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Search result: Catalogue data in Autumn Semester 2018

Mathematics Master Information
Electives
For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.
Electives: Pure Mathematics
Selection: Analysis
NumberTitleTypeECTSHoursLecturers
401-4115-00LIntroduction to Geometric Measure TheoryW6 credits3VU. Lang
AbstractIntroduction to Geometric Measure Theory from a metric viewpoint. Contents: Lipschitz maps, differentiability, area and coarea formula, rectifiable sets, introduction to the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, normal currents, relation to BV functions, slicing, compactness theorem for integral currents and applications.
Objective
ContentExtendability and differentiability of Lipschitz maps, metric differentiability, rectifiable sets, approximate tangent spaces, area and coarea formula, brief survey of the (de Rham-Federer-Fleming) theory of currents, currents in metric spaces after Ambrosio-Kirchheim, currents with finite mass and normal currents, relation to BV functions, rectifiable and integral currents, slicing, compactness theorem for integral currents and applications.
Literature- Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces, 1995
- Herbert Federer, Geometric Measure Theory, 1969
- Leon Simon, Introduction to Geometric Measure Theory, 2014, web.stanford.edu/class/math285/ts-gmt.pdf
- Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta math. 185 (2000), 1-80
- Urs Lang, Local currents in metric spaces, J. Geom. Anal. 21 (2011), 683-742
401-4463-62LFourier Analysis in Function Space TheoryW4 credits2VT. Rivière
AbstractIn the most important part of the course, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
Objective
ContentDuring the first lectures we will review the theory of tempered distributions and their Fourier transforms. We will go in particular through the notion of Fréchet spaces, Banach-Steinhaus for Fréchet spaces etc. We will then apply this theory to the Fourier characterization of Hilbert-Sobolev spaces.
In the second part of the course we will study fundamental pro­perties of the Hardy-Littlewood Maximal Function in relation with L^p spaces. We will then make a digression through the notion of Marcinkiewicz weak L^p spaces and Lorentz spaces. At this occa­sion we shall give in particular a proof of Aoki-Rolewicz theorem on the metrisability of quasi-normed spaces. We will introduce the preduals to the weak L^p spaces, the Lorentz L^{p',1} spaces as well as the general L^{p,q} spaces and show some applications of these dualities such as the improved Sobolev embeddings.
In the third part of the course, the most important one, we will present the notion of Singular Integrals and Calderón-Zygmund theory as well as its application to the analysis of linear elliptic operators.
This theory will naturally bring us, via the so called Littlewood-Paley decomposition, to the Fourier characterization of classical Hilbert and non Hilbert Function spaces which is one of the main goals of this course.
If time permits we shall present the notion of Paraproduct, Para­compositions and the use of Littlewood-Paley decomposition for estimating products and general non-linearities. We also hope to cover fundamental notions from integrability by compensation theory such as Coifman-Rochberg-Weiss commutator estimates and some of its applications to the analysis of PDE.
Literature1) Elias M. Stein, "Singular Integrals and Differentiability Proper­ties of Functions" (PMS-30) Princeton University Press.
2) Javier Duoandikoetxea, "Fourier Analysis" AMS.
3) Loukas Grafakos, "Classical Fourier Analysis" GTM 249 Springer.
4) Loukas Grafakos, "Modern Fourier Analysis" GTM 250 Springer.
Prerequisites / NoticeNotions from ETH courses in Measure Theory, Functional Analysis I and II (Fun­damental results in Banach and Hilbert Space theory, Fourier transform of L^2 Functions)
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