Francesca Da Lio: Catalogue data in Autumn Semester 2022

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
E-mail	francesca.dalio@math.ethz.ch
URL	http://www.math.ethz.ch/~fdalio
Department	Mathematics
Relationship	Adjunct Professor

Number	Title	ECTS	Hours	Lecturers
401-2283-00L	Analysis III (Measure Theory)	6 credits	3V + 2U	F. Da Lio
Abstract	Measure and integration theory, including: Caratheodory's theorem, Lebesgue measure, Radon measure, Hausdorff measure, convergence theorems, L^p spaces, Radon-Nikodym theorem, product measure and Fubini's theorem
Learning objective	Basics of abstract measure and integration theory
Content	Measure Spaces (Lebesgue Measure, Hausdorff Measure, Radon Measure) • Measurable Functions: definition and properties • Integration: definition, properties, theorems of convergence, Lebesgue L^p spaces • Product Measures and Multiple Integrals. Fubini and Tonelli Theorems, Convolutions • Differentiation of measures (if time permits)
Lecture notes	Die Vorlesung folgt dem Skript von der Dozentin (https://people.math.ethz.ch/~fdalio/Measuremainfile.pdf)
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
Prerequisites / Notice	Analysis 1 & 2 und basic notions of topology
401-3350-72L	Elliptic Partial Differential Equations Number of participants limited to 12.	4 credits	2S	F. Da Lio, L. Kobel-Keller
Abstract
Learning objective
401-5350-00L	Analysis Seminar	0 credits	1K	F. Da Lio, A. Figalli, N. Hungerbühler, M. Iacobelli, T. Ilmanen, L. Kobel-Keller, T. Rivière, J. Serra, University lecturers
Abstract	Research colloquium
Learning objective
406-0253-AAL	Mathematics I & II 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.	13 credits	28R	A. Cannas da Silva, F. Da Lio
Abstract	Mathematics I covers mathematical concepts and techniques necessary to model, solve and discuss scientific problems, notably through linear algebra and calculus, with an emphasis on ordinary differential equations. The main focus of Mathematics II is multivariable calculus.
Learning 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 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. 4. Multivariable Differential Calculus: functions of several variables, partial differentiation, curves and surfaces in space, scalar and vector fields, gradient, curl and divergence. 5. Multivariable Integral Calculus: multiple integrals, line and surface integrals, work and flow, Green, Gauss and Stokes theorems, applications.
Literature	- Bretscher, O.: Linear Algebra with Applications (Pearson Prentice Hall). - Thomas, G. B.: Thomas' Calculus, Part 1 - Early Transcendentals (Pearson Addison-Wesley). - Thomas, G. B.: Thomas' Calculus, Parts 2 (Pearson Addison-Wesley).
Prerequisites / Notice	Prerequisites: familiarity with the basic notions from Calculus, in particular those of function, derivative and integral. Assistance: Tuesdays and Wednesdays 17-18h, in Room HG E 41.