# Search result: Catalogue data in Autumn Semester 2016

Mathematics Master | ||||||

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 | ||||||

Number | Title | Type | ECTS | Hours | Lecturers | |
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401-3536-11L | Geometric Aspects of Hamiltonian Dynamics | W | 6 credits | 3V | P. Biran | |

Abstract | The course will concentrate on the geometry of the group of Hamiltonian diffeomorphisms introduced by Hofer in the early 1990's and its relations to various topics in symplectic geometry such as capacities, Lagrangian submanifolds, holomorphic curves, as well as recent algebraic structures on the group of Hamiltonian diffeomorphisms such as quasi-morphisms. | |||||

Objective | ||||||

Literature | Books: * L. Polterovich: "The geometry of the group of symplectic diffeomorphisms" * H. Hofer & E. Zehnder: "Symplectic invariants and Hamiltonian dynamics" | |||||

Prerequisites / Notice | Prerequisites. Good knowledge of undergraduate mathematics (analysis, complex functions, topology, and differential geometry). Some knowledge of elementary algebraic topology would be useful. | |||||

401-4767-66L | Partial Differential Equations (Hyperbolic PDEs) | W | 7 credits | 4V | D. Christodoulou | |

Abstract | The course begins with characteristics, the definition of hyperbolicity, causal structure and the domain of dependence theorem. The course then focuses on nonlinear systems of equations in two independent variables, in particular the Euler equations of compressible fluids with plane symmetry and the Einstein equations of general relativity with spherical symmetry. | |||||

Objective | The objective is to introduce students in mathematics and physics to an area of mathematical analysis involving differential geometry which is of fundamental importance for the development of classical macroscopic continuum physics. | |||||

Content | The course shall begin with the basic structure associated to hyperbolic partial differential equations, characteristic hypersurfaces and bicharacteristics, causal structure, and the domain of dependence theorem. The course shall then focus on nonlinear systems of equations in two independent variables. The first topic shall be the Euler equations of compressible fluids under plane symmetry where we shall study the formation of shocks, and second topic shall be the Einstein equations of general relativity under spherical symmetry where we shall study the formation of black holes and spacetime singularities. | |||||

Prerequisites / Notice | Basic real analysis and differential geometry. | |||||

401-4831-66L | Mathematical Themes in General Relativity I | W | 4 credits | 2V | A. Carlotto | |

Abstract | First part of a one-year course offering a rigorous introduction to general relativity, with special emphasis on aspects of current interest in mathematical research. Topics covered include: initial value formulation of the Einstein equations, causality theory and singularities, constructions of data sets by gluing or conformal methods, asymptotically flat spaces and positive mass theorems. | |||||

Objective | Acquisition of a solid and broad background in general relativity and mastery of the basic mathematical methods and ideas developed in such context and successfully exploited in the field of geometric analysis. | |||||

Content | Lorentzian geometry; geometric review of special relativity; the Einstein equations and their basic classes of special solutions; the Einstein equations as an initial-value problem; causality theory and hyperbolicity; singularities and trapped domains; Penrose diagrams; asymptotically flat spaces: ADM invariants, positive mass theorems, Penrose inequalities, geometric properties. | |||||

Lecture notes | Lecture notes written by the instructor will be provided to all enrolled students. | |||||

Prerequisites / Notice | The content of the basic courses of the first three years at ETH will be assumed. In particular, enrolled students are expected to be fluent both in Differential Geometry (at least at the level of Differentialgeometrie I, II) and Functional Analysis (at least at the level of Funktionalanalysis I, II). Some background on partial differential equations, mainly of elliptic and hyperbolic type, (say at the level of the monograph by L. C. Evans) would also be desirable. | |||||

401-4497-66L | Free Boundary Problems | W | 4 credits | 2V | A. Figalli | |

Abstract | ||||||

Objective | ||||||

401-4463-62L | Fourier Analysis in Function Space Theory | W | 6 credits | 3V | T. Rivière | |

Abstract | In 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 | ||||||

Content | During 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. | |||||

Literature | 1) 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 / Notice | Notions 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) | |||||

401-4475-66L | Partial Differential Equations and Semigroups of Bounded Linear Operators | W | 4 credits | 2G | A. Jentzen | |

Abstract | In this course we study the concept of a semigroup of bounded linear operators and we use this concept to investigate existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs) of the evolutionary type. | |||||

Objective | The aim of this course is to teach the students a decent knowledge (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs. | |||||

Content | The course includes content (i) on semigroups of bounded linear operators, (ii) on solutions of partial differential equations (PDEs) of the evolutionary type, and (iii) on the analytic concepts used to formulate and study such semigroups and such PDEs. Key example PDEs that are treated in this course are heat and wave equations. | |||||

Lecture notes | Lecture Notes are available in the lecture homepage (please follow the link in the Learning materials section). | |||||

Literature | 1. Amnon Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1983). 2. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations. Springer-Verlag, New York (2000). | |||||

Prerequisites / Notice | Mandatory prerequisites: Functional analysis Start of lectures: Friday, September 23, 2016 For more details, please follow the link in the Learning materials section. | |||||

401-3303-00L | Special Topics in One Complex Variable | W | 6 credits | 3V | H. Knörrer | |

Abstract | Hypergeometric Functions, Boundary values of holomorphic functions, Nevanlinna Theory and other special topics. | |||||

Objective | Advanced methods of one complex variables | |||||

Literature | R. Remmert: Funktionentheorie II. Springer Verlag E.Titchmarsh: The Theory of Functions. Oxford University Press C.Caratheodory: Funktionentheorie. Birkhaeuser E.Hille: Analytic Function Theory. AMS Chelsea Publishing A.Gogolin:Komplexe Integration. Springer | |||||

Prerequisites / Notice | Funktionentheorie |

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