substitution, 1st and 2nd order linear ODEs, homogeneous systems of linear ODEs with constant coefficients, introduction to 2-dimensional dynamical systems. | ||||

Literature | - Bretscher, O.: Linear Algebra with Applications (Pearson Prentice Hall). - Thomas, G. B.: Thomas' Calculus, Part 1 - Early Transcendentals (Pearson Addison-Wesley). | ||||

Prerequisites / Notice | Prerequisites: familiarity with the basic notions from Calculus, in particular those of function and derivative. Assistance: Tuesdays and Wednesdays 17-19h, in Room HG E 41. | ||||

406-2284-AAL | Measure and IntegrationEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | 6 credits | 13R | F. Da Lio | |

Abstract | Introduction to the abstract measure theory and integration, including the following topics: Lebesgue measure and Lebesgue integral, Lp-spaces, convergence theorems, differentiation of measures, product measures (Fubini's theorem), abstract measures, Radon-Nikodym theorem, probabilistic language. | ||||

Objective | Basic acquaintance with the theory of measure and integration, in particular, Lebesgue's measure and integral. | ||||

Literature | 1. Lecture notes by Professor Michael Struwe (http://www.math.ethz.ch/~struwe/Skripten/AnalysisIII-SS2007-18-4-08.pdf) 2. L. Evans and R.F. Gariepy "Measure theory and fine properties of functions" 3. Walter Rudin "Real and complex analysis" 4. R. Bartle The elements of Integration and Lebesgue Measure 5. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis. http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf |